Disipación Controlada con Cúbits Superconductores

Por Joe Kitzman

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Autores: P.M. Harrington, M. Naghiloo, D. Tan, K.W. Murch

Afiliación Primaria del Primer Autor: Departamento de Física, Universidad de Washington, Saint Louis, Missouri 63130, USA

Original: Publicado en Physical Review A

Introducción

Los sistemas cuánticos son generalmente muy sensibles y al interaccionar con el entorno, sus propiedades cuánticas pueden perder coherencia. Esto esencialmente hace que un determinado sistema cuántico se disipe en un comportamiento puramente clásico. No obstante, en ciertos contextos es posible usar esta disipación de una manera controlada para incrementar el control sobre los sistemas cuánticos. Algunos ejemplos de esta disipación controlada incluyen el enfriamiento de átomos por láser, el enfriamiento de osciladores mecánicos a bajas frecuencias y el control de los circuitos cuánticos. En esta reciente publicación [1], los autores son capaces de demostrar la estabilización de los estados de superposición en un cúbit superconductor usando un canal de pérdidas de cristal fotónico hecho a medida. Considerando cómo el cristal fotónico induce pérdidas en el sistema, los autores proporcionan un enfoque con ecuación maestra que explica cómo la combinación de un impulso especializado aplicado al cúbit, además de la disipación dada por el cristal fotónico, permite un control preciso del estado del cúbit por tiempos mucho mayores que los tiempos de coherencia estándar de un cúbit.

Detalles Experimentales

Este experimento consiste en un cúbit superconductor cuyo momento dipolar se acopla al campo eléctrico dentro de una cavidad de guía de ondas tridimensional. En este experimento, el rol de la cavidad de guía de ondas es facilitar el control por microondas del cúbit así como leer el estado del cúbit. El cúbit superconductor consiste en dos uniones de Josephson en paralelo, formando un dispositivo interferométrico cuántico superconductor (a menudo conocido como “SQUID” por sus siglas en inglés: superconducting quantum interference device). Esto permite a los autores cambiar la frecuencia de resonancia del cúbit haciendo pasar un campo magnético externo a través de la espira del SQUID. En la salida de la cavidad de guía de ondas, los autores conectan un cristal fotónico al circuito. Este cristal fotónico está hecho de un cable coaxial usual que está mecánicamente deformado de una manera concreta para cambiar su impedancia. El resultado de la impedancia que varía con el espacio en el cable causa la apertura de una banda prohibida – llegando a energías fotónicas (o frecuencias) donde la densidad fotónica de los estados es cero (ver Fig. 1 para un esquema del montaje experimental). Cambiando la densidad fotónica de los estados en función de la energía, el decaimiento del cúbit también cambiará en función de la frecuencia.

Figura 1
Izquierda: Esquema del sistema experimental. El cúbit superconductor se monta en una cavidad de cobre que se usa para controlar y leer el estado del cúbit. Haciendo pasar corriente a través del cable superconductor enroscado alrededor de la cavidad se genera un campo magnético perpendicular al sustrato que contiene al cúbit, permitiendo a los autores sintonizar la frecuencia de resonancia del cúbit. El cristal fotónico se conecta al puerto de salida de la cavidad, cambiando la densidad de los estados sobre los que puede decaer el cúbit. Derecha: Medidas a temperatura ambiente de la reflexión del cristal fotónico. En la banda de corte (de 5.5 – 6.4 GHz) la mayor parte de la señal enviada al cristal fotónico se refleja, verificando que hay una baja densidad de estados a esas frecuencias. Por encima de 6.4 GHz, la banda fotónica prohibida se estrecha y los fotones se pueden transmitir a través del cristal fotónico.

Tasas de Decaimiento del Cúbit

Para medir la tasa de decaimiento del cúbit, los autores primero llevan el cúbit a su estado excitado aplicando un pulso de energía al sistema que es resonante con el cúbit. Luego, miden la probabilidad de que el cúbit permanezca en su estado excitado en función del tiempo después de haber aplicado el pulso. Ajustando la probabilidad medida a un decaimiento exponencial y extrayendo la constante de decaimiento, se puede determinar la tasa de decaimiento del cúbit. La frecuencia de resonancia del cúbit se ajusta luego cambiando el flujo magnético externo que circula por la espira del SQUID y midiendo la tasa de decaimiento del cúbit en función de la frecuencia del cúbit para investigar el impacto del cristal fotónico sobre la vida media del cúbit. La tasa de decaimiento total del cúbit se puede expresar como

\gamma_1 = \gamma_d + \rho(\omega_q)(g/\Delta_q)^2 \kappa.

En la Ec. 1, \gamma_1 es la tasa de decaimiento medida del cúbit, \kappa/2\pi = 18~\textrm{MHz} es el ancho de banda de la cavidad de microondas, g/(2\pi) = 200~\textrm{MHz} es la fuerza de acoplamiento entre el cúbit y la cavidad, \Delta_q = \omega_c - \omega_q es la diferencia en frecuencia de resonancia entre el cúbit y la cavidad, \rho(\omega_q) es la densidad de estados del cristal fotónico a la frecuencia del cúbit, y \gamma_d representa el decaimiento del cúbit en canales de disipación aparte del cristal fotónico. Midiendo la tasa de decaimiento total del cúbit para varios valores de \omega_q, ¡debería ser posible extraer información acerca de la densidad de estados del cristal fotónico! Ver Fig. 2 a continuación para la medida resultante.

Figura 2
Medida de las tasas de decaimiento del cúbit sobre un rango amplio de frecuencias. Dado que la pérdida del cúbit varía rápidamente con la frecuencia del cúbit, llevando el flujo de polarización al punto donde la derivada de la pérdida del cúbit es grande, las bandas laterales del triplete de Mollow pueden muestrear frecuencias tanto con muy altas como con muy bajas pérdidas. Midiendo las frecuencias generalizadas de Rabi a lo largo de los mismos valores de frecuencia del cúbit, los autores verifican la variable de acoplamiento del cúbit con el cristal fotónico.

Dinámica y Emisión de un Cúbit Controlado

Después de verificar que la densidad de estados en el cristal fotónico puede modificar la tasa de decaimiento del cúbit, los autores ahora consideran más cuidadosamente cómo emite el cúbit energía realmente. Específicamente, se considera un fuerte impulso aplicado con amplitud \Omega que es desintonizado de la energía del cúbit una cantidad \Delta = \omega_d - \omega_q, donde \omega_d es la frecuencia del impulso y \omega_q es la energía del cúbit. Si la amplitud del impulso es mucho mayor que la tasa de pérdida del cúbit, el cúbit emitirá energía a tres frecuencias diferentes \omega_d y \omega_d~\pm~\Omega_R, donde \Omega_R = \sqrt{\Omega^2 + \Delta^2} se conoce como frecuencia de Rabi generalizada. Este espectro de emisión se llama triplete de Mollow [2]. Ver Fig. 3 para un esquema de emisión del triplete de Mollow.

Figura 3
Esquema que representa la emisión del sistema de dos niveles controlado. Bajo la presencia de un impulso fuerte, el cúbit emite radiación a frecuencias correspondientes a la frecuencia del impulso \omega_d así como a frecuencias \omega_d \pm \Omega_R. Debido al espectro de pérdidas modificado, el área de una banda lateral puede disminuir, indicando que el cúbit emitirá radiación a esta frecuencia en menor medida que la otra banda lateral. Arriba a la derecha: bajo la presencia del impulso, el eje de cuantización del cúbit también rota, lo cual cambia los estados del cúbit que se pueden preparar / estabilizar.

Dado que los autores han observado que el cristal fotónico modifica la tasa de pérdida del cúbit en una escala de energía comparable a los valores experimentales accesibles de \Omega_R, es posible que una de las bandas laterales del triplete de Mollow experimente una tasa de pérdidas grande mientras que la otra banda lateral experimenta una tasa baja.

Lo siguiente a considerar es cómo afecta la presencia de un impulso aplicado al espectro de energía del cúbit. En un sistema de referencia en rotación con la frecuencia del impulso, el hamiltoniano del cúbit viene dado por

H_q = \frac{\Delta}{2}\sigma_z + \frac{\Omega}{2}\sigma_x,

donde \sigma_z y \sigma_x son matrices de Pauli. Dado que este hamiltoniano no es diagonal, es conveniente rotar la base de manera que el hamiltoniano se pueda escribir de la forma

\tilde{H}_q = \frac{\Omega_R}{2}\tilde{\sigma}_z,

donde la matriz de Pauli Z rotada puede expresarse como \tilde{\sigma}_z = \sin{2\theta}\sigma_x - \cos{2\theta}\sigma_z y el ángulo de rotación se define como \tan{2\theta} = -\Omega/\Delta con 0<\theta<\pi/2. Dado que hemos escrito el hamiltoniano en una base rotada, debemos considerar también cómo rotan los nuevos autoestados del sistema con respecto a los autoestados originales, que llamaremos |g\rangle y |e\rangle para los estados fundamental y excitado, respectivamente.

|\tilde{g}\rangle = \cos{\theta}|g\rangle - \sin{\theta}|e\rangle

|\tilde{e}\rangle = \sin{\theta}|g\rangle + \cos{\theta}|e\rangle

Llegados a este punto, ¡probablemente sea conveniente considerar un ejemplo útil! En el caso de un impulso resonante, \Delta = 0, que inmediatamente nos informa de que \theta = 45^{\circ}, por lo que podemos reescribir los autoestados rotados del sistema como |\tilde{g}\rangle = \frac{1}{\sqrt{2}}(|g\rangle - |e\rangle) \equiv |-x\rangle y |\tilde{e}\rangle = \frac{1}{\sqrt{2}}(|g\rangle + |e\rangle) \equiv |+x\rangle, los cuales tienen la propiedad especial de que \sigma_x|\pm x\rangle = \pm 1 |\pm x\rangle. Dado que el estado |-x\rangle tiene una energía menor, emitirá energía correspondiente a la banda lateral de menor energía del triplete de Mollow y viceversa para el estado |+x\rangle. Si la pérdida del cúbit es muy diferente para cualquiera de estos estados, ¡fomentará el decaimiento hacia los estados |-x\rangle o |+x\rangle! Específicamente, si el cúbit está a una frecuencia de resonancia cercana a 6.4766 GHz (ver Fig. 2), el estado de mayor energía (correspondiente a |+x\rangle en este ejemplo) tiene una tasa de pérdida menor, por lo que deberíamos esperar que mientras el impulso esté activo, ¡el cúbit preferentemente decaerá hacia este estado! ¡Esto significa que el valor esperado \langle \sigma_x \rangle tenderá a +1 en este supuesto! En el caso de un espectro de pérdidas uniforme, no habría un decaimiento preferido para el cúbit y sería de esperar que todos los valores esperados decayeran a cero.

La Ecuación Maestra de Lindblad

En presencia del impulso combinado y la disipación sufrida por el cúbit, la dinámica de la matriz de densidad reducida que describe el cúbit puede ser expresada de acuerdo a la ecuación maestra de Lindblad [3]:

\dot{\rho} = i[\rho,H] + \gamma_0 \cos{(\theta)}\sin{(\theta)}\mathcal{D}[\tilde{\sigma}_z]\rho + \gamma_{-} \sin{^4\left(\theta\right)} \mathcal{D}[\tilde{\sigma}_{+}\rho + \gamma_{+}\cos{^4\left(\theta\right)} \mathcal{D}[\tilde{\sigma}_{-}]\rho.

Aquí, \rho es la matriz de densidad reducida para el cúbit, el superoperador de disipación también se introduce como \mathcal{D}[A]\rho = \left( 2 A \rho A^{\dagger} - A^{\dagger}A\rho - \rho A^{\dagger}A\right)/2. La tasa \gamma_0 representa un desfase del cúbit en la base rotada de \theta, que se acopla al operador \tilde{\sigma}_z y las transiciones entre autoestados en la base rotada son controlados por los operadores de “salto” \tilde{\sigma}_{\pm}, que están relacionados con las tasas \gamma_{\mp}. Similar al ejemplo anterior, si el cristal fotónico modifica la pérdida del cúbit tal que \gamma_{\pm} \gg \gamma_{\mp}, el autoestado del sistema de referencia en rotación correspondiente se estabilizará.

Resultados experimentales

Para verificar que los autores pueden usar la combinación de impulsos y disipación para preparar y estabilizar estados de cúbits, implementan el siguiente protocolo de bath engineering. Primero, se lleva el flujo de polarización del cúbit a la frecuencia de resonancia de 6.4766 GHz (como en nuestro ejemplo). Después, se aplica un impulso coherente al sistema durante casi 16~\mu s (¡que es mucho más largo que el tiempo de coherencia del cúbit en ausencia del impulso!). Durante este tiempo, el cúbit debería decaer preferiblemente a un autoestado del sistema rotado si las bandas laterales del triplete de Mollow tienen pesos diferentes. Una vez se corta el impulso, se mide el valor esperado \langle \sigma_x \rangle para varias combinaciones de parámetros del impulso. Los resultados se muestran a continuación también en la Fig. 4 como comparaciones con los resultados numéricos a la ecuación maestra, los autores no solo ven que los valores esperados del cúbit no decaen a 0, ¡sino que hay una concordancia fantástica entre la teoría y el experimento!

Figura 4
(a) Medida de \langle \sigma_x \rangle mientras que los parámetros del impulso van cambiando. Para ciertos parámetros del impulso, el valor de \langle \sigma_x \rangle es negativo. Así como van cambiando los parámetros del impulso, el sistema pasa por una región de “cero coherencia”, donde el protocolo de bath engineering ya no funciona antes de estabilizar \langle \sigma_x \rangle a valores positivos . (b) Soluciones numéricas a la ecuación maestra bajo los mismos parámetros de impulso que (a). Los autores observan una excelente correspondencia entre los experimentos y las soluciones numéricas. (c) Comparación de las líneas de corte horizontales de (a,b). Los autores observan una concordancia entre los valores medidos (puntos) y los valores simulados (líneas) cuando se consideran todos los valores esperados para el estado del cúbit (\langle \sigma_{x,y,z} \rangle).

Conclusión

En conclusión, los autores son capaces de demostrar la fabricación de un cable coaxial con impedancia que varía en el espacio que actúa como un cristal fotónico y a cambio controlan el espectro de pérdidas de un cúbit superconductor. Los autores luego hacen uso de este espectro de emisión modificado en el contexto de la ecuación maestra para preparar y estabilizar estados no triviales del cúbit por tiempos mucho mayores que los tiempos de coherencia del cúbit.

Referencias

[1] P. M. Harrington, M. Naghiloo, D. Tan, and K. W. Murch, Bath engineering of a fluorescing artificial atom with a photonic crystal, Phys. Rev. A 99, 052126 (2019)

[2] B. R. Mollow, Power spectrum of light scattered by two-level systems, Phys. Rev. 188, 1969 (1969)

[3] G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976).

What can quantum information tell us about the foundations of statistical mechanics?

By Mauro E.S. Morales

Title: Entanglement and the foundations of statistical mechanics

Authors: Sandu Popescu1,2, Anthony J. Short1, Andreas Winter3.

Institutions: 1H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK

2Hewlett-Packard Laboratories, Stoke Gifford, Bristol BS12 6QZ, UK

3Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK

Manuscript: Published in Nature [1], Open Access on arXiv [2]

It is sometimes easy to forget, that in addition to the impact it has had on the development of new technologies, the ongoing development of quantum information theory has had implications on the foundations of Physics itself. In fact, based on insights from quantum information, in [1] the authors argue for re-framing a fundamental principle that lies is at the very basis of statistical mechanics, namely the equal probability postulate.

