Speeding up Control-Z gates on a fluxonium quantum computer

Title: Fast logic with slow qubits: microwave-activated controlled-Z gate on low-frequency fluxoniums

Authors: Quentin Ficheux, Long B. Nguyen, Aaron Somoroff, Haonan Xiong, Konstantin N. Nesterov, Maxim G. Vavilov, and Vladimir E. Manucharyan

First Authors’ Institution: Department of Physics, Joint Quantum Institute, and Center for Nanophysics and Advanced Materials, University of Maryland

Status: Preprint: https://arxiv.org/abs/2011.02634

Introduction
We exist in the era of noisy intermediate-scale quantum (NISQ) processors [1], currently available in the form of two 53-qubit processors made by IBM and Google. These are very promising for simulating many-body quantum physics, as was recently demonstrated when Google’s Sycamore processor claimed a “quantum advantage” [2]. NISQ processors, however, are still limited in processing power due to their small size and the presence of noise in quantum gates.

The commonality in current NISQ processors is their qubit implementation: the transmon. The transmon is composed of a capacitor in series with a Josephson junction (Fig 1a), effectively a weakly-anharmonic electromagnetic oscillator (Fig 1b). First demonstrated in 2007 [3], it has been widely adopted in many-qubit processors as it is a very simple design to implement. The transmon’s weak anharmonicity, however, is its limiting factor for current performance and further scaling.

**Figure 1** (a) Transmon circuit schematic and (b) potential structure with overlaid energy levels [4], as compared to (c) Fluxonium circuit schematic and (d) potential structure with overlaid energy levels [4]. (e) Frequency vs. flux bias for a two-fluxonium system [this work], where the minimum in 00 – 10 transition corresponds to the “sweet spot” in flux bias.

There are many promising alternatives to the transmon, Fluxonium being a favorite because of its incredible high anharmonicity and subsequent long coherence times (with observed $T_{2}$ up to 500 $\mu$ s [5, 6]). Fluxonium has similar elements to the transmon but is additionally shunted with a large inductor (Fig 1c) attributing to its highly anharmonic spectrum (Fig 1d) and allowing it to be insensitive to offset charges [4]. Further, you can tune its resonant frequency to the so-called “sweet spot” in flux bias where it is first-order insensitive to flux noise (Fig 1e). Such a coherent and noise insensitive qubit would be ideal for scaling up quantum processors, right? So, why doesn’t a fluxonium quantum computer exist yet? It turns out, this noise insensitivity is exactly the problem. Let me explain!

Let us first consider the simplest form of circuit-circuit coupling: mutual capacitance (Fig 2a). With two fluxonium circuits, the coupling term is proportional to $n_{a}n_{b}$ , where $n_a$ and $n_b$ are the amount of charge across the Josephson junctions in each circuit. The capacitive coupling produces little effect on the computational states $|00\rangle, |01\rangle, |10\rangle, |11\rangle$ , since transition matrix elements of $n_a$ vanish with the transition frequency. The $|10\rangle$ and $|20\rangle$ states will also remain unaffected due to parity selection rules. The states that will be affected are the higher energy non-computational states $|12\rangle$ and $|21\rangle$ which have higher transition frequencies, meaning the $n_a$ transition matrix elements should be more dominant, causing a significant level repulsion, $\Delta$ (see Fig. 2b). This level repulsion becomes key in connecting the two fluxonium subspaces, inducing an on-demand qubit-qubit interaction.

Figure 2 (a) Image of two fluxonium coupled on-chip by a mutual capacitance; see also a zoomed in image of the large inductive element composed of >100 Josephson junctions [this work]. (b) The lowest energy levels for this two-fluxonium system. Pink arrows describe transitions at frequencies $f_{10-20}$ and $f_{11-21}$ , which are detuned by $\Delta$ . This detuning is due to repulsion between $|12\rangle$ and $|21\rangle$ caused by the $n_{A}n_{B}$ coupling.

