The first trapped-ion quantum computer proposal

Title: Quantum Computations with Cold Trapped Ions

Authors: Ignacio Cirac, Peter Zoller

Status: Published 1994 in Physical Review Letters

In 1994, theorists Ignacio Cirac and Peter Zoller published a paper that marked the birth of a new field in experimental physics: trapped-ion quantum computing.

The idea that we could use quantum systems to solve some problems more efficiently than classical computers had been around for a while already, but Cirac and Zoller proposed a key component to the physical realization of an actual quantum computer on a trapped-ion system: the two-qubit gate.

Trapped ions were a natural choice for quantum computers because the technology for controlling these systems at the quantum level was already advanced. Laser cooling, a staple technique in atomic physics, was first demonstrated on a cloud of ions, and quantum jumps were first observed in single trapped-ion systems.

So, when buzz about universal quantum computers began, the ion trappers tuned in. They thought they had (or could develop) all of the tools necessary to build the first quantum computer.

There are a few requirements for making a quantum computer, but two of the most fundamental are:

Good qubits with long coherence times relative to the calculation time. This means that:
- If the qubit is in state $|1\rangle$ it will remain so without decaying to state $|0\rangle$ and vice versa. (In the field of quantum computing, the time it takes for this decay to happen is known as “T1 coherence time” or “energy coherence time.”)
- If the system is in a superposition state $|\psi\rangle = a|1\rangle + b|0\rangle$ then the phase relationship between the two terms will remained well defined, i.e. there is no “dephasing” noise. The time for an equal superposition state, $|+\rangle$ , to completely dephase to an orthogonal state, $|-\rangle$ , is known as “T2 coherence time” or “dephasing time.”
A way to implement multi-qubit gates. These are the basic building blocks of any computational algorithm. In classical computing this would be like an AND or an OR gate. The quantum version of these gates are a little more complex, however, since the outcome of these gates is often an entangled state among the qubits involved. But you need just one two-qubit gate combined with single-qubit rotations to build a universal quantum computer.

The first point is easy. We just have to define two states in the ion to be the qubit states $|1\rangle$ and $|0\rangle$ . As long as the upper state is long-lived and the qubit is sufficiently isolated from the environment, trapped-ion qubits can have extremely long coherence times (the record is over 10 minutes! [1]).

But point two wasn’t quite so obvious when people first started considering a trapped-ion quantum computer. You can’t directly couple the electronic levels of two different ions to share their quantum information, so they needed an indirect way to mediate coupling between two qubits. This ended up being the shared motion of the ions in a trap.

Let me explain. An ion confined in a harmonic trap will have its motional energy quantized into harmonic oscillator levels $n \hbar \omega$ . If there are $N$ ions in this trap, then, just like coupled harmonic oscillators, the system is defined by the $3N$ normal modes of motion shared among the ions in the trap. This means that, because the ions are electrically charged and thus through their Coulombic repulsion the motion of one ion affects the motion of another, we can use this interaction to couple qubits together—as an information bus for multi-qubit gates.

But this only works if we have a way to couple the qubit to the motion. In 1994, when this paper was written, this coupling had already been demonstrated. Through laser cooling, physicists showed that light could be used to control the motion of an atom [2]. And through an extension of the general laser cooling concept, physicists showed that they could use light to couple the electronic degree of freedom to a single, particular harmonic oscillator energy level, provided the transition linewidth is narrow enough that these harmonic levels can be resolved. This is known as a resolved sideband interaction [3].

If an ion is in state the ground qubit state and the n^th motional energy level, $\psi = |0\rangle |n\rangle$ , then we can drive this sideband transition by applying a laser whose frequency $\omega_l = \omega_0 \pm \omega_m$ , where $\omega_0$ is qubit frequency splitting and $\omega_m$ is one of the shared motion normal mode frequencies. Depending on whether we choose a positive or negative detuning, this will cause a blue sideband transition up to $\psi = |1\rangle |n+1\rangle$ or a red sideband transition to $\psi =| 1\rangle |n-1\rangle$ , respectively. In this way we can add and subtract single phonons from the trapped ion system, which can allow us to cool the system to the ground state of motion and also move information from the electronic state of one ion to the electronic state of another by transferring it through their shared motional mode.

