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arXiv:1905.09292 · npj Quantum Information

Power-optimal, stabilized entangling gate between trapped-ion qubits

R. Blümel, Nikodem Grzesiak, Neal C. Pisenti, Kenneth Wright, Yunseong Nam

Published Oct 8, 2021• 52 citations
arXivHTMLPDF

Abstract

Abstract To achieve scalable quantum computing, improving entangling-gate fidelity and its implementation efficiency are of utmost importance. We present here a linear method to construct provably power-optimal entangling gates on an arbitrary pair of qubits on a trapped-ion quantum computer. This method leverages simultaneous modulation of amplitude, frequency, and phase of the beams that illuminate the ions and, unlike the state of the art, does not require any search in the parameter space. The linear method is extensible, enabling stabilization against external parameter fluctuations to an arbitrary order at a cost linear in the order. We implement and demonstrate the power-optimal, stabilized gate on a trapped-ion quantum computer.

DOI
10.1038/s41534-021-00489-w

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Abstract To achieve scalable quantum computing, improving entangling-gate fidelity and its implementation efficiency are of utmost importance. We present here a linear method to construct provably power-optimal entangling gates on an arbitrary pair of qubits on a trapped-ion quantum computer. This method leverages simultaneous modulation of amplitude, frequency, and phase of the beams that illuminate the ions and, unlike the state of the art, does not require any search in the parameter space. The linear method is extensible, enabling stabilization against external parameter fluctuations to an arbitrary order at a cost linear in the order. We implement and demonstrate the power-optimal, stabilized gate on a trapped-ion quantum computer.
In order to scale up quantum processors and achieve a quantum advantage, it is crucial