The concept of a thermodynamic “equilibrium” is central to classical statistical mechanics. In such an equilibrium, one can assume that there are no macroscopic changes in a given system. Consider a box full of solid particles inside, and take this box to be connected to a heat bath of temperature T and isolated from everything else. For a given temperature, we know that the probability that the system is in a state with energy E_i is given by

P(E_i)=\frac{1}{Z}e^{-E_i /kT} ,

where Z is the well-known partition function, which roughly tells us how many different ways one can partition a system into subsystems having the same energy, and k is the Boltzmann constant which relates absolute temperature to the kinetic energy of each microscopic particle in any given system.

A key assumption in this is that all possible states of the “total system”, which encapsulates the box and the bath, have equal probability. This assignment of probabilities to each energy is known as the canonical ensemble. Physicists also work with other types of ensembles, for instance, the micro-canonical ensemble, where the total energy is fixed and all states have equal probability. It is important to stress that this is an assumption on the total system, not something that is proven from other postulates. In other words, we postulate this a priori.

A general canonical principle

In [1], the authors propose a way to derive probabilities assigned by the canonical ensemble by explicitly considering quantum systems. In fact, their methods prove a more general canonical principle than the classical one, and we shall elaborate on this general principle further.

First, let us consider a large isolated quantum system R described by a Hilbert space \mathcal{H}_R which is decomposed into a system S with Hilbert space \mathcal{H}_S and an environment E with Hilbert space \mathcal{H}_E. In principle \mathcal{H}_R could be described by \mathcal{H}_S\otimes \mathcal{H}_E, but we can consider restrictions over the space as shown in the picture below.

This restriction would make the space R smaller and would be analogous to the system presented in the introduction with a fixed temperature T. In a quantum setting, such restrictions are described by considering constraints on the possible joint states of system S and E. We note that such restrictions need not be related solely to temperature, it can in fact be any type of constraint whatsoever on the total system, a feature that will turn out to be important for the generality of the proof.

We can consider as in classical thermodynamics, the state that gives equal probability to all states in \mathcal{H}_R, which can be represented using the identity matrix. This state gives equal probability to all states of R, assuming that R is in this state is akin to the assumption in the combined box/bath example in the introduction.

In this case, the canonical state would be obtained by tracing out the degrees of freedom from the bath. We denote this state as \Omega_S . If we had taken the system R as the box/bath combined with the restriction of the temperature T, then \Omega_S would correspond to the Gibbs state, which describes an equilibrium probability distribution that remains invariant under any future evolution of the system, with the probabilities given in the introduction. So far, we have just rewritten everything in the language of quantum mechanics but the authors take a step further. It’s important to remark that we could have taken any other restriction for \mathcal{H}_R and the canonical state would be different from the canonical thermal canonical state defined earlier in the introduction.

What if the state of R is not the identity?

If the state of R corresponds instead to some state |\phi\rangle \langle \phi | and defines the state of the system S as \rho_S=Tr_{E}\left( |\phi\rangle \langle \phi |\right), then the authors show that \rho_S is close to the state \Omega_S for almost all possible states |\phi\rangle \langle \phi |. This implies that there is no need to assume equal probability for all states since, as we will see, most of the states in system R will give the correct canonical state in S.

In quantum information, we can measure how close two states are from each other using the so-called trace distance. We will denote the distance between \rho_S and \Omega_S as D(\rho_S,\Omega_S).

This distance represents the maximal difference between the two states in the difference of obtaining any measurement. In other words, the trace distance tells us how hard is to tell apart \rho_S and \Omega_S apart under measurements (the greater the distance, the harder to tell apart).

Distance between density matrices is defined by D

To understand what the authors prove let’s set some notation. Let \rho_S (\phi)=Tr_{E}\left( |\phi\rangle \langle \phi |\right) be the state obtained by tracing out the environment and define the set of states at a distance of the canonical state equal or greater to \eta as \mathcal{S} . The radius defined by \eta is shown below.

Note that \mathcal{S}  is a set in the Hilbert space \mathcal{H}_R (different from the one pictured above, which is the space of density matrices). We picture below the set \mathcal{S}

The set \mathcal{S} fills a volume in the Hilbert space, we denote the fraction of states at distance equal or greater to \eta of the canonical state as

\frac{V\left(\mathcal{S}\right)}{V\left(\mathcal{H}_R\right)}

where V(\cdot) refers to the “volume” of the set in the argument. Another way of interpreting this ration is as the probability of picking a random state |\phi\rangle such that the distance of \rho_S to the canonical state is equal or greater to \eta.

What the authors prove rigorously is that this probability gets smaller (in fact exponentially smaller) as \eta grows. More precisely they prove that for \epsilon>0 we have that

\frac{V\left(\mathcal{S}\right)}{V\left(\mathcal{H}_R\right)}\leq \eta'

with \eta\approx\epsilon and \eta'=4 \exp(-Cd_R \epsilon^2) .

Note that as \epsilon grows, the probability, of picking a state such that the distance is big enough, decays exponentially.

We won’t go into the full intricacies of the proof for this statement, but we will mention that a key ingredient is Levy’s lemma (for those curious about this Lemma, see [2]). This lemma has in fact seen use in other areas of quantum information. Those familiar with variational quantum algorithms may have heard of barren plateaus, which limit the trainability of variational circuits [3]. Levy’s lemma is a key ingredient in proving that under certain conditions barren plateaus become inevitable when training these quantum circuits.

References

[1] Popescu, S., Short, A. & Winter, A. Entanglement and the foundations of statistical mechanics. Nature Phys 2, 754–758 (2006). https://doi.org/10.1038/nphys444

[2] Popescu, S., Short, A. & Winter, A. The foundations of statistical mechanics from entanglement: Individual states vs. averages. arXiv:0511225 [quant-ph], Oct. 2006.

[3] McClean, J.R., Boixo, S., Smelyanskiy, V.N. et al. Barren plateaus in quantum neural network training landscapes. Nat Commun 9, 4812 (2018). https://doi.org/10.1038/s41467-018-07090-4

Control cuántico del movimiento

Por Akash Dixit

Título: Preparación de estados cuánticos, tomografía y entrelazamiento de osciladores mecánicos

Autores: E. Alex Wollack, Agnetta Y. Cleland, Rachel G. Gruenke, Zhaoyou
Wang, Patricio Arrangoiz-Arriola, y Amir H. Safavi-Naeini

Institución: Departamento de Física Aplicada y Laboratorio de Ginzton, Universidad de Stanford 348 Via Pueblo Mall, Stanford, California 94305, USA

Original: Publicado en Nature [1], Acceso libre en arXiv

Introducción
El campo de las ciencias de la información cuántica contiene multitud de diferentes tecnologías, incluyendo átomos, espines y defectos en los centros de diamante. Este trabajo se centra en dos tecnologías emergentes: circuitos superconductores y osciladores mecánicos. Cada sistema tiene sus ventajas, pero no es obvio que ninguna sea la mejor plataforma para construir un ordenador cuántico, desarrollar sensores cuánticos o facilitar la comunicación cuántica. Para alcanzar estos objetivos es necesario desarrollar un sistema cuántico híbrido que pueda utilizar los puntos fuertes de diversas tecnologías cuánticas.

En este trabajo, los autores demuestran la posibilidad de acoplar cúbits superconductores a movimiento mecánico. Esto establece los cimientos para un sistema cuántico híbrido que pueda aprovechar lo mejor de los dos sistemas. El cúbit es personalizable y fácil de comunicarse con él, haciéndolo ideal para la inicialización y caracterización del estado. Los modos mecánicos se fabrican con escasa huella espacial y tienen tiempos de vida largos, haciendo posible escalarlos a sistemas más grandes y mantener la información cuántica durante largos períodos de tiempo. A continuación, describiré cómo los autores usan estos acoplamientos entre los dos sistemas tanto para preparar como para medir los estados de movimiento mecánico usando el cúbit. Primeramente, describo el sistema cuidadosamente diseñado que acopla un cúbit a dos osciladores mecánicos. Después, hablo de los dos modos de operación, donde el cúbit es usado tanto para preparar estados de movimiento mecánico como para medir el estado cuántico del modo mecánico. Finalmente, muestro cómo los autores usan el cúbit como un intermediario para preparar estados mecánicamente entrelazados entre los dos osciladores.

Mecanismo

Figura 1: Acoplamiento de un cúbit a la mecánica. a. El esquema del mecanismo muestra un único cúbit acoplado a dos osciladores mecánicos a distintas frecuencias. b. Imagen óptica de un cúbit con las uniones de Josephson mostradas en el recuadro. El cable que viene por la izquierda lleva corriente para aplicar un flujo a la espira de la unión. El panel rectangular a la derecha del diagrama es el panel capacitivo que vincula el cúbit con los osciladores mecánicos. c. Dos osciladores mecánicos armados en una estructura periódica de LiNbO3. Figura adaptada de la Ref. 1.

El dispositivo usado en este trabajo consiste en dos osciladores mecánicos y un cúbit superconductor. Los osciladores mecánicos se fabrican en una lámina delgada de niobato de litio (LiNbO3). Estos osciladores se forman provocando un defecto en una estructura periódica del material, llamado cristal fonónico. El defecto es un desajuste en la periodicidad de la estructura y confina el movimiento mecánico, impidiendo la radiación acústica y permitiendo períodos de integridad largos. Al igual que ocurre con la radiación electromagnética, el movimiento mecánico puede ser cuantizado. Los quantum del movimiento mecánico se llaman fonones y el oscilador mecánico puede ser caracterizado como un oscilador armónico con niveles de energía equiespaciados. El cúbit se hace fabricando un oscilador LC con materiales superconductores. El elemento clave de este circuito es la unión de Josephson, que está hecha de óxido de aluminio intercalado entre capas de aluminio superconductor. La unión actúa como un inductor no lineal que modifica la distancia entre los niveles de energía del oscilador LC. Los niveles de energía del oscilador LC usual (que es un oscilador armónico) están equiespaciados, lo que significa que la energía de transición entre dos niveles cualesquiera es la misma. Sin embargo, con el inductor no lineal en el circuito, ya no hay niveles de energía equiespaciados, haciendo posible distinguir los dos niveles de menor energía del sistema, el fundamental (\left| g \right\rangle) y el excitado (\left| e \right\rangle). Los dos niveles forman un bit cuántico (cúbit). El cúbit está diseñado para que se pueda ajustar su frecuencia poniendo dos uniones de Josephson en paralelo. Aplicando un campo magnético mediante un cable que lleve corriente, se produce un flujo magnético a través de la espira que permite cambiar la frecuencia del cúbit.

El cúbit y los osciladores mecánicos se fabrican en chips separados que se colocan a una distancia \sim \mu m. Para acoplar el cúbit con los osciladores mecánicos, los autores usan la piezoelectricidad de la lámina de niobato de litio. El movimiento mecánico de este material produce una acumulación de carga eléctrica sobre los paneles de aluminio situados en ambos chips, que están diseñados para ser el elemento capacitivo del cúbit. El cúbit capacitor se carga con el movimiento de los osciladores mecánicos, garantizando que los dos sistemas están conectados.

Inicializando un estado mecánico
Los autores diseñan el cúbit para que interaccione de dos maneras diferentes con los osciladores mecánicos. En el primer modo, el cúbit está sintonizado para entrar en resonancia con un oscilador mecánico en concreto (\omega_q = \omega_1, \omega_2). Nótese que las frecuencias mecánicas de los dos osciladores son diferentes, así que el cúbit sólo puede estar en resonancia con una a la vez. Esto permite el intercambio directo de energía entre el cúbit y cada oscilador a una tasa relacionada con el acoplamiento capacitivo entre los dos, g_1 = 2 \pi \times 9.5 MHz, g_2 = 2 \pi \times 10.5 MHz. El hamiltoniano que describe la interacción entre el cúbit y el oscilador mecánico en resonancia es la interacción de Jaynes-Cummings:

\mathcal{H}_{\mathrm{on}} = g(a^{\dagger} \sigma^{-} + a \sigma^{+})
[Ecuación 1].

a^{\dagger}, a y \sigma^{+}, \sigma^{-} son los operadores creación y destrucción para el oscilador mecánico y el cúbit, respectivamente. Cuando están en resonancia, el cúbit y el oscilador mecánico intercambian sus respectivos estados en un tiempo de \pi/g \sim 24-26 ns dependiendo del oscilador en cuestión.

Figura 2: Intercambio de estados entre cúbit y oscilador. El cúbit (Q) se inicializa en el estado excitado. Una vez el cúbit se lleva a resonancia con el oscilador mecánico, los dos intercambian sus estados coherentemente. Esto significa que el estado del sistema oscila entre \left| 0,e \right\rangle y \left| 1,g \right\rangle, donde \left| m,g \right\rangle representa el estado de la mecánica y del cúbit. Entre un intercambio completo, el cúbit y los osciladores mecánicos están entrelazados con función de estado \left| 1,g \right\rangle + \left| 0,e \right\rangle.

Este intercambio puede ser usado como un método de preparación de estados mecánicos. Los autores primero sintonizan el cúbit para que no esté en resonancia con ninguno de los osciladores mecánicos. Luego, con el modo mecánico vacío de quanta, el cúbit se inicializa con estados \left| 0,g \right\rangle, \left| 0,e \right\rangle o \left| 0,g \right\rangle + \left| 0,e \right\rangle. El estado \left| m, q \right\rangle describe el número de fonones de un oscilador mecánico concreto, m = 0, 1, 2…, y si el cúbit está en su estado fundamental o excitado q = g, e. La frecuencia del cúbit se sintoniza para que esté en resonancia con alguno de los modos mecánicos durante el tiempo correspondiente a un intercambio completo. Cuando la operación de intercambio es aplicada al estado \left| 0,g \right\rangle, el sistema permanece inalterado dado que ambos subsistemas están en su estado fundamental y no hay energía que intercambiar. Durante el intercambio, el estado \left| 0,e \right\rangle se convierte en \left| 1,g \right\rangle como se muestra en la Figura 1. Cuando el cúbit se inicializa en un estado de superposición, el estado es \left| 0,g \right\rangle + \left| 0,e \right\rangle. La operación de intercambio actúa sobre ambas partes de esta superposición dando lugar al estado final \left| 0,g \right\rangle + \left| 1,g \right\rangle. El oscilador mecánico está ahora en un estado de superposición, pero el estado del oscilador mecánico no está entrelazado con el estado del cúbit.

Midiendo un estado mecánico
En el segundo modo de operación, el cúbit no está en resonancia con ninguno de los osciladores, lo cual se conoce como interacción dispersiva. La tasa de interacción dispersiva entre el cúbit y el oscilador mecánico, \chi, se determina ahora por el acoplamiento capacitivo directo, g, la desintonización entre el cúbit y la mecánica, \Delta, y otros parámetros del cúbit. En el límite en que la desintonización entre el cúbit y la mecánica es mayor que la tasa de interacción capacitiva (\Delta \gg g), la interacción mostrada en la Ecuación 1 es aproximada por el hamiltoniano fuera de resonancia:

\mathcal{H}_{\mathrm{off}} = \chi a^{\dagger} a \sigma_z
[Ecuación 2].

La combinación a^{\dagger}a es la versión en operadores del número de fonones, m, en el oscilador mecánico. \sigma_z es la versión en operadores del estado del cúbit, bien \left| g \right\rangle o bien \left| e \right\rangle.