Figure 2 (a) Image of two fluxonium coupled on-chip by a mutual capacitance; see also a zoomed in image of the large inductive element composed of >100 Josephson junctions [this work]. (b) The lowest energy levels for this two-fluxonium system. Pink arrows describe transitions at frequencies $f_{10-20}$ and $f_{11-21}$ , which are detuned by $\Delta$ . This detuning is due to repulsion between $|12\rangle$ and $|21\rangle$ caused by the $n_{A}n_{B}$ coupling.

In a previous paper [7], the authors describe how one can use these coupled subspaces to perform a microwave activated control-Z (CZ) gate between two fluxoniums by applying a 2pi-pulse between the $|11\rangle$ and $|21\rangle$ states.

When one applies a CZ gate, if qubit 2 is in the ground (0) state, this transition on qubit 1 will not occur. However, if qubit 2 is in the excited (1) state, this transition on qubit 1 will occur! You can now readout the state of qubit 2 and infer the state of qubit 1! Read more about quantum logic gates here.

The nearby transition $|10\rangle$ to $|20\rangle$ will stay unexcited as long as the gate pulse is much longer than $1/\Delta$ . If the gate pulse is applied over a short duration, one will have unwanted leakage to the nearby transition. Although the prospect of a CZ gate between fluxoniums is attractive, for the device parameters used in this work, $\Delta$ = 22 MHz and a high-fidelity CZ gate would require ~450 ns. For perspective, a transmon-transmon CZ gate is on the order of 10s of ns. We can now see that insensitivity to offset charges makes fluxonium relatively insensitive to capacitive coupling, ie. the repulsion $\Delta$ is very small. This causes gate times between two fluxonium to be very long, making them less attractive for large scale processors.

Is it possible to speed up this gate without significantly decreasing gate fidelity via leakage to the nearby $|20\rangle$ state? This question leads us to the most impressive result of this work: exact leakage cancellation by synchronized Rabi oscillations!

If we apply a constant drive tone $f_{d}$ at the transition frequency $f_{11-21}$ , we will observe Rabi oscillations between the $|10\rangle$ and $|20\rangle$ states (the state vector traces a circle along the Bloch sphere from the bottom of the sphere, to the top, and to the bottom again). If our drive is slightly detuned from the transition frequency, $f_{d} = f_{11-21} - \delta$ , the circle traced by the Bloch vector shifts such that it doesn’t make it all the way to the excited state (a review of Rabi oscillations can be found here). Since the detuning between the $f_{11-21}$ and $f_{10-20}$ transitions, $\Delta$ , is very small, the authors were able to choose a drive frequency $f_{d}$ which is near both transition frequencies (Fig. 3a). Since we are driving near both of these transitions, we see Rabi oscillations in both of these two level systems (Fig. 3b).

Fig. 3 (a) signal vs. drive frequency. Note a strong signal response for transition frequencies $f_{11-21}$ and $f_{10-20}$ , which are separated by $\Delta$ . The Rabi drive tone $f_{d}$ is detuned from $f_{11-21}$ by $\delta$ such that full Rabi oscillations for both transitions are completed in the same amount of time. (b) Cones traced by each state vector on the Bloch sphere due to the applied Rabi drive tone, $f_{d}$ . Note the projection of these paths trace circles in opposite directions. (c) Optimal gate time of ~61 ns was chosen to minimize infidelity, leakage error, and phase error.

What is really interesting is that the detuning $d$ from $f_{11-21}$ can be chosen such that the circles traced by both Rabi oscillations are completed in the same amount of time, $t$ , which is exactly equal to $1/\Delta$ . Now, we are able to perform a 2 $\pi$ rotation on both states simultaneously, but how does this help us perform a selective CZ gate on just $|11\rangle$ to $|21\rangle$ ? The key is to visualize how these two Rabi oscillations are evolving the state vector on the Bloch sphere.

As observed from the center of the Bloch sphere, one will notice that the two oscillating state vectors travel in different directions and define two distinct cones inside the spheres (see Fig. 3b again, noting the projection of the paths). These cones define the solid angles $\Theta_{10} = 2 \pi (1-(\Delta-\delta)/ \Omega )$ and $\Theta_{11} = 2\pi (1+\delta/ \Omega )$ , corresponding to a “geometric phase” accumulation $\Phi_{ij}=\Theta_{ij}/2$ for each system (you can read more about geometric phase here, but essentially it arises from the fact that the state vector traces a closed loop!).