One important thing to note: if we start in $|0\rangle |n=0\rangle$ , then applying a red sideband will do nothing, since there is no motional energy level lower than $n=0$ , which is necessary to satisfy energy conservation in this case. The same reasoning can be applied for the case where we try to apply a blue sideband pulse on a starting state $|1\rangle |n=0\rangle$ —there is no motional state below $n=0$ , so the blue sideband does nothing to this state. See the figure below for a pictorial representation:

So how do you make a two-qubit gate out of this interaction? Starting with ions with all modes cooled to the ground state of motion, and three relevant internal energy levels, $|g\rangle$ , $|e_0\rangle$ , and $| e_1\rangle$ (where $|g\rangle$ and $|e_0\rangle$ are the qubit levels and $| e_1\rangle$ is an auxiliary level) Cirac and Zoller proposed the following three steps:

Red sideband $\pi$ -pulse between $|g\rangle _1$ and $|e_0\rangle _1$ on ion 1. This will move the population in state $|e_0\rangle_1$ to state $|g\rangle _1$ and add a quantum of shared motion to the system. It will do nothing to state $|g\rangle _1$ . (The subscript outside of the ket denotes which ion.)
Red sideband $2\pi$ -pulse between $|g\rangle _2$ and $|e_1\rangle _2$ on ion 2. If a quantum of motion was added in step 1, then this will cause a transition between $|g\rangle _2 |n=1\rangle$ and $|e_1\rangle _2|n=0\rangle$ . Since it is a $2\pi$ -pulse, the population won’t change, but it will acquire a $\pi$ phase shift.
Red sideband $\pi$ -pulse between $|g\rangle _1$ and $|e_0\rangle _1$ on ion 1. This transfers anything in $|g\rangle _1 |n=1 \rangle$ back to $|e_0\rangle _1 |n=0\rangle$ , leaving the system back in the ground state of motion.

Now, let’s look at a truth table of the results of these pulses on two qubits. From the original paper we get:

If we combine this gate with single qubit rotations (and reverting back to standard qubit state labels $|0\rangle$ and $|1\rangle$ ), then the truth table can be simplified to:

This is the controlled-NOT (CNOT) gate. The first ion acts as the “control” qubit. If it is in state $|1\rangle$ , then a NOT gate is performed on the “target” qubit, or ion 2, which flips the state of the qubit. If the control qubit is $|0\rangle$ , then nothing happens.

The fact that this proposal enabled quantum computing on trapped ions with such a simple series of pulses created a ton of excitement among ion trappers. However, it had one fatal flaw: if the ions’ motion heats up during the gate, then it will fail. Keeping ions in the ground motional state for long periods of time unfortunately was an unrealistic expectation, since their motion is extremely sensitive to electric field noise. So, while this is a very important paper from a historical perspective, the Cirac-Zoller gate is not used in any modern trapped-ion quantum computers. In fact, it was never experimentally realized with the originally proposed setup, since a few years after this proposal, Klaus Mølmer and Anders Sørenson came up with their scheme for a two-qubit gate that was more robust to ion heating [4]. The Mølmer-Sørenson gate is still commonly used today.

[1] Wang, Y., Um, M., Zhang, J. et al. Single-qubit quantum memory exceeding ten-minute coherence time. Nature Photon 11, 646–650 (2017). https://doi.org/10.1038/s41566-017-0007-1

[2] D. J. Wineland, R. E. Drullinger, and F. L. Walls. Radiation-Pressure Cooling of Bound Resonant Absorbers. Phys. Rev. Lett. 40, 1639 (1978).

[3] Diedrich, F., Bergquist, J., et al. Laser cooling to the zero-point energy of motion. Phys Rev Lett. 62:403-406 (1989).

[4] Sørensen, A., Mølmer, K. Quantum Computation with Ions in Thermal Motion. Phys. Rev. Lett. 82 (9): 1971–1974. (1999).
arXiv:quant-ph/9810039

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