Sin la interacción entre el cúbit y la mecánica, el hamiltoniano de sólo el cúbit quedaría \mathcal{H}_{q} = \omega_q \sigma_z, donde \omega_q es la frecuencia de transición del cúbit. Cuando añadimos la interacción fuera de resonancia, el hamiltoniano se puede escribir como \mathcal{H}_{q} + \mathcal{H}_{\mathrm{off}} = (\omega_q - \chi a^{\dagger}a )\sigma_z. Comparando el hamiltoniano combinado con el de sólo el cúbit, vemos que el efecto de la interacción es modificar la frecuencia de transición del cúbit (representado por todo lo anterior a \sigma_z). Por lo que ahora la frecuencia de transición del cúbit depende del número de fonones en el oscilador mecánico (m = a^{\dagger}a). Por cada fonón adicional en el oscilador mecánico, la frecuencia de transición del cúbit cambia \chi.

Esta interacción es crucial para poder caracterizar el estado del oscilador mecánico. Dado que un distinto número de fonones imparte un cambio de frecuencia diferente en el cúbit, el estado mecánico está impreso en la frecuencia del cúbit. Para solventar la probabilidad de distinto número de fonones en los osciladores mecánicos, se realiza una medida interferométrica sobre los cúbits. El oscilador mecánico se prepara en un estado de Fock con 0 o 1 fonones o en una superposición de varios fonones 0, 1, 2… Luego, el cúbit se coloca en un estado de superposición \left| g \right\rangle + \left| e \right\rangle y se deja precesar por un tiempo variable, t. Durante este tiempo, el estado superpuesto acumula una fase de \chi si hay un fonón, 2\chi para dos fonones y así sucesivamente. La fase acumulada refleja la probabilidad (A_n) de que el oscilador mecánico contenga cero, uno, dos, etc. fonones. El estado del cúbit evoluciona a \left| g \right\rangle + e^{i\phi} \left| e \right\rangle, donde la fase acumulada es \phi = \sum_n A_n n \chi t. Los autores rotan el cúbit de vuelta a su base de medición y monitorizan la población final del estado excitado como función del tiempo de interacción, t, y ajustan la trayectoria a la forma funcional

S(t) = \sum_n A_n e^{-\kappa t/2} \cos [(2 n \chi t) + \phi_n]
[Ecuación 3]

Esta función incluye las probabilidades del número de fonones, A_n, así como la precesión dependiente de este número n \chi. También incluye el desfase dependiente del número, \phi_n y la constante de decaimiento de fonones, \kappa. Esto captura la dinámica de la trayectoria del cúbit incluso cuando las probabilidades de los fonones van cambiando debido al decaimiento de la energía. La figura a continuación muestra una traza de interferometría y el ajuste que se usó para extraer la población de fonones en el oscilador mecánico. La traza contiene una combinación de varias oscilaciones de frecuencia, cada una de ellas correspondiente a un número de fonones distinto. El peso de una frecuencia particular en la combinación representa la probabilidad de que el correspondiente número de fonones esté presente en el estado mecánico que se vaya a medir.

Figura 3. Caracterización del estado mecánico. La interferometría de cúbits se realiza en presencia de fonones en el oscilador mecánico. La trayectoria resultante del estado del cúbit contiene información sobre la distribución de probabilidad del número de fonones. Una traza típica se muestra aquí y se ajusta a la forma funcional de la Ecuación 3 para determinar el índice de fonones del estado mecánico. Figura adaptada de [1].

Entrelazando dos osciladores mecánicos
Con la habilidad de controlar y medir el estado de cada oscilador mecánico, el siguiente paso es preparar un estado del sistema donde el movimiento de los dos osciladores esté entrelazado. Escribimos el estado del cúbit y los dos osciladores como \left| m_1, q, m_2 \right\rangle, donde el oscilador mecánico contiene m_1, m_2=0,1,2,.. fonones y el cúbit puede estar tanto en el estado fundamental (g) como en el excitado (e). Primero, se prepara el cúbit en su estado excitado con \left| 0,e,0 \right\rangle. Medio intercambio entre el cúbit y el primer oscilador mecánico los entrelaza, \left| 1, g, 0 \right\rangle + \left| 0, e, 0 \right\rangle. Esto se consigue llevando el cúbit a resonancia con el oscilador mecánico sólo durante la mitad del tiempo requerido para llevar a cabo un intercambio completo, como se puede ver en la Figura 2. Finalmente, el estado del cúbit se intercambia por completo con el del segundo oscilador mecánico, resultando en el estado \left| 1, g, 0 \right\rangle + \left| 0, g, 1 \right\rangle. Esto deja al cúbit en su estado fundamental con los dos osciladores mecánicos completamente entrelazados entre sí (\left| 1,0 \right\rangle + \left| 0,1 \right\rangle) \bigotimes \left| g \right\rangle.

Perspectiva de futuro
Los autores construyen un dispositivo que acopla el movimiento mecánico a un cúbit superconductor. El cúbit es usado para preparar y medir los modos de un modo mecánico individual. Los autores presentan un protocolo que prepara dos modos mecánicos, ambos acoplados al mismo cúbit, en un estado entrelazado. Este trabajo demuestra los cimientos que se necesitan para construir un sistema cuántico híbrido combinando dos sistemas cuánticos dispares. Los autores emparejan el control preciso del cúbit superconductor con las largas vidas medias de los modos mecánicos para construir un dispositivo que aproveche los puntos fuertes de ambos sistemas. Este tipo de diseño permitirá futuros avances en la computación cuántica, la detección y la comunicación partiendo de muchas tecnologías diferentes.

Referencias

[1] Wollack, E.A., Cleland, A.Y., Gruenke, R.G. et al. Quantum state preparation and tomography of entangled mechanical resonators. Nature 604, 463–467 (2022).

Akash Dixit construye cúbits superconductores y los acopla a cavidades 3D para desarrollar novedosas arquitecturas cuánticas y buscar materia oscura.

Gracias a Joe Kitzman por sus grandes aportaciones y comentarios a la hora de editar este artículo.

Quantum routing with teleportation

This post was sponsored by Tabor Electronics. To keep up to date with Tabor products and applications, join their community on LinkedIn and sign up for their newsletter.

Authors: Dhruv DevulapalliEddie SchouteAniruddha BapatAndrew M. ChildsAlexey V. Gorshkov

arXiv: https://arxiv.org/abs/2204.04185

Background and motivation

When we write quantum circuits on paper or in software, it’s often convenient to assume that any pair of qubits are connected. It’s convenient both (i) as a level of abstraction – we sometimes don’t want to think about low-level hardware details when thinking about algorithms – and (ii) because it’s in some sense true – even if there’s not a direct edge between two qubits, as long as there is some connected path the qubits can interact. This is exhibited in the figure below.

In the left panel, this figure shows a five-qubit superconducting processor from IBMQ, and highlights the qubit connections in the right panel. Qubit Q0 and Q1 are directly connected, but qubit Q0 and Q3 are not. However, there is a connected path from qubit Q0 to Q3, namely the path Q0 – Q2 – Q3. Because there is a connected path, two-qubit operations can be performed between qubits Q0 and Q3.

How is this possible? Swapping two qubits is a unitary operation – indeed a self-inverse operation – and so a permissible quantum operation. Furthermore, it’s safe to assume that this is a readily available operation on a quantum computer. Indeed, we can compose a swap operation out of three controlled-not (CNOT) operations, and CNOTs are commonly assumed to be a primitive operation on a quantum computer. A CNOT is defined as

where a and b are bits and ⊕ denotes addition modulo two. In words, the second qubit is flipped if the first qubit is in the |1⟩ (excited) state. The subscript “12” indicates that qubit 1 is the control and qubit 2 is the target. If we swap this indices, then

From this, a little algebra shows that the composition of three CNOTs implements a swap operation:

So, we can assume we have such a swap operation (SWAP) available between connected qubits.

In the above figure, qubits Q0 and Q3 weren’t directly connected, but they were both connected to qubit Q2. If we swap the state of Q0 and Q2, then there is now a direct connection between Q0 and Q3, and we can perform a two-qubit gate. If we’d like, after the two-qubit gate we can SWAP Q0 and Q2 again to restore the previous configuration. It’s easy to generalize this to any pair of qubits which have a connected path between them. Such a sequence of SWAPs is known as a SWAP network, and the general task of “getting qubits where they need to be” is known as qubit routing. The word “routing” is used in reference to packet switching on networks, for example the internet, a task with many common features.

Thus it’s safe to assume that we can perform a two-qubit gate between any pair of qubits. The downside is the additional SWAP operations needed to do so. Quantum computers are noisy and each operation has some probability of error, so the more operations there are the more likely it is for an error to occur. It is thus of great interest and practical importance to develop procedures to perform qubit routing with the fewest possible resources, i.e., with the shortest possible depth.

Main idea and results of the paper

This paper focuses on performing qubit routing with the fewest possible resources, and in particular considers a clever qubit routing procedure based on teleportation. These authors weren’t the first to consider teleportation for qubit routing, but they analyze it in novel ways. As we will discuss below, teleportation requires local operations (including measurements) and classical communication, abbreviated LOCC. As such, the author’s scheme can be referred to as LOCC routing in general and teleportation routing in particular. Here, we use “TELE routing” to mean teleportation-based routing and “SWAP routing” to mean SWAP-based routing.

The authors’ main strategy is to define metrics for how well qubit routing algorithms perform, then compare TELE routing to SWAP routing in three main categories. What are the three categories? A routing problem is defined by a quantum computer you want to run on and a circuit you want to run. More abstractly, we represent a quantum computer by a graph G where nodes (vertices) are qubits and edges are connections between qubits, and we represent a circuit as a permutation π of the graph. (We don’t care about the operations here, only how to route the qubits, so it’s sufficient to represent the circuit as a permutation.) So, a routing problem is defined by a graph G and a permutation π. The three categories the authors consider are:

  1. A specific graph G and a specific permutation π.
  2. A specific graph G and any permutation π.
  3. Any graph G and any permutation π.

The main results in each category, colloquially stated, are:

  1. There exists a graph G with N nodes and a permutation π such that SWAP routing takes depth of order N and TELE routing takes constant depth independent of N.
  2. There exists a graph G with N nodes such that, for any permutation π on G, SWAP routing takes depth log N and TELE routing takes constant depth independent of N.
  3. For any graph G with N nodes and any permutation π on G, the maximum advantage of TELE routing over SWAP routing is of order (N log N)½.

The remainder of this article is an invitation to understanding these results, starting with a review of teleportation then walking through the simpler results while providing intuition for the others.

Teleportation

Since we are going to use teleportation as a subroutine for qubit routing, let’s quickly (re-)analyze the protocol. The quantum circuit for teleportation is shown below.

This circuit “teleports” an arbitrary quantum state |𝜓⟩ = α|0⟩ + β|1⟩ on the first qubit to the third qubit by means of local operations (both unitary operations and measurements) and classical communication. Concretely, “classical communication” means performing operations conditional on the measurement outcomes (classical information) of the first two qubits. Because sending this classical information cannot be instantaneous, the name “teleportation” is not to be taken in a literal sense.

We can understand the above circuit for teleportation as follows. The Hadamard and CNOT create a Bell state on the last two qubits. (We omit normalization here and throughout.)

After, we measure the top two qubits in the Bell basis, which corresponds to the Bell state preparation circuit in reverse. Before the measurements, one can show with a little algebra that the final state of the three qubits is as follows (again omitting normalization):

Written this way, it’s easy to see how to always obtain the state |𝜓⟩ on the third qubit after measuring the first two qubits:

  • If we measure |00⟩ (the first term in the above equation), the state of the third qubit is |𝜓⟩
  • If we measure |01⟩ (the first term in the above equation), the state of the third qubit is X|𝜓⟩. Perform X to obtain |𝜓⟩.
  • If we measure |10⟩ (the first term in the above equation), the state of the third qubit is Z|𝜓⟩. Perform Z to obtain |𝜓⟩.
  • If we measure |11⟩ (the first term in the above equation), the state of the third qubit is XZ|𝜓⟩. Perform XZ to obtain |𝜓⟩.

Thus we always obtain |𝜓⟩ on the third qubit. Now that we have one three-qubit “gadget” for teleportation, we can consider chaining several of these gadgets together to teleport a qubit a greater distance. This is illustrated in Figure 1 of the paper:

Notice the very nice property that the depth of this seven-qubit teleportation circuit is the same as the depth of the three-qubit teleportation circuit. Specifically, both circuits have a depth of four. This is different from using SWAP routing in which the SWAPs have to be sequential as shown below.

Here, the depth of the circuit grows with the number of qubits. This observation is key to understanding why and when teleportation-based routing may be advantageous.

Routing time, and bounds

Let rt(G, π) denote the routing time (minimum circuit depth) to perform the permutation π on the graph G. Let rt(G) denote the worst-case routing time taken over all permutations on G.

Note that any SWAP routing procedure can be “mimicked” by a TELE routing procedure which simply substitutes each SWAP operation with a teleportation gadget, using the same (constant) depth. But, it’s possible for TELE routing to be faster. Therefore, the time for TELE routing is at most the time for SWAP routing.

In prior work, it has been shown that SWAP routing on a graph G with N nodes takes O(N) time. Combining this with the previous argument, we also have that TELE routing takes O(N) time.

In summary, so far we have TELE routing time ≤ SWAP routing time = O(N) on a graph G with N nodes.

It’s also possible to show lower bounds. Since swapping two nodes at a distance d requires at least d SWAPs, we have that the SWAP routing time is at least diam(G). (The diameter of a graph G is the maximum shortest-path distance between any pair of nodes.) This is referred to as the “diameter lower bound” in the paper.

The diameter bound doesn’t apply to TELE routing, but it is possible to provide a lower bound for this. Leaving the proof to an unpublished article by the same authors, the authors provide the bound

where c(G) is the vertex expansion of G and, for connected graphs, is between 2 / N and 1. LOCC routing is the most general, so this implies SWAP routing ≥ TELE routing ≥ 2 / c(G) – 1.

TELE routing vs SWAP routing

Define the teleportation advantage adv(G, π) as the ratio of SWAP routing to TELE routing, i.e.

Category 1: A specific graph G and a specific permutation π

The first case the authors consider is shown below.

Here we have G as a line graph (hollow black nodes with black lines as edges) and π as the permutation which swaps the left-most and right-most qubits. If there are N nodes in G, SWAP routing takes depth of order N because each SWAP must be in parallel. However, as we have seen above, the depth of TELE routing is constant in N. Therefore the teleportation advantage adv(G, π) is of order N, a significant advantage!

The second case the authors consider is similar, shown below.

Here we have the same graph G but a “rainbow permutation”  π, so-called because the red lines form a rainbow as drawn above. The parameter 0 < α < 1 quantifies how many nodes appear in the rainbow permutation. By the diameter bound, SWAP routing as depth N. For TELE routing, one can swap each pair of nodes sequentially with a constant depth circuit. Since there are Nα / 2 pairs of nodes in the permutation, TELE routing takes depth Nα / 2. So, the teleportation advantage in this case is O(N1 – α). This is sublinear for nonzero α, so less than the linear advantage for the first case, but still advantageous.

One might suppose TELE routing is only advantageous because the diameter of the line graph in the above examples was of order N (the number of nodes). But now consider wrapping the line graph around so two end nodes are connected in a circle. Further, place an additional node in the center of the circle with an edge to every node on the circumference, as shown below.

The diameter of this graph, the so-called “wheel graph” or WN, is constant, independent of the number of nodes N. (Specifically, the diameter is two.) Now consider the permutation shown in red on this graph. This permutation swaps qubits at a distance l along the “rim” of the wheel. As the authors argue, the SWAP routing time for this case is min(3l, N / l – 1). The 3l corresponds to using the central node to SWAP every pair of qubits sequentially, and the N / l – 1 corresponds to swapping qubits along the rim of the wheel in parallel. Now, for TELE routing, this permutation on G can be done in constant depth by simply teleporting each pair of qubits along the rim in parallel. If we set l to be the square root of N / 2, this yields the maximum teleportation advantage of

Thus, teleportation routing enables super-diametric speedups.

Category 2: A specific graph G and any permutation π

In the above example we got to hand-pick the permutation π. Now let’s consider the more general case of any permutation π, and ask if we can find some graph G where TELE routing is advantageous.

The authors show the answer to this question turns out to be yes: there exists a graph G with N nodes where SWAP routing takes depth at least logarithmic in N, and TELE routing takes constant depth independent of N. The graph G which achieves this is shown below.

This graph has n layers of subgraphs vertically stacked on top of each other. As such, the authors call it L(n). The nth layer is a complete graph on 2n nodes, shown with blue edges above. These layers are stacked by connecting every node in the current layer to the layer below it, shown with black edges above. For example, the first layer K1 has one node, and an edge to each node in the layer K2 below. The layer K2 has two nodes, and each node is connected to every node in the layer K4 below it. And so on. The total number of nodes in L(n) is 2n – 1. Imagine building a quantum computer with this topology!

The proofs of the SWAP routing and TELE routing depths quoted above are somewhat involved, so we omit them here and refer the interested reader to the paper (see Sec. V). 

Category 3: Any graph G and any permutation π

Last, the authors consider the most general category of any graph G and any permutation π. For this case, the relevant metric is the “maximum teleportation advantage”

The authors prove (Theorem 6.4) that 

Thus, for any quantum computer with N qubits, no matter what the topology is or the specific quantum computation we wish to execute, the maximum advantage we can obtain using teleportation-based routing over swap-based routing is of order (N log N)½. A careful reader may question if this result disagrees with the first example in Category 1 where a specific graph G and specific permutation π admitted a teleportation advantage of order N. There is, however, no disagreement: the present result considers the ratio of worst-case permutations, but the result in Category 1 considers a specific permutation. 

While it’s theoretically interesting to consider any graphs G, there are common patterns to which quantum computers are currently built based on engineering and other considerations. For example, superconducting qubits are often arranged in a two-dimensional plane with nearest-neighbor connectivity. The authors specialize the above result to this case of planar graphs and show that there is at most a constant factor advantage to using TELE routing. We remark again that this result considers the ratio of worst-case permutations and does not disagree with previous results concerning specific permutations. Indeed, you may construct or encounter a quantum circuit you wish to run on a planar quantum computer for which teleportation-based routing is significantly more practical, even as a constant factor improvement.

Summary and conclusions

The qubit routing problem is well-motivated by practical considerations and interesting to study. A swap-based routing approach is always possible and bears similarity to similar classical problems. However, just as there are clever, uniquely quantum strategies for subroutines like addition on a quantum computer, there is a clever, uniquely quantum strategy to qubit routing based on teleportation. It’s easy to construct examples where teleportation-based routing is advantageous, and the authors provide general statements about its performance relative to swap-based routing. Although in the most general sense the advantage is at most (N log N)½ for a quantum computer with N qubits – and even at most constant for planar graphs – there are very likely practical scenarios in which teleportation-based routing is likely to be advantageous. So, next time you are pondering practicalities of how an algorithm may run on a quantum computer, keep teleportation as a strategy for qubit routing in the back of your head!

Controlled Dissipation with Superconducting Qubits

This post was sponsored by Tabor Electronics. To keep up to date with Tabor products and applications, join their community on LinkedIn and sign up for their newsletter.

Authors: P.M. Harrington, M. Naghiloo, D. Tan, K.W. Murch

First Author’s Primary Affiliation: Department of Physics, Washington University, Saint Louis, Missouri 63130, USA

Manuscript: Published in Physical Review A

Introduction

Quantum systems are generally very sensitive, and upon interacting with the environment, their quantum properties can decohere. This essentially makes a given quantum system dissipate into purely classical behavior. However, in certain contexts it is possible to use dissipation in a controlled fashion to increase the control of quantum systems. A few examples of controlled dissipation in this way include laser cooling of atoms, cooling of low frequency mechanical oscillators, and for the control of quantum circuits. In this recent publication [1], the authors are able to demonstrate stabilization of superposition states in a superconducting qubit using a custom made photonic crystal loss channel. By considering how the photonic crystal induces loss on the system, the authors provide a master equation treatment which explains how the combination of a specialized drive applied to the qubit in addition to the dissipation provided via the photonic crystal allows for precise control of the qubit state for times much longer than standard qubit coherence times.

Experimental Details

This experiment consists of a superconducting qubit whose dipole moment is coupled to the electric field inside of a three dimensional waveguide cavity. In this experiment, the role of the waveguide cavity is to provide microwave control of the qubit as well as reading out the state of the qubit. The superconducting qubit consists of two Josephson junctions in parallel, forming a superconducting quantum interference device (often referred to as a “SQUID”). This allows the authors to change the resonant frequency of the qubit by threading an external magnetic field through the SQUID loop. On the output of the waveguide cavity, the authors connect a photonic crystal to the circuit. This photonic crystal is made out of a regular coaxial cable which is mechanically deformed in a specific way in order to change the impedance of the cable. The result of the spatially varying impedance in the cable leads to the opening of a bandgap – leading to photon energies (or frequencies) where the photonic density of states is zero (see Fig. 1 for a schematic of the experimental setup). By changing the photonic density of states as a fucntion of energy, the decay of the qubit will also change as a function of frequency.

Figure 1
Left: Schematic of the experimental system. The superconducting qubit is mounted in a copper cavity which is used for control and readout of the qubit state. By passing current though a superconducting wire wrapped around the cavity a magnetic field is generated perpendicular to the substrate containing the qubit, allowing the authors to tune the resonant frequency of the qubit. The photonic crystal is connected to the output port of the cavity, changing the density of states that the qubit can decay into. Right: Room temperature measurements of the reflection off of the photonic crystal. In the stop-band (from 5.5 – 6.4 GHz) most of the signal sent into the photonic crystal is reflected, verifying that there is a low density of states at those frequencies. Above 6.4 GHz, the photonic band gap closes and photons can transmit through the photonic crystal.

Qubit Decay Rates

In order to measure the decay rate of the qubit, the authors first excite the qubit into its excited state by applying a pulse of energy which is resonant with the qubit to the system. They then measure the probability of the qubit remaining in its excited state as a function of time after the pulse is applied. By fitting the measured probability to an exponential decay and extracting the decay constant, one is able to determine the qubit decay rate. The resonant frequency of the qubit is then adjusted by changing the external magnetic flux threading the SQUID loop, and measuring the qubit decay rate as a function of qubit frequency in order to investigate the impact of the photonic crystal on the qubit lifetime. The total decay rate of the qubit can be written as

\gamma_1 = \gamma_d + \rho(\omega_q)(g/\Delta_q)^2 \kappa.

In Eq. 1, \gamma_1 is the measured decay rate of the qubit, \kappa/2\pi = 18~\textrm{MHz} is the linewidth of the microwave cavity, g/(2\pi) = 200~\textrm{MHz} is the coupling strength between the qubit and the cavity, \Delta_q = \omega_c - \omega_q is the difference in resonant frequency between the qubit and the cavity, \rho(\omega_q) is the density of states of the photonic crystal at the qubit frequency, and \gamma_d represents decay of the qubit into dissipation channels other than the photonic crystal. By measuring the total qubit decay rate for various values of \omega_q, it should be possible to extract information about the density of states of the photonic crystal! See Fig. 2 below for the resulting measurement

Figure 2
Measurement of qubit decay rates over a broad range in frequencies. Because the qubit loss varies quickly with qubit frequency, by flux biasing the qubit to a point where the derivative of the qubit loss is large, it is possible for Mollow triplet sidebands to sample frequencies with both very high and very low loss. By measuring the generalized Rabi frequencies across the same values of qubit frequency, the authors verify the variable couple of the qubit to the photonic crystal.

Dynamics and Emission of a Driven Qubit

After verifying that the density of states in the photonic crystal can shape the decay rate of the qubit, the authors now consider more carefully how the qubit actually emits energy. Specifically, a strong drive applied with amplitude \Omega which is detuned from the qubit energy by \Delta = \omega_d - \omega_q, where \omega_d is the frequency of the drive and \omega_q is the qubit energy is considered. If the amplitude of the drive is much larger than the loss rate of the qubit, the qubit will emit energy at three different frequencies \omega_d, and \omega_d~\pm~\Omega_R, where \Omega_R = \sqrt{\Omega^2 + \Delta^2} is called the generalized Rabi frequency. This emission spectrum is called the Mollow triplet [2]. See Fig. 3 for a schematic of the Mollow triplet emission.

Figure 3
Schematic which represents the emssion of the driven two level system. Under the presence of a strong drive, the qubit emits radiation at frequencies corresponding to the drive frequency \omega_d as well as at the frequencies \omega_d \pm \Omega_R. Due to the shaped loss spectrum, the area of one sideband can be supressed, indicating that the qubit will emit radiation at this frequency at a lower rate compared to the other sideband. Top right: under the presence of the drive, the quantization axis of the qubit also rotates, which changes which qubit states can be prepared/stabilized.

Because the authors have observed that the photonic crystal shapes the loss rate of the qubit on a energy scale comparable to values of experimentally accessible \Omega_R, it is possible for one of the sidebands of the Mollow triplet to experience a high loss rate while the other sideband of the Mollow triplet experiences a low loss rate.

The next thing to consider is what the presence of an applied drive does to the energy spectrum of the qubit. In a frame rotating with the drive frequency, the qubit Hamiltonian is given by

H_q = \frac{\Delta}{2}\sigma_z + \frac{\Omega}{2}\sigma_x,

where \sigma_z and \sigma_x are Pauli matrices. Since this Hamiltonian is not diagonal, it is convenient to rotate basis such that the Hamiltonian can be written in a new form

\tilde{H}_q = \frac{\Omega_R}{2}\tilde{\sigma}_z,

where the rotated Pauli Z matrix can be written as \tilde{\sigma}_z = \sin{2\theta}\sigma_x - \cos{2\theta}\sigma_z, and the rotation angle is defined as \tan{2\theta} = -\Omega/\Delta with 0<\theta<\pi/2. Because we have written the Hamiltonian in a rotated basis, we must also consider how the new eigenstates of the system rotate relative to the original eigenstates, which we will call |g\rangle and |e\rangle for ground and excited state, respectively.

|\tilde{g}\rangle = \cos{\theta}|g\rangle - \sin{\theta}|e\rangle

|\tilde{e}\rangle = \sin{\theta}|g\rangle + \cos{\theta}|e\rangle

At this point it’s probably useful to consider a useful example! In the case of a resonant drive, \Delta = 0, which immediately informs us that \theta = 45^{\circ}, so we can rewrite the rotated eigenstates of the system as |\tilde{g}\rangle = \frac{1}{\sqrt{2}}(|g\rangle - |e\rangle) \equiv |-x\rangle, and |\tilde{e}\rangle = \frac{1}{\sqrt{2}}(|g\rangle + |e\rangle) \equiv |+x\rangle, which have the special property that \sigma_x|\pm x\rangle = \pm 1 |\pm x\rangle. Because the state |-x\rangle has a lower energy, it will emit energy corresponding to the lower energy sideband of the Mollow triplet and vice versa for the state |+x\rangle. If the loss of the qubit is vastly different for either of these states, that will promote decay into either the state |-x\rangle or |+x\rangle! Specifically, if the qubit is at a resonant frequency near 6.4766 GHz (see Fig. 2), the state at higher energy (corresponding to |+x\rangle in this example) has a lower loss rate, so we should expect that while the drive is turned on, the qubit preferentially would decay into this state! This means that the expectation value \langle \sigma_x \rangle would tend towards +1 in this scenario! In the case of a uniform loss spectrum, there would be no preferred decay of the qubit and one would expect that all of the qubit expectation values would decay to zero.

Lindblad Master Equation

In the presence of the combined drive and dissipation experienced by the qubit, the dynamics of the reduced density matrix which describes the qubit can be written according to the Lindblad Master equation [3]:

\dot{\rho} = i[\rho,H] + \gamma_0 \cos{(\theta)}\sin{(\theta)}\mathcal{D}[\tilde{\sigma}_z]\rho + \gamma_{-} \sin{^4\left(\theta\right)} \mathcal{D}[\tilde{\sigma}_{+}\rho + \gamma_{+}\cos{^4\left(\theta\right)} \mathcal{D}[\tilde{\sigma}_{-}]\rho.

Here, \rho is the reduced density matrix for the qubit, the dissipation superoperator is also introduced as \mathcal{D}[A]\rho = \left( 2 A \rho A^{\dagger} - A^{\dagger}A\rho - \rho A^{\dagger}A\right)/2. The rate \gamma_0 represents dephasing of the qubit in the basis rotated by \theta, which couples to the \tilde{\sigma}_z operator and transitions between eigenstates in the rotated basis are driven by the “jump” operators \tilde{\sigma}_{\pm} which are related to the rates \gamma_{\mp}. Similar to the previous example, if the photonic crystal shapes the qubit loss such that \gamma_{\pm} \gg \gamma_{\mp}, a corresponding rotating frame eigenstate will be stabilized.

Experimental Results

In order to verify that the authors can use the combination of drive and dissipation to prepare and stabilize qubit states, they implement the following bath engineering protocol. First, the qubit is flux biased to a resonant frequency of 6.4766 GHz (as in our example). Then, a coherent drive is applied to the system for nearly 16~\mu s (which is much longer than the qubit coherence times in the absence of drive!). During this time, the qubit should preferentially decay to an eigenstate of the rotated system if the Mollow triplet sidebands have different weights. Once the drive is shut off, the expectation value \langle \sigma_x \rangle is measured for various combinations of drive parameters. Results are shown below as well in Fig. 4 as comparisons to numerical solutions to the master equation, the authors not only see that the qubit expectation values don’t decay to 0, but also fantastic agreement between the theory and the experiment! Additionally, we can recall our earlier example, and we see that along the linecut of \Delta = 0, the expectation value \langle \sigma_x \rangle approaches the value of +1 as we expected!

Figure 4
(a) Measurement of \langle \sigma_x \rangle as the parameters of the drive change. For certain drive parameters the value of \langle \sigma_x \rangle is negative. As the drive parameters are changed, the system crosses through a region of “zero coherence” where the bath engineering protocol no longer works before stabilizing \langle \sigma_x \rangle to positive values. (b) Numeric solutions to the master equation under the same drive parameters as (a). The authors observe excellent agreement between the experiments and the numeric solutions. (c) Comparison of horizontal linecuts from (a,b). The authors observe agreement between the measured values (dots) and simulated values (lines) when considering all expectation values for the qubit state (\langle \sigma_{x,y,z} \rangle)

Conclusion

In conclusion, the authors are able to demonstrate the fabrication of a spatially changing impedance coaxial cable which acts as a photonic crystal, and in turn controlling the loss spectrum of a superconducting qubit. The authors are then able to leverage this shaped emission spectrum in the context of the master equation to prepare and stabilize non-trivial states of the qubit for times much longer than the coherence times of the qubit.

References

[1] P. M. Harrington, M. Naghiloo, D. Tan, and K. W. Murch, Bath engineering of a fluorescing artificial atom with a photonic crystal, Phys. Rev. A 99, 052126 (2019)

[2] B. R. Mollow, Power spectrum of light scattered by two-level systems, Phys. Rev. 188, 1969 (1969)

[3] G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976).

Quantum control of motion

By Akash Dixit

Title: Quantum state preparation, tomography, and entanglement of mechanical oscillators

Authors: E. Alex Wollack, Agnetta Y. Cleland, Rachel G. Gruenke, Zhaoyou
Wang, Patricio Arrangoiz-Arriola, and Amir H. Safavi-Naeini

Institution: Department of Applied Physics and Ginzton Laboratory, Stanford University 348 Via Pueblo Mall, Stanford, California 94305, USA

Manuscript: Published in Nature [1], Open Access on arXiv

Introduction
The field of quantum information sciences contains a multitude of different technologies, including atoms, spins, and defect centers in diamond. This work focuses on two emerging technologies: superconducting circuits and mechanical oscillators. Each system has its advantages, but it is not obvious that any one is the best platform for building a quantum computer, developing quantum sensors, or facilitating quantum communication. To achieve these goals, it is necessary to develop hybrid quantum systems that can utilize the strengths of various quantum technologies.

In this work, the authors demonstrate the ability to couple superconducting qubits to mechanical motion. This establishes the building blocks for a hybrid quantum system that can take advantage of the the best of both systems. The qubit is customizable and easy to communicate with, making it ideal for state initialization and characterization. The mechanical modes are fabricated with small spatial footprints and have long lifetimes, making it possible to scale to larger systems and hold quantum information for long timescales. I will describe how the authors use the coupling between these two systems to both prepare and measure states of mechanical motion using the qubit. I first describe the carefully engineered device that couples one qubit to two mechanical oscillators. Then I discuss the two modes of operation, where the qubit is used to both prepare states of mechanical motion and measure the quantum state of the mechanical mode. Finally, I show how the authors use the qubit as an intermediary to prepare entangled mechanical states across two oscillators.

Device

Figure 1: Coupling a qubit to mechanics. a. Schematic of the device shows a single qubit coupled to two mechanical oscillators at distinct frequencies. b. Optical image of the qubit with the Josephson junctions shown in the inset. The wire coming in from the left carries current to apply a flux to the junction loop. The rectangular pad on the right of the image is the capacitive pad used to link the qubit to the mechanical oscillators. c. Two mechanical oscillators formed in a periodic structure of LiNiO3. Figure adapted from Ref 1.

The device used in this works consists of two mechanical oscillators and a superconducting qubit. The mechanical oscillators are fabricated in thin film lithium niobate (LiNiO3). These oscillators are formed by embedding a defect in a periodic structure of the material, called a phononic crystal. The defect is a mismatch in the periodicity of the structure and confines mechanical motion, preventing acoustic radiation and enabling long mechanical lifetimes. Like electromagnetic radiation, mechanical motion can be quantized. The individual quanta of mechanical motion are called phonons, and the mechanical oscillator can be characterized as a harmonic oscillator with equal energy level spacing. The qubit is made by fabricating an LC oscillator with superconducting materials. The key element of this circuit is a Josephson junction, which is made of aluminum oxide sandwiched between layers of superconducting aluminum. The junction acts as a nonlinear inductor that modifies the energy level spacing of the LC oscillator. The energy levels of the usual LC oscillator (which is a harmonic oscillator) are equally spaced, meaning the transition energy between any two levels is the same. However, with the nonlinear inductor in the circuit, there are no longer equally spaced energy levels, making it possible to uniquely address the two lowest energy levels of the system, ground (\left| g \right\rangle) and excited (\left| e \right\rangle). The two levels form a quantum bit (qubit). The qubit is designed to be tunable in frequency by placing two Josephson junctions in a parallel with each other. By applying a magnetic field using a wire carrying current, a magnetic flux is threaded through the loop to change the qubit frequency.

The qubit and mechanical oscillators are fabricated on separate chips that are placed \sim \mu m apart. To couple the qubit and mechanical oscillators, the authors use the piezoelectricy of the lithium niobate film. The mechanical motion of this material produces an accumulation of electric charges onto aluminum pads located on both chips, which are designed to be the capacitive element of the qubit. The qubit capacitor is charged by the motion of the mechanical oscillators, ensuring that the two systems are linked together.

Initializing a mechanical state
The authors design the qubit to interact in two different ways with the mechanical oscillators. In the first mode, the qubit is tuned to be on resonance with a particular mechanical oscillator (\omega_q = \omega_1, \omega_2). Note that the mechanical frequencies of the two oscillators are different, so the qubit can only be in resonance with one at a time. This allows for the direct exchange of energy between qubit and either oscillator at a rate related to the capacitive coupling between the two, g_1 = 2 \pi \times 9.5 MHz, g_2 = 2 \pi \times 10.5 MHz. The Hamiltonian that describes the interaction between a qubit and mechanical oscillator on resonance the Jaynes-Cummings interaction:

\mathcal{H}_{\mathrm{on}} = g(a^{\dagger} \sigma^{-} + a \sigma^{+})
[Equation 1].

a^{\dagger}, a and \sigma^{+}, \sigma^{-} are the creation, annihilation operators for the mechanical oscillator and qubit respectively. When on resonance, the qubit and mechanical oscillator swap their respective states in time \pi/g \sim 24-26 ns depending on the particular oscillator.

Figure 2: Swapping qubit and mechanical state. The qubit (Q) is first initialized in it excited state. Once the qubit is brought into resonance with the mechanical oscillator, the two coherently exchange their states. This means the joint state oscillates between \left| 0,e \right\rangle and \left| 1,g \right\rangle where \left| m,q \right\rangle represents the state of the mechanics and the qubit. In between a full swap, the qubit and mechanical oscillators are entangled together with the joint state \left| 1,g \right\rangle +\left| 0,e \right\rangle.

This swap can be used as a method of mechanical state preparation. The authors first tune the qubit so that it is off resonant from either mechanical oscillator. Then with the mechanical mode containing no quanta, the qubit is initialized so the joint states are \left| 0,g \right\rangle, \left| 0,e \right\rangle, or \left| 0,g \right\rangle + \left| 0,e \right\rangle state. The joint state \left| m, q \right\rangle, describe the phonon number of a particular mechanical oscillator, m = 0, 1, 2…, and whether the qubit is in the ground or excited state, q = g, e. The qubit frequency is tuned to be on resonance with either mechanical mode for a time corresponding to a full swap. When the swap operation is applied to the joint state \left| 0,g \right\rangle, the system remains unchanged since both subsystems are in their ground state and there is no energy to exchange. Under the swap, the state \left| 0,e \right\rangle becomes \left| 1,g \right\rangle as shown in Figure 1. When the qubit is initialized in a superposition state, the joint state is \left| 0,g \right\rangle + \left| 0,e \right\rangle. The swap operation acts on both parts of this superposition leading to the final state \left| 0,g \right\rangle + \left| 1,g \right\rangle. The mechanical oscillator is now in a superposition state, but the state of the mechanical oscillator is not entangled with the qubit state.

Measuring a mechanical state
In the second mode of operation, the qubit is off resonance from either mechanical oscillator, usually called a dispersive interaction. The dispersive interaction rate between qubit and mechanical oscillator, \chi, is now set by the direct capactive coupling, g, the detuning between qubit and mechanics, \Delta, and other qubit parameters. In the limit that the detuning between qubit and mechanics is larger than the the capacitive interaction rate (\Delta \gg g), the interaction shown in Equation 1 is approximated by the off resonant Hamilitonian:

\mathcal{H}_{\mathrm{off}} = \chi a^{\dagger} a \sigma_z
[Equation 2].

The combination a^{\dagger}a is the operator version of the number of phonons, m, in the mechanical oscillator. \sigma_z is the operator version of the qubit state, either \left| g \right\rangle or \left| e \right\rangle.

Without the interaction between the qubit and mechanics, the Hamiltonian of the just the qubit would look like \mathcal{H}_{q} = \omega_q \sigma_z, where \omega_q is the transition frequency of the qubit. When we add in the off resonant interaction, the Hamiltonian can be expressed as \mathcal{H}_{q} + \mathcal{H}_{\mathrm{off}} = (\omega_q - \chi a^{\dagger}a )\sigma_z. By comparing the combined Hamiltonian with the one of just the qubit, we see that the effect of the interaction is to modify the transition frequency of the qubit (represented by everything before the \sigma_z). So now, the qubit transition frequency is dependent on the number of phonons in the mechanical oscillator (m = a^{\dagger}a). For every additional phonon in the mechanical oscillator, the qubit transition frequency shifts by \chi.

This interaction is crucial to being able to characterize the state of the mechanical oscillator. Since the different phonon numbers impart a different frequency shift on the qubit, the mechanical state is imprinted on the frequency of the qubit. To resolve the probabilities of different phonon numbers in the mechanical oscillator, a qubit interferometry measurement is performed. The mechanical oscillator is prepared in a Fock state with 0 or 1 phonons or in a superposition of many phonon 0, 1, 2,… Then the qubit is placed in a superposition state \left| g \right\rangle + \left| e \right\rangle and allowed to precess for a variable time, t. During this time, the superposition state accumulates a phase at rate \chi if there is one phonon, 2\chi for two phonons, and so on. The phase accumulated then reflects the probability (A_n) that the mechanical oscillator contained zero, one, two, etc… phonons. The qubit state evolves to \left| g \right\rangle + e^{i\phi} \left| e \right\rangle, where the phase accumulated is \phi = \sum_n A_n n \chi t. The authors rotate the qubit back into its measurement basis and monitor the final population of the qubit excited state as a function of the interaction time, t, and fit the trajectory to the functional form

S(t) = \sum_n A_n e^{-\kappa t/2} \cos [(2 n \chi t) + \phi_n]
[Equation 3]

This form includes the phonon number probabilities, A_n, as well as the number dependent precession rate, n \chi. It also includes a number dependent phase offset, \phi_n, and the phonon decay constant, \kappa. This captures the dynamics of the qubit trajectory even when the phonon probabilities are changing due to energy decay. The figure below shows an interferometry trace and the fit used to extract the phonon population in the mechanical oscillator. The trace contains a combination of various frequency oscillations each corresponding to a different phonon number. The weight of a particular frequency in the combination represents the probability of the corresponding phonon number to be present in the mechanical state being measured.

Figure 3: Mechanical state characterization. Qubit interferometry is performed in the presence of phonons in the mechanical oscillator. The resulting qubit state trajectory contains information about the probability distribution of phonon numbers. A typical trace is shown here and is fit with the functional form in Equation 3 to determine the phonon content of the mechanical state. Figure adapted from [1].

Entangling two mechanical oscillators
With the ability to control and measure the state of each mechanical oscillator, the next step is to prepare a joint state where the motion of the two oscillators is entangled together. We write the joint state of the qubit and two mechanical oscillators as \left| m_1, q, m_2 \right\rangle, where the mechanical oscillators can contain m_1, m_2=0,1,2,.. phonons, and the qubit can be in either the ground (g) or excited (e) state. First the qubit is prepared in its excited state with \left| 0,e,0 \right\rangle. A half swap between the qubit and the first mechanical oscillator entangles the two, \left| 1, g, 0 \right\rangle + \left| 0, e, 0 \right\rangle. This is accomplished by bringing the qubit into resonance with the mechanical oscillator for only half the time required the perform a full swap as seen in Figure 2. Finally, the qubit state is fully swapped with the second mechanical state resulting in the state \left| 1, g, 0 \right\rangle + \left| 0, g, 1 \right\rangle. This leaves the qubit in the ground state with the two mechanical state fully entangled together (\left| 1,0 \right\rangle + \left| 0,1 \right\rangle) \bigotimes \left| g \right\rangle.

Future outlook
The authors construct a device that couples mechanical motion to a superconducting qubit. The qubit is used to prepare and measure the modes of individual mechanical modes. The authors present a protocol that prepares two mechanical modes, both coupled to the same qubit, in an entangled state. This work demonstrates the building blocks needed to construct a hybrid quantum system by combining two disparate quantum systems. The authors match the precise control of a superconducting qubit with the long lifetimes of the mechanical modes to construct a devices that engages the strengths of both systems. This kind of design will enable future advances in quantum computing, sensing, and communication by drawing from many different technologies.

References

[1] Wollack, E.A., Cleland, A.Y., Gruenke, R.G. et al. Quantum state preparation and tomography of entangled mechanical resonators. Nature 604, 463–467 (2022).

Akash Dixit builds superconducting qubits and couples them to 3D cavities to develop novel quantum architectures and search for dark matter.

Thanks to Joe Kitzman for great discussions and feedback in editing this article.

Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics

Authors: Uwe von Lüpke, Yu Yang, Marius Bild, Laurent Michaud, Matteo Fadel, and Yiwen Chu

First Author’s Primary Affiliation: Department of Physics, ETH Zurich, Zurich, Switzerland

Manuscript: Published in Nature Physics

Introduction

Superconducting qubits are a promising candidate for functional quantum computation as well as investigating fundamental physics of composite quantum systems where superconducting qubits are coupled to other quantum degrees of freedom. The most common example of this is circuit quantum electrodynamics (cQED), where a superconducting qubit is coupled to an electromagnetic resonator, and the resonator can be used to control and read out the quantum state of the qubit. In an analog to cQED, it is possible to replace this electromagnetic resonator with a mechanical resonator – this now allows for the study the quantum limits of mechanical excitations in a field commonly known as circuit quantum acoustodynamics (cQAD). By coupling a superconducting qubit to a mechanical resonator in this fashion, physicists are able to draw upon the rich and developed field of cQED to study not only further applications in quantum information science using cQAD as a building block, but also the fundamental physics of mechanical resonators in their quantum limit. In addition to the ability to study new physics, acoustic resonators are much more compact due to the slow speed of sound (relative to the speed of light which would be used in an electromagnetic cavity!) leading to much smaller wavelengths at high frequencies. In cQED/cQAD the interaction between the qubit and the resonator is often described by the Jaynes-Cummings Hamiltonian:

\hat{H}/ \hbar = \omega_c \hat{a}^{\dagger} \hat{a} + \frac{\omega_q}{2} \hat{\sigma}_{z} + g\left( \hat{a}\hat{\sigma}_{+} + \hat{a}^{\dagger} \hat{\sigma}_{-}\right)

Here the first term in the Hamiltonian describes the resonator as a harmonic oscillator with a transition frequency \omega_c, and the second term describes the qubit as a two level system with transition frequency \omega_q. The interesting physics described by this Hamiltonian is contained in the third term, which contains the interaction between the qubit and the resonator. Because the terms \hat{a} \hat{\sigma}_{+} and \hat{a}^{\dagger} \hat{\sigma}_{-} conserve total excitation number, we can think of this interaction term as the qubit and the resonator “trading” excitations with a rate g!

In this recent paper published in Nature Physics, the authors demonstrate strong coupling between a superconducting qubit and an HBAR (high bulk overtone acoustic resonator)[1]. HBAR devices launch mechanical excitations (called phonons) by using the piezoelectric effect. This means that the polarization and the mechanical strain in the material are not independent – by applying an electric field to a piezoelectric material it is possible to create mechanical excitations! The device in this experiment uses a thin film of piezoelectric aluminum nitride (AlN) patterned onto a small sapphire chip. This substrate is then sandwiched together with another chip containing a superconducting qubit which acts as an anharmonic oscillator. By carefully aligning the two chips relative to each other, the authors are able to couple the electric field of the qubit to the piezoelectric material on the chip containing the HBAR and thus couple the degrees of freedom of the qubit to the phonon modes in the HBAR (see Fig. 1 for a description of the device). The joint quantum acoustics system is then loaded into an electromagnetic cavity, which will also couple to the qubit and allow for the control and measurement of the device.

Figure 1
(a) View of the hybrid flip-chip device. Both large substrates are sapphire, which is transparent and thus makes it much easier to align the two chips to a high precision. The substrate that contains the superconducting qubit is on the bottom and slightly larger than the top chip which contains the HBAR resonator. (b) Optical microscopy image of the small disc of piezoelectric AlN which is responsible for the creation of phonons.

Measurement of Phonon Lifetime

By applying strong microwave signals into the system, the qubit frequency is able to be moved around by a small amount such that the qubit’s resonant frequency can be equal to the resonant frequency of the phonon mode. In this case, the qubit and mechanical system will transfer exctations to each other in the time \pi/2g. This can be used as a tool to measure how long phonons will remain in the HBAR device by first promoting the qubit to its excited state and then shifting the qubit’s frequency so that it’s resonant frequency is the same as that of the mechanical mode for a time \pi/2g. This is often called a “swap” operation. Once the excitation has been fully transferred to the mechanical mode, the qubit’s resonant frequency is then quickly moved far away in frequency so that the two systems stop exchanging energy. After a variable amount of time the qubit is then brought back to the mechanical resonator and another swap operation is performed. Then, by measuring the probability of the qubit being in its ground or excited state, experimentalists are able to measure whether or not the phonon was lost to the environment during the time the qubit was not resonant with the HBAR device. Another similar measurement is preformed to measure the phase coherence of the phonon mode, this is done by preparing a superposition state in the qubit and measuring the evolution of its phase (see Fig. 2).

Figure 2
Measurement of phonon lifetime in the HBAR resonator using the experimental protocol described in the section above (top). A measurement of the phonon dephasing rate is also measured by preparing a qubit superposition state and then preforming the swap operation into the phonon resonator (bottom).

Measurement of Phonon Coherent States

By applying a strong tone to the system which is resonant with the HBAR device, the HBAR device will be placed into a coherent state which can be written down as a sum of Fock states:

|\alpha\rangle = \sum_{n = 0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n\rangle

In order to determine how this will impact the spectral features of the qubit, it can be helpful to look at the probability of having $m$ phonons given a certain coherent state |\alpha\rangle, which is found to be |\langle m | \alpha\rangle|^2 = e^{-\overline{n}}\frac{\overline{n}^m}{m!}, where the mean phonon number \overline{n} = |\alpha|^2 has been introduced. This is simply a Poisson distribution in phonon number, and interestingly by measuring the mean phonon number it’s possible to learn about the quantum mechanical fluctuations in the phonon resonator!

The Hamiltonian which describes the interaction between the qubit and mechanical modes in the regime where the detuning (\Delta = \omega_q - \omega_m is the difference in resonant frequency between the qubit and mechanical mode) is much larger than the coupling rate, g \ll |\Delta| can be approximated as:

\hat{H}_{dispersive}/\hbar = \omega_m \hat{a}^{\dagger}\hat{a} + \frac{1}{2}\left(\omega_q + \chi \hat{a}^{\dagger}\hat{a}\right)\hat{\sigma}_z

Where here the dispersive shift \chi \simeq 2g^2 / \Delta has been introduced. Writing the system Hamiltonian down in this from is typically called the dispersive regime, and this allows us to see that the effective qubit frequency \omega_q' = \omega_q +\chi\hat{a}^{\dagger}\hat{a} is now shifted by the number of excitations in the mechanical resonator! Oftentimes, in order to investigate the dispersive interaction between a qubit and a resonator, the authors will measure the qubit’s absorption spectrum, which is the frequency at which the qubit absorbs energy and is driven from its ground to excited state. This is also often called the qubit spectrum. If the qubit and resonator both have extremely low loss (both loss rates must be much less than \chi), the system is said to be in the strong dispersive regime, and the qubit spectrum is split into many peaks where the transition energy between the ground and excited states is shifted by \chi for each phonon.

By changing the amplitude of the signal, the authors are able to vary the mean phonon number injected. This is measured by observing the qubit spectra split into multiple peaks each representing different phonon numbers, with each peak. Then, by comparing the relative height of each peak, the authors are able to determine the corresponding phononic coherent state. See Fig. 3 for the resulting measurement. Additionally, the authors see that there is a linear relationship between the mean phonon number and the strength of the signal generating the phononic coherent state, as expected.

Figure 3
(a) Measurement of the qubit absorption spectrum as a function of drive amplitude for the signal displacing the HBAR’s coherent state. At low drive amplitudes, the qubit spectra has a single peak, while at high drive powers, the qubit spectra splits into many peaks, each one indicating a different phononic Fock state. (b) Extraction of the probability of each Fock state at different drive amplitudes. The extracted probability distribution in Poissonian in the phonon number, as would be expected for a coherent state. The authors also find that the mean phonon number in the HBAR is linear in drive amplitude (inset).

Parity Measurement of Phonon Number

After investigating the qubit’s response to phonon states in the frequency domain, the authors look to the qubit response in time to learn about how the presence of phonons impacts the qubit. By repeatedly preparing the qubit into its excited state and preforming multiple swap operations between the qubit and the HBAR device, it is possible to prepare higher number Fock states (by quickly adding many excitations into the HBAR device one at a time). This is done by first exciting the qubit, swapping the excitation into the mechanical resonator, and repeating to add more excitations to the HBAR. After preparing the mechanical resonator’s state, the authors put the qubit into a superposition state: |\psi_q\rangle = \frac{1}{\sqrt{2}}\left(|g\rangle + |e\rangle\right). As a function of time, the qubit will accumulate a phase on the component of its wavefunction corresponding to the excited state of: \phi = -n \chi t , where n is the number of excitations in the HBAR device. It’s important to note here that because the HBAR is in a Fock state, there is not a distribution of phonon numbers now as there would be for a coherent state, but rather one single Fock state describes the quantum state of the HBAR! After allowing the qubit state to accumulate phase for some amount of time, the qubit’s state is then rotated with the same phase as the pulse that prepared the original superposition state. This means that if the qubit accumulated no extra phase, it would be repositioned to the excited state (assuming that there are no losses). In reality the probability of measuring the qubit in its excited state will always decay in time, but the presence of phonons in the HBAR device can be measured from the frequency of oscillation from this measurement. The frequency of oscillation can be calculated to be equal to M|\chi_{Ramsey}|/ (2 \pi), where M is the phonon number in the HBAR resonator. Fig. 4 details this measurement as a function of time for several different swap operations. At a time of approximately t = 7\mu s, which corresponds to the time \pi/\chi, the authors are able to tell whether or not the resonator has an odd or even number of phonons based on whether or not the Ramsey decay is at a maximum or minimum! At this time, if there are an even number of phonons in the HBAR, the qubit phase has accumulated an even integer multiple of \pi so that qubit superposition states are re projected to the excited state prior to measurement. Similarly, an odd number of phonons in the HBAR results in an odd multiple of \pi phase accumulation so that the qubit is re projected to its ground state prior to measurement. This measurement allows the authors to quickly measure the parity of the phonon resonator in a single shot, rather than measuring the entire qubit spectra, which takes much more time.

Figure 4
(a) Measurement of qubit absorption spectra when the HBAR is prepared in different Fock states. Rather than seeing the spectra split into several different peaks here (as in Fig. 3a), the authors find that the qubit spectra remains largely constant, only the center frequency of the qubit is shifted proportional to the number of phonons in the HBAR resonator. (b) Ramsey measurements of the qubit. Here, the frequency of oscillation of the Ramsey decay allows the authors to extract the phonon number in the HBAR resonator. Importantly, at t\simeq 7 \mu s, the authors are able to determine the parity (even-ness or odd-ness) of the phonon number in the HBAR resonator. (c) Comparison of measured parity based on two measurement schemes described by spectroscopic measurements (panel (a)), as well as measured parity by Ramsey measurements (panel (b)), the authors find very good agreement in measured parity between the two measurement schemes!

Conclusion

In this experiment, the authors demonstrate a hybrid quantum acoustics experiment which operates in the strong dispersive regime, where the dispersive interaction between a superconducting qubit is much stronger than either the loss of the qubit or the loss of the HBAR resonator. By entering this special regime of circuit quantum acoustodynamics (cQAD), the authors are able to perform experiments which allow them to probe the quantum properties of high frequency sound. By using special experimental techniques, the authors are able to create non-classical phonon states in the HBAR resonator (Fock states) and determine phonon parity based on two separate measurement schemes.

References:

[1] U. von Lupke, Y. Yang, M. Bild, L. Michaud, M. Fadel, and Y. Chu, Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics, Nature Physics 10.1038/s41567-022-01591-2 (2022)

Many thanks to Akash Dixit for his many helpful comments and suggestions in the writing of this summary!

Could Metastable States Be the Answer?

Title: omg blueprint for trapped ion quantum computing with metastable states

Authors: D. T. C. Allcock, W. C. Campbell, J. Chiaverini, I. L. Chuang, E. R. Hudson, I. D. Moore, A. Ransford, C. Roman, J. M. Sage, and D. J. Wineland

First Author’s Institution: University of Oregon

Status: Published in Applied Physics Letters

Background Info

This section is intended to be a (very) brief overview of atomic ion qubits for the newly initiated. If you would like to skip ahead to the new stuff from the journal article, click here.

When looking for candidates for quantum bits (qubits), you want a quantum system that has at least two states whose separation is unique (so that you can convert from one state to the other without risking converting to a different third state). Atomic ions are natural choices for qubits since atoms have energy levels whose separations are naturally unequal to one another (see Figure 1 for an example of an ion qubit). Atomic ions also have some of the longest coherence times of any type of qubit, meaning they remain in the state you put them in for a long time (typically anywhere from on the order of seconds to years depending on the atomic states being used).

FIG 1 Simplified diagram of 40Ca+ energy level structure. The 42S1/2 and 32D5/2 states form a two-level quantum system that can be used as a qubit. This qubit is addressable via a 729 nm laser, and has a lifetime of about 1.2 s. A 397 nm laser is used to Doppler cool the ions via the 42S1/2 to 42P1/2 transition, and an 866 nm laser is used to repump electrons out of the metastable 32D3/2 state (otherwise, they can become trapped there).

Furthermore, ions can be trapped, shuttled, addressed, and otherwise manipulated with electromagnetic fields and waves. When trapped and cooled together, a group of ions form a crystal-like structure referred to as a Coulomb crystal (so-called because the ions are held in this crystal-like structure by the Coulomb force of repulsion between each other and the electric and magnetic fields used to trap them).

FIG 2 Photographs of Be+ Coulomb crystals. The left grouping of 6 images is taken from [2], and the right image is taken from [3].

Despite all of these advantages, using atomic ions as qubits in a quantum computer poses some challenges which must be overcome. They are error prone due to interactions with stray photons, background gases in the vacuum system, or stray electromagnetic fields from outside interference. Furthermore, care must be taken to avoid crosstalk, an unwanted affect where light being used to perform an operation on one qubit scatters and affects a nearby qubit. It is also difficult to scale up to larger numbers of qubits.

In order to build a quantum computer with atomic ion qubits, the authors list four key needs:

  1. The ability to perform an operation on a qubit without affecting other nearby qubits (aka crosstalk)
  2. The ability to read qubits’ states without disturbing nearby qubits
  3. The ability to entangle two different groups of qubits
  4. The ability to quickly re-arrange and/or move ion-qubits within a Coulomb crystal without heating the ions

All of this needs to be accomplished in large arrays of ions while maintaining the same high fidelities that experiments with small numbers of ions have demonstrated.

One approach designed to address the problem of errors due to crosstalk is the dual-species approach. As its name implies, this approach makes use of two different species of atoms that are trapped together. Generally, at least one of the species will be easy to laser cool and can be used to sympathetically cool the other species of ion it is co-trapped with. (As one species is Doppler cooled, the other species which cannot be Doppler cooled will be “sympathetically cooled” due to Coulomb repulsion between it and the laser cooled species.) The two different species of atoms should also be close in mass to enable efficient sympathetic cooling as well as to minimize the difference in response to both applied and stray electric and magnetic fields [4].

By arranging the atomic ions in the trap such that the species of atom alternates every other ion, you can prevent crosstalk between neighboring qubits. This allows for much easier addressing of individual qubits without worrying about accidentally affecting its nearest neighbors.

However, dual-species brings its own challenges, one of which is needing twice as many laser systems to be able to address the two different atomic species. Perhaps the biggest challenge, however, is the difference in mass between the two species. Because the acceleration an ion experiences is proportional to its charge to mass ratio, a difference in mass means that the two species will experience a different acceleration from the same electromagnetic field. This is problematic since ion traps use electromagnetic fields to trap ions. It also makes it difficult to re-arrange/shuttle qubits around within the trap.

This is where the authors’ proposed omg architecture comes in. The omg architecture aims to keep the advantages of the dual species architecture while eliminating the difference in mass (and thus all of the difficulties associated with having two different masses).

omg Architecture

The omg architecture uses two different types of electronic qubits within the same species of atomic ion (nobody said we had to use the exact same two energy levels in every atom as our qubit states, did they?). The authors name this architecture omg after the three types of electronic qubits housed within a single species of atomic ion:

  • o for optical-frequency qubits
  • m for metastable-state qubits
  • g for ground-state qubits

The optical-frequency qubit consists of a ground state and a metastable state whose energy difference corresponds to a visible wavelength of light. These qubits are addressed with lasers.

The metastable-state qubit consists of two metastable states (e.g. hyperfine levels or Zeeman levels) typically in the 2D5/2 or 2F7/2 state. These states must have long lifetimes compared to the length of time that information is stored in them (but don’t need to be as long as ground state qubits). These qubits are addressed with RF magnetic fields and gradients or stimulated Raman transitions.

The ground-state qubit consists of two ground states (e.g. hyperfine levels or Zeeman levels) in the 2S1/2 state. These qubits are addressed with microwaves.

FIG 2 Simplified energy level structure of alkaline earth ions that have hyperfine structure and metastable states. The three types of qubits are shown as colored circles with arrows (o-type is in white, m-type is in red, and g-type is in blue). Figure taken from [1].

By utilizing a species of atomic ion that has all three types of qubits (hereafter referred to as omg ions), you can have dual species functionality without having a difference in mass to contend with. This really is the best of both worlds, since it means having the ability to address individual qubits without interfering with neighboring qubits while retaining the ability to easily trap, re-arrange, and shuttle ions with electromagnetic fields. Several species that the authors give as omg candidates are 43Ca+, 87Sr+, 133Ba+, 135Ba+, 137Ba+, 171Yb+, 173Yb+.

The three key ingredients for quantum computation are state preparation, gate operations, and storage. State preparation depends on the laser cooling mechanisms that are available in that particular species of atomic ion. Gate operations depend on having wavelengths that are “technologically convenient.” By technologically convenient, I mean wavelengths for which it is easy to interface to existing computer hardware (think telecom wavelengths). m-type qubits could be ideal candidates for gate operations given their longer wavelength transitions (in the MHz and Low GHz frequencies). Storage requires qubits with long lifetimes (g-type qubits have the longest lifetimes, but m-type qubits are also sufficiently long-lived for this job). o-type qubits are ideal for state readout because of their visible fluorescence.

Thus, an omg ion houses within a single atomic species everything you need to meet the three architectural requirements of a quantum computer. The authors go on to outline three possible schemes for building a quantum computer using the omg architecture. These three modes are denoted by the notation {state preparation, gate, storage} with the corresponding symbol (o, m, g) for each purpose. In all three modes, o-type qubits are used for the readout of states and g-type qubits are used for sympathetically cooling the ion array. I have summarized the three different modes below:

{m, m, m} Mode

  • Uses metastable-state qubits for all operations
  • Uses g-type ions for laser cooling and o-type ions for state readout of info
  • Main Advantages:
    • Since all operations are performed with m-type qubits, there is no need to convert a qubit from one type to another
    • Laser cooling and g-qubit state preparation can be performed during gate operations on other ions within the crystal
  • Main Disadvantages:
    • Storage is limited by the lifetime of the metastable state
    • Because m-type qubits are used for both storage and gate operations, this mode requires focused laser beams (or physically shuttling the ions away from neighbors) to avoid disturbing the storage qubits while performing gate operations

{g, m, g} Mode

  • Uses m-type qubits for gate operations and g-type qubits for state preparation and storage
  • Uses g-type ions for laser cooling and o-type ions for state readout of info
  • Main Advantages:
    • The long lifetimes of ground-state qubits enable excellent storage of information
    • The storage qubits are protected from laser light used to perform gate operations
  • Main Disadvantages:
    • Requires the ability to convert between m-type and g-type qubits without loss of information
    • This mode is likely the most difficult for readout of information as well as sympathetic cooling while an algorithm is being run (since doing so requires all g-qubits involved in the algorithm to be converted to m-qubits to protect them during these operations)

{m, g, m} Mode

  • Uses m-type qubits for state preparation and storage and g-type qubits for gate operations
  • Uses g-type ions for laser cooling and o-type ions for state readout of info
  • Main Advantages:
    • Protects the storage qubits from laser light used to perform gate operations
    • Only the qubits involved in an active process (gate operations, cooling, or state readout) need to be converted (storage qubits are protected from such operations)
  • Main Disadvantages:
    • Storage is limited by the lifetime of the metastable state
    • Requires the ability to convert between m-type and g-type qubits without loss of information
FIG 3 Pictographic representation of the three modes discussed. Each circle represents a qubit of the type that corresponds to the letter. Large arrows represent laser beams. Each row is a single mode. The first three columns depict state preparation, gate operations, and storage, respectively. The fourth column (labeled type cast) represents the conversion of a qubit to another type. The fifth column (labeled read enable) represents the conversion of a qubit to an o-type so it can be excited by a laser and fluoresce (for readout of state). Figure taken from [1].

Tl;dr

The omg architecture is an architecture proposed by the authors that would utilize multiple types of qubits within the same type of atomic ion. Doing so enables various tasks to be performed on qubits more easily without scattered light or cross talk between neighboring qubits causing decoherence during the process. It also avoids the issues arising from mass-mismatch that the dual-species architecture must grapple with.

References

[1] Allcock, D. T., et al. “Omg Blueprint for Trapped Ion Quantum Computing with Metastable States.” Applied Physics Letters, vol. 119, no. 21, 2021, p. 214002., https://doi.org/10.1063/5.0069544.

[2] Heinrich, Johannes, et al. “A Be+ Ion Trap for H2+ Spectroscopy.” Thèse de doctorat: Physique: Sorbonne université.

[3] Thompson, Richard C. “Ion Coulomb Crystals.” Contemporary Physics, 2015, pp. 1–17., https://doi.org/10.1080/00107514.2014.989715.

[4] Home, Jonathon P. “Quantum Science and Metrology with Mixed-Species Ion Chains.” Advances In Atomic, Molecular, and Optical Physics, 2013, pp. 231–277., https://doi.org/10.1016/b978-0-12-408090-4.00004-9.

Quantum Entanglement of Macroscopic Mechanical Objects

Title: Direct observation of deterministic macroscopic entanglement

Authors: Shlomi Kotler, Gabriel A. Peterson, Ezad Shojaee, Florent Lecocq, Katarina Cicak, Alex Kwiatkowski, Shawn Geller, Scott Glancy, Emanuel Knill, Raymond W. Simmonds, José Aumentado, John D. Teufel

Institution: National Institute of Standards and Technology (NIST)

Manuscript: Published in Science, open access on arXiv

Quantum entanglement is one of the most bizarre and powerful phenomena in quantum mechanics. Over the years, physicists have created and observed entanglement of a wide range of systems, from the spin states of atoms to the polarization of photons. Most experiments to date, however, have studied quantum entanglement in the smallest of microscopic systems, the regime where quantum mechanics is most easily observed. It is much more difficult to observe quantum entanglement in macroscopic objects, where environmental disturbances seemingly destroy their quantum behavior. A recent paper from researchers at NIST reports observation of such entanglement: namely, the position and momentum of two physically separate mechanical oscillators. Entanglement of mechanical oscillators isn’t exactly new: position entanglement was first observed in the vibrational states of two atomic ions back in 2009. But this entanglement explores an entirely different regime, where the vibrations are not just of singular atoms, but the collective motion of billions of atoms in a macroscopic object.

SEM image of the two aluminum drums, and the complete LC circuit.

The study analyzes the mechanical oscillations of two drum-like membranes. The drums are patterned out of aluminum on a sapphire chip, are roughly 20 microns in length, and weigh roughly 70 picograms. While the drums are tiny to us- each drum is smaller than the width of a human hair- they contain several billion atoms, large enough to be considered ‘macroscopic’ for a quantum system. The membranes are designed to oscillate at 11MHz and 16MHz frequencies, respectively (they are purposefully designed to oscillate at different frequencies, so that each membrane can be identified). There is a metal base below each drumhead, so that the drumhead and the metal base act like a parallel-plate capacitor. When the drum vibrates, the distance between the plates changes, thereby changing the capacitance of the drum. By wiring up the drum to a large spiral inductor, we form an LC circuit, which oscillates at a resonant frequency given by 1/\sqrt{LC} . The LC circuit in this work is designed to oscillate at 6GHz. As the drum vibrates, the changing capacitance of the drum changes the resonant frequency of the LC circuit. By probing the circuit frequency, we gain information about the motion of the drum. The device is placed inside a dilution refrigerator which cools the device down to temperatures below 10mK. At this temperature, aluminum becomes a superconductor and both the circuit and drums have very few energy loss mechanisms. Once energy enters either one of the cavities, it can remain for milliseconds. This gives the cavities narrow resonances in frequency space, making them well-suited to behave quantum mechanically.

Quantum Electromechanics- The Basics

We can measure the quantum properties of this electromechanical system by noting that both the microwave circuit and the mechanical drums are harmonic oscillators, which we can treat quantum mechanically with creation and annihilation operators: \hat{a} for the LC circuit, and \hat{b}_1 and \hat{b}_2 for the two drums. Then a quantum measurement of drum i ‘s position is given by

\hat{x}_i = x_{0, i}(\hat{b}^{\dagger}_i + \hat{b}_i) ,

and momentum by

\hat{p}_i = ip_{0, i}(\hat{b}^{\dagger}_i - \hat{b}_i) .

Quantum mechanically, the energies of these two oscillators are quantized. The average energy of the circuit is given by \hbar\omega_c (n_c + 1/2) , where n_c is the average number of microwave-frequency photons inside the circuit. The drum energies are given by \hbar\omega_m (n_{m, i} + 1/2) , where n_{m, i} is the average number of phonons in drum i . Basic statistical mechanics tells us that the circuit and drums are naturally in a thermal state, with average photon/phonon numbers given by the Bose-Einstein occupation factor:

n(\omega) = \frac{1}{e^{\hbar\omega/kT} - 1}

At 10mK, the 6GHz circuit is naturally in the ground state, with n_c \approx 0 photons. The lower-frequency drums are more occupied with n_m \approx 20 phonons in each drum. With careful engineering, the authors can control and measure the two-drum system with single-phonon level precision.

Schematic drawing of the three peaks in frequency space: center frequency, red sideband at f_c - f_m, and blue sideband at f_c + f_m.

Let’s take a closer look at the circuit frequency measurement. As the vibrations of the drums modulate the LC circuit frequency, this shows up in frequency space as sidebands, two peaks which are separated from the circuit frequency f_c by exactly the mechanical frequency f_m of the oscillators (see image above). We call the peak at (f_c - f_m) the red sideband, and the peak at (f_c + f_m) the blue sideband. By sending a sequence of microwave pulses at these sideband frequencies, the authors are able to initialize, entangle, and readout the motional states of the two drums.

To see how this works, let’s focus on a single drumhead \hat{b} coupled to an LC circuit \hat{a} . If a red sideband pulse is applied, the interaction Hamiltonian is given by

\hbar g(\hat{a}^{\dagger}\hat{b} + \hat{a}\hat{b}^{\dagger}) .

(See derivation here. It’s straightforward but too long for this article.)

This acts like a phonon-photon swap operation, where a phonon of energy in the drum is converted into a photon of energy in the LC circuit at rate g and vice versa. For example, when applied to the state |1_m, 0_c \rangle (1 phonon, 0 photons), for a time t = \pi/2g , the resulting evolution gives |0_m, 1_c\rangle . If a blue sideband pulse is applied, the interaction is very different :


\hbar g(\hat{a}^{\dagger}\hat{b}^{\dagger} + \hat{a}\hat{b})

(See derivation here. It’s straightforward but too long for this article.)

This interaction serves to generate an entangled photon-phonon pair. For example, when applied to the state |0_m, 0_c \rangle , the resulting state takes the form (no normalization for simplicity) |0_m, 0_c\rangle + \sqrt{p} |1_m, 1_c \rangle + \mathcal{O}(p) , where p is the probability of generating an entangled pair.

Experimental Sequence

The experimental sequence in this work is in three steps: state preparation, where the drums are actively cooled to their motional ground state, entanglement, in which the motional state of the drums are entangled, and readout, in which the position and momentum fluctuations of the drums are measured. This sequence is repeated a large number of times, and the study looks at the correlations between x_1 , x_2 , p_1 , and p_2 .

State Preparation

Recall that at 10mK, the ~10MHz drums have an average of n_m \approx 20 phonons of vibrational energy. The drums should ideally be in their motional ground state (n_m = 0) to maximize the fidelity of the entanglement protocol. A red sideband pulse can be used to cool the drums to their quantum ground state. Due to the swap interaction described above, a phonon of energy in the drum is converted into a photon of energy in the LC circuit. If the decay rate of the circuit is fast enough (which it is in this experiment), the converted photon is emitted out of the circuit before it can be swapped back into the drum. If the pulse is applied for a long enough time, phonons are continually removed from the drum until there are nearly 0. This ground-state cooling technique was first demonstrated in macroscopic objects 10 years ago, using microwave radiation and even optical radiation, and has worked remarkably well since.

Entanglement

To perform entanglement, the authors implement two pulses in parallel: a blue sideband pulse on drum 1, and a red sideband pulse on drum 2. The blue sideband pulse entangles a phonon in drum 1 and a photon in the LC circuit, then the red sideband converts the photon into a phonon in drum 2. The net effect is to generate a phonon in each of drum 1 and drum 2 which are entangled.

Readout

A blue sideband pulse can be used to measure the position and momentum of the drums (a red sideband pulse can be used for this too, but this work uses a blue sideband scheme). By sending a blue sideband pulse and looking at the reflected signal, the position and momentum of the oscillator can be indirectly probed.

It can be shown that the position and momentum of the drums are imprinted in the two quadratures of the reflected signal. For those unfamiliar, the quadratures of an oscillating signal s(t) refer to the cosine and sine components of the signal:

s(t) = I(t) \cos(\omega t) + Q(t)\sin(\omega t)

I(t) represents one quadrature, Q(t)  represents the other. In a blue sideband measurement, I(t) \propto \hat{b}^\dagger + \hat{b}  is proportional to position fluctuations and Q(t) \propto \hat{b}^\dagger - \hat{b}  is proportional to momentum fluctuations. The authors send in a blue sideband pulse and look at the reflected I and Q signals to extract the position and momentum of each drum. These I and Q measurements can be done relatively easily using standard microwave electronics.

Pulse sequence for entangling drums 1 and 2. Red indicates a red sideband pulse at frequency f_c - f_m, whereas blue indicates a blue sideband pulse at frequency f_c + f_m.

The full pulse sequence is shown above: this implements ground state preparation, entanglement, and readout of the two-drum mechanical state. The authors perform this pulse sequence a large number of times and record the values of {x_1, x_2, p_1, p_2}  , and plot the results. To show how the position and momentum of the drums are correlated, the authors plot each data point in phase space where the (x, y) axes represent different combinations of {x_1, x_2, p_1, p_2}  . The authors do this for two different cases: no entangling pulse, and with entangling pulse, and examine the differences with each case.

Results

Position/momentum data for the ground state with no entangling pulse applied.

As expected, the position and momentum of the two drums showed no significant correlations for the data with no entangling pulse. The circular shape of the data in phase space indicates the fluctuations are randomly distributed and uncorrelated. From the magnitude of the fluctuations, the authors can also extract the average energy of the drums at n_{m, 1} = 0.79  and n_{m,2} = 0.6  phonons respectively, which indicates that the ground-state cooling is pretty successful.

Position/momentum data after an entangling pulse is applied.

The entangling pulse data tells a different story. The positions x_1  and x_2  are clearly correlated, while momenta p_1  and p_2  are clearly anti-correlated. This is a remarkable result as the two drums are physically separated and yet are moving in a coordinated way.

While the position/momentum data is impressive, these correlations could still be classical in nature. To verify that the correlated motion is a result of entanglement, the authors use the covariance matrix C_{ij}  , with elements defined by

C_{ij} = \langle \Delta s_i \Delta s_j \rangle = \langle(s_i - \langle s_i \rangle)(s_j - \langle s_j\rangle)\rangle

where s_i  can represent x_1  , p_1  , x_2  or p_2  . For example, C_{x_1, x_2} = \langle \Delta x_1 \Delta x_2 \rangle   . If two variables, say x_1  and x_2  are not correlated with one another, then C_{x_1, x_2} = 0  . If they are correlated, then C_{x_1, x_2}  will have some nonzero value.

According to the Simon-Duan criterion for entanglement, if the smallest eigenvalue \nu  of the partial transpose of the covariance matrix satisfies \nu < 1/2  , then the two-drum mechanical state is entangled. Covariance matrices for the two cases are shown below:

Covariance matrix for position/momentum data of the ground state and entangled state.

In the case with no entangling pulse, the position/momentum measurements for drums 1 and 2 were not correlated. Therefore the off-diagonal elements are nearly zero, and the covariance matrix is purely diagonal. After applying the entangling pulse, the covariance matrix looks quite different. The correlated nature of x_1/x_2  and p_1/p_2  creates off-diagonal elements in the covariance matrix. The authors find that by varying the entangling pulse time, the value of \nu  decreases below 1/2  , verifying quantum entanglement for long enough entangling pulses. At the longest entangling time measured, \nu  is an order of magnitude below the entanglement threshold.

Pesky Pesky Noise

What makes observing quantum properties in macroscopic objects so difficult in the first place is the presence of environmental noise which corrupts the state of a macroscopic object. Ideally, one would like the measurements {x_1, x_2, p_1, p_2}   to reflect only position/momentum fluctuations, without any additional unwanted fluctuations. In practice, however, the I and Q measurements also contain vacuum noise, so that the position/momentum measurements take the form


x_i = \sqrt{\eta_i}X_i + \sqrt{1 - \eta_i}\xi_i ,


p_i = \sqrt{\eta_i}P_i + \sqrt{1 - \eta_i}\xi_i

where X_i  , P_i  are the true values of position/momentum, \xi_i  is the vacuum noise of each I/Q measurement (basically just a random variable with variance 1/2), and \eta_i  is the measurement efficiency. If the value of \eta  is small enough, then the measurements of {x_1, x_2, p_1, p_2}  become corrupted with noise, and true entanglement becomes hard to verify. The measured value of \nu  differs from the true value by


\nu_{\mathrm{meas}} = \eta \nu + (1 - \eta)\cdot 1/2

where \eta = \sqrt{\eta_1\eta_2}  is the geometric mean of the efficiencies. The smaller the value of \eta  , the closer \nu_{\mathrm{meas}}  is to 1/2  and the harder it is to verify the \nu<1/2  threshold. The authors show the calculated value of \nu   as a function of entangling pulse time:

Measured values of \nu (left) and extracted true values of \nu (right) vs. entangling pulse duration. \nu < 0.5 indicates the threshold for quantum entanglement.

The authors find that even without calibrating out the noise in their measurements, they obtain values of \nu_{\mathrm{meas}}  that are >40% below the entanglement threshold for the longest pulse time in this work. This is a remarkable result: the authors are able to observe macroscopic entanglement directly from the measured data, even in the presence of noise!

To summarize, this work demonstrates the ground-state cooling, entanglement, and measurement of the quantum motional states of two mechanical oscillators. The authors observe quantum behavior of the collective motion of billions of atoms, further confirming that even large objects can be described with a quantum-mechanical wavefunction. The results of this work pave the way for many unanswered questions: how large can a system get and still behave quantum-mechanically? Will gravity destroy quantum states at some intermediate size? Can we use entanglement in large objects as a resource for quantum computing? This work is an exciting step in the long road ahead towards answering these questions.

Quantum Communication with itinerant surface acoustic wave phonons

Authors: E. Dumur, K.J. Satzinger, G.A. Peairs, M-H. Chou, A. Bienfait, H.-S. Chang, C.R. Conner, J. Grebel, R.G. Povey, Y.P. Zhong, A.N. Cleland

First Author’s Primary Affiliation: Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA

Manuscript: Published in NPJ Quantum Information

Introduction

Superconducting qubits are among the state of the art architectures in the development of quantum processors. In order to successfully build a functioning quantum computer, it is essential to be able to transfer information about quantum states amongst multiple qubits while maintaining the “quantum” properties of these states. Typically, one would couple two or more superconducting qubits via a transmission line where the signal travels at the speed of light. Importantly, because superconducting qubits operate in the GHz frequency range, the wavelength of light with this frequency is large relative to the size of the qubit, which is approximately (1mm)^{2}. The wavelength of light at these frequencies is given by \lambda = 6 \textrm{cm} for a signal with frequency 5 GHz. This means that the structures which couple our qubits together must be (of order) this size and are much larger than the qubits themselves! For a simple case, like coupling two qubits together this does not present any challenges[2], but as superconducting processors become larger in quantum volume (and therefore spatial size), it becomes more and more important to think critically about how we can create a smaller spatial structure with which to couple two or more qubits.

Surface acoustic wave (SAW) devices utilize the “slow” speed of surface sound waves in crystals (typically about 4000 m/s) in order to create high frequency resonant structures with a small spatial footprint. For example, in order to create a structure with a resonant frequency of 4 GHz, one would need a wavelength of \lambda = (4000 \textrm{m/s})/(4\textrm{GHz}) = 1 \mu \textrm{m}, which is approximately 5 orders of magnitude smaller than the wavelength of a signal which travels at the speed of light! SAW devices are created by fabricating metal strips called interdigitated transducers (IDT for short) on a piezoelectric substrate. In a piezoelectric material, the electric fields in the material induce mechanical strain and vice versa so that an AC voltage applied across the metal strips launches a strain wave propagating across the substrate at the same frequency (see Fig. 1 for a schematic). Here, the wavelength of the surface wave is defined by the periodicity of the metal finger structure, so we are able to create high frequency resonators using standard nano-fabrication techniques.

FIg. 1
Schematic of an IDT structure (red) which is driven by an AC voltage and launches surface acoustic waves (green)

In addition to using IDT structures to launch SAWs, we can also add periodic metallized structures on either side of the IDT launcher which act to reflect phonons emitted from the IDT (called mirrors). See Fig. 2 (adapted from [3]) for a schematic which details both the IDT as well as the mirror structures.

Figure 2 Schematic of a SAW resonator which contains both the IDT which launches SAWs (center), and the mirrors on either side of the IDT which form an acoustic cavity.

Together, the IDT and mirror structure create an acoustic cavity for phonons, where the spatial size is much smaller than a cavity for microwave photons at the same frequency!
GHz-frequency SAW resonators have been coupled to superconducting qubits before, sometimes in a “flip-chip” configuration[4]. This allows the experimentalist to fabricate a standard superconducting qubit on one substrate (typically on silicon or sapphire) and the SAW resonator on a separate piezoelectric substrate (LiNbO_3 is very common for these types of experiments). The chip containing the SAW resonator is then fastened on top of the substrate where the qubit is fabricated. Using an experimental setup like this also allows one to tune the coupling between the qubit and SAW via on-chip inductors, which can allow us to study each system independent from one another. By coupling the qubit to a SAW device, we can transfer excitations from the qubit to the SAW (and vice versa). For example, one can often write the interaction between the SAW and the qubit to be:

\hat{H}_{int} = \hbar g\left(\hat{\sigma}_{+} \hat{m} + \hat{\sigma}_{-} \hat{m}^{\dagger} \right)

Here, {\sigma}_{\pm} are the creation and destruction operators for excitations in the qubit, and \hat{m} and \hat{m}^{\dagger} are bosonic operators for the phonon modes in the SAW. If we prepare the qubit in the excited state and have no phonons in the SAW resonator, then after a time \pi/g, the excitation will be transferred to the SAW! As an equation:

|e,0\rangle \rightarrow |g,1\rangle

Here the quantum state is written as a product of both the qubit state and the state of the SAW, where |e(g)\rangle is the excited (ground) state of the qubit and the number in the state vector denotes the number of phonons excited in the SAW device.

Experimental Details and Preliminary Results

In this set of experiments, the primary goal is to use two SAW resonators to mediate the quantum state transfer between two qubits which are separated spatially by using a phonon based communication channel. Here, the previously mentioned flip-chip configuration will be used. On the sapphire substrate, the two qubits are fabricated. Each qubit contains a SQUID loop, which means that the resonant freuquency of the qubit is tunable via an external magnetic flux threading the SQUID loop. Additionally extra control lines are added near each qubit which can manipulate the quantum state of the qubit. The control lines which manipulate the individual qubit states are known as XY lines, while control lines which provide local magnetic flux control to each qubit are known as Z lines. On the “top” LiNbO_3 chip, two IDT devices with the same resonant frequency (near 4GHz) are fabricated. These two IDT are separated by 2mm, which means it takes a phonon approximately 500ns to traverse from one IDT to the other. An acoustic mirror structure is added on one side of each IDT so that phonons are preferentially launched in one direction at certain frequencies (this specific design is called a unidirectional transducer, or UDT for short). This allows for constructive interference of phonons at some frequencies, which we will call the UDT regime. At all other frequencies phonons will not constructively interfere, and we will call this the IDT regime. Two tunable couplers are added on each chip so that the interaction strength between each qubit and each SAW resonator can be independently tuned. See Fig. 3 for a full schematic of the composite device.

Figure 3
(a) Schematic of the composite device and a description of each piece. (b) Effective circuit diagram which describes the circuitry necessary to manipulate the qubit states as well as couple the qubits to the acoustic resonators. For each qubit Q_i, the control lines Q_{xyi} control the state of that qubit and the control line labeled Q_{zi} controls the local magnetic field which tunes the resonant frequency of that qubit. The coupler labeled G_i controls the coupling to the acoustic channel on a separate chip, and the control line G_{zi} allows for the control of the coupling strength via an external magnetic flux.

The first experiment that can be done with this device is the independent characterization of a single qubit, for example qubit Q1, when it is weakly coupled to the phononic quantum channel. This characterization allows the authors to verify that the qubits have long enough coherence to take full advantage of the communication channel. This means that we need the qubit to maintain its state much longer than the phonon travel time of 500ns, otherwise we won’t be able to measure any effects due to phonons traversing the communication channel! In order to measure how long the qubit can maintain its state, a T_1 measurement is performed, where the qubit is put into its excited state via a microwave pulse, and then the probability of the qubit remaining in its excited state as a function of time is measured. The result is shown in Fig. 4.

Figure 4
T_1 data for qubit Q1 across a broad range of qubit frequencies. Interestingly, we see that when the qubit is near-resonant with the SAW device, its lifetime drops dramatically!

At first glance, many striking features of this measurement are apparent. First, over the frequency range of approximately 3.85GHz to 4.15GHz, the qubit does not remain in its excited state for very long. This is because over this frequency range, the SAW resonator has a high conductance, and therefore the qubit excitation is transferred into a phonon. Finally, and perhaps most interestingly, in the range where the qubit excitation is lost to a phonon, the qubit excited state actually increases after roughly 1\mus. This is because the qubit excitation is lost to a phonon, and the phonon travels to the other end of the phonon channel, then it is reflected back to the original SAW where it is in turn converted back to a qubit excitation! A similar, yet weaker feature is also noticeable near 2\mus. Because we can see these features, this is an indication that the qubit coherence is long enough such that we can use the full potential of the phonon communication channel in this device!
After significantly long coherence is verified, the authors attempt a quantum state transfer between the qubits. The experimental protocol is as follows: prepare qubit Q1 into its excited state, then turn on the coupling between qubit Q1 and a SAW resonator. This will allow for a phonon to be launched across the phonon channel. Then approximately 500ns later, the authors turn on the coupling between the other SAW resonator and qubit Q2. This will allow for the transiting phonon to be converted into an excitation in qubit Q2. The results are shown in Fig. 5.

Quantum state transfer in the UDT regime (left) and the IDT regime (right). We see that the state transfer is possible in both regimes, but much more effective in the UDT regime. The pulse sequence in the right panel demonstrates the measurement protocol: a pulse applied to qubit 1 via the control line Q_{xy1} prepares qubit 1 into its excited state. Then the coupling between the acoustic channel and qubit 1 is turned on (represented by K_1). After a set amount of time, the coupler between the acoustic channel and qubit 2 is turned on (represented by K_2), and the state of qubit 2 is measured.

Here we can see that when the SAW is operated in the UDT regime, the probability of Q2 being excited via a phonon is near 68%, while in the IDT regime it is much lower (only about 15%). This is an indication that operating in the UDT regime allows for highly efficient state transfer from one qubit to another mediated by phonons!!

Entanglement

Now that we know we can transfer a quantum state from one qubit to the other using phonons as an intermediate step, a logical next step is to attempt to create a non trivial multi-qubit state, specifically a Bell state! In order to do this experiment, the authors harness the utility of the tunable couplers mentioned previously. If we load an excitation into a qubit and turn on the coupling between the qubit and SAW resonator for a specific amount of time, the qubit excited state probability will decay to approximately 50% (see Fig. 6, approximately 175ns). At this time, there is a 50% chance the qubit has lost its excitation to the emission of a phonon in the communication channel, and we will call this launching “half” a phonon. Of course, we can write the process quantum mechanically:

|e,0,g\rangle \rightarrow \frac{1}{\sqrt{2}}\left(|e,0,g \rangle+ |g,1,g\rangle\right)

Here we have labeled the quantum states as the following |Q1,\gamma,Q2\rangle, where the first index denotes the state of qubit 1, \gamma represents the number of phonons in the acoustic channel, and the final index labels the state of qubit 2. Upon the arrival of the phonon on the other side of the channel, the authors turn coupler 2 on and “catch” the traveling phonon so that the total process is:

|e,0,g\rangle \rightarrow \frac{1}{\sqrt{2}}\left(|e,0,g \rangle+ |g,1,g\rangle\right) \rightarrow \frac{1}{\sqrt{2}}\left(|e,g\rangle + e^{i\phi}|g,e\rangle\right)\otimes|0\rangle

Here, we have introduced a relative phase difference \phi, as well as factored out the index which denotes the phonon number. Because we can factor out the phonon number here, we can write the two qubit wavefunction after this process as |\psi\rangle = \frac{1}{\sqrt{2}}\left(|e,g\rangle + e^{i\phi}|g,e\rangle\right), which we recognize to be a Bell state, which is entangled! Results from this experimental protocol are shown in Fig. 6a. Additionally, a reconstruction of the two qubit density matrix allows the authors to verify that the state they have prepared is actually a Bell state! See Fig. 6b for a comparison with theory.

Figure 6
(a) Experimental results for the generation of the Bell state. We see that we have approximately 50% chance of measuring each qubit in its excited state. (b) A reconstruction of the two qubit density matrix. Here the red boxes represent the expectation for a perfect Bell state, and the dashed boxes are simulation results which take into account all of the losses in the system.

Phonon-Qubit Dispersive Interaction

The final set of experiments performed with this remarkable device uses phonons as a probe of the state of one of the qubits. For example, the phase change of a phonon will be different if it interacts with a qubit in its excited state rather than its ground state. In order to test this, again the authors launch half a phonon using qubit Q1. When this phonon is traveling, the resonant frequency of qubit Q1 is changed so that the quantum state of Q1 is changed. When the phonon reaches qubit Q2, the coupler is turned on for a fixed amount of time (200 ns), and the phonon and qubit are allowed to interact. The phonon then reflects back to qubit Q1 and the coupler is turned back on so that the excitation is transferred back to Q1. If the phase of the qubit and the phase of the phonon interfere constructively, the qubit will return to its excited state. However, if they interfere destructively, the qubit will emit its remaining energy and relax to its ground state. Therefore, a measurement of the excited state probability of Q1 will tell us about the phase interference between the phonon and Q1! As we sweep the relative phase of Q1, we should expect to see oscillations in the excited state probability of Q1, where the peaks are constructive interference conditions and the valleys are destructive interference conditions. The relevant pulse sequences are shown in the right panel of Fig. 7.

Figure 7
A measurement of qubit Q1’s excited state probability as a function of its induced phase. There is a discrete phase change (salmon) when the qubit Q2 is prepared in its excited state prior to the measurement.

The experimental process can then be repeated, with the only difference being we have first excited qubit Q2 into its excited state, which means that the phonon should pick up an additional phase shift! This is read out as a discrete phase shift in the left panel of Fig. 7 (the salmon dots are shifted in phase relative to the blue dots by \Delta\phi = 0.40\pi). Here, we say that Q1 probes the state of Q2 via phonons.

Finally, the authors swap the roles of the two qubits and perform one final measurement. They prepare qubit Q2 in a superposition of its ground and excited states, with some variable phase \theta. As an equation: |\psi\rangle = \frac{1}{\sqrt{2}}\left(|g\rangle + e^{i\theta}|e\rangle\right). Experimentally, the phase \theta is set by the phase of a microwave pulse. Once the state is prepared, they wait a fixed amount of time and apply another pulse which rotates the state by \pi/2 radians about the x-axis of the Bloch sphere and measure the state of qubit Q2. As we sweep the phase of the first pulse, we should expect an oscillation in the excited state probability of qubit Q2. As a contrast, they repeat the measurement where the only change is they have first excited qubit Q1 and turned on the relevant couplers. If a phonon is released via qubit Q1, this will again manifest as a phase change relative to the first measurement. The relevant pulse sequence and results are shown in Fig. 8.

Figure 8
A measurement of qubit Q2’s excited state probability as a function of its induced phase via a microwave drive. There is a discrete phase change (salmon) when the qubit Q2 is prepared in its excited state prior to the measurement.

Again, there is a discrete phase shift in the excited state probability of qubit Q2, this time of \Delta \theta = 0.95\pi. This means that they can use the phonon channel to perform phase sensitive measurements of “arbitrary” quantum systems (where of course here that system is another qubit)!

Conclusion

In conclusion, this remarkable set of experiments shows that it is possible to use a phonon-based communication channel to not only transfer a quantum state from one qubit to another, but it is also possible to perform more complex operations, such as preparing a two qubit Bell state! Finally, we can harness the power of traveling phonons to probe the characteristics of other quantum systems and learn about them by measuring a separate qubit!

References

[1] E. Dumur et al, npj Quantum Information 7, 1734 (2021)

[2] J. Majer et. a, Nature 449, 443–447 (2007)

[3] T. Aref et. al, Quantum acoustics with surface acoustic waves, in Super-
conducting Devices in Quantum Optics, edited by R. H. Hadfield and G. Johansson (Springer International Publishing, Cham, 2016) pp. 217–244.

[4] K. J. Satzinger et. al, Nature 563, 7733 (2018)

Many thanks to Piero Chiappina for his helpful comments, edits, and suggestions!