Since these two transitions are now distinguishable by their geometric phase, the authors can then apply a unitary operation U = diag(1, 1, 1, $e^{i\Delta\Phi})$ to the states. This is effectively the same as assigning a phase difference $\Delta\Phi$ between two trajectories to realize a control-Phase operation (again, you can review quantum logic gates here)! Therefore, a CZ gate is obtained when $\Delta\Phi = -(\Theta_{11}-\Theta_{10})/2 = -\pi\Delta/\Omega = \pm\pi$ !

Basically, the $f_{11-21}$ and $f_{10-20}$ transitions are very close in frequency space, but when their Rabi oscillations are synchronized in time, they become distinguishable by their geometric phase accumulation. A CZ gate becomes possible in a time as short as $1/\Delta$ if a control-Phase operation is also applied! In this work, the theory is verified by simulation of the complete system hamiltonian and verified by experiment! The authors state that this procedure can readily be extended to any other phase accumulation, an exciting result that can be further studied in other systems.

Now that this leakage cancellation has been performed, the authors determine the shortest gate time possible with sufficient fidelity by using Optimized Randomized Benchmarking over a variety of pulse parameters. Given their specific device parameters for their two fluxoniums (see their paper for these details), this results in an optimal gate time of ~ 61 ns. This is a huge improvement over the ~ 450 ns required without any leakage cancellation! Further, the ratio of coherence time : gate time (347 $\mu$ s : 61 ns) is unmatched across quantum computing platforms (to the best of the authors’ knowledge).

Final Remarks
Typically, a two-fluxonium system would have very long gate times. While it can have very high coherence, slow gates make it less ideal for large scale quantum processors. However, one can engineer the system using synchronized Rabi oscillations and a control-Phase operation to significantly shorten CZ gate times. By doing this, the authors demonstrated the best ratio of gate speed to coherence time that we know of to-date! Even though these fluxonium are about fifty times slower than transmons (ie. their coherence is about fifty times longer), the two-qubit gate is faster than microwave-activated gates on transmons, with gate error comparable to the lowest reported. Further work can be done by testing this procedure on other phase accumulation processes and in other two-qubit systems.

The only remaining factor preventing the development of large scale fluxonium processors is fabrication. A simple meandering inductor is physically limited by its maximum impedance. Instead, one can either use an array of hundreds of Josephson Junctions (as in this paper, Fig. 2a) or a NbTiN nanowire [8] to create this large inductive element. Current limitations in fabrication make fluxonium much more difficult to create than a transmon; however, we can expect fabrication techniques and equipment to improve in the future, making fluxonium a more viable option for scaling up NISQ processors!

References
[1] Preskill, John, “Quantum Computing in the NISQ era and beyond”, https://arxiv.org/abs/1801.00862

[2] Arute, F., Arya, K., Babbush, R. et al., “Quantum supremacy using a programmable superconducting processor”, https://www.nature.com/articles/s41586-019-1666-5

[3] Koch, J., Yu, T. M. et al, “Charge insensitive qubit design derived from the Cooper pair box”, https://arxiv.org/abs/cond-mat/0703002

[4] Masluk, Nicholas Adam, “Reducing the loss of the fluxonium artificial atom”, https://qulab.eng.yale.edu/documents/theses/Masluk,%20Nicholas%20A.%20-%20Reducing%20the%20losses%20of%20the%20fluxonium%20artificial%20atom%20(Yale,%202012).pdf

[5] Nguyen, L. B. et al., “The high-coherence fluxonium qubit”, https://arxiv.org/abs/1810.11006

[6] Zhang, H. et al., “Universal fast flux control of a coherent, low-frequency qubit”, https://arxiv.org/abs/2002.10653

[7] Nesterov, K. N., Pechenezhskiy, I. V., Wang, C., Manucharyan, V. E., and Vavilov, M. G., “Microwave-activated controlled- Z gate for fixed-frequency fluxonium qubits”, https://arxiv.org/abs/1802.03095

[8] Hazard, T. M. et al., “Nanowire superinductance fluxonium qubit”, https://arxiv.org/abs/1805.00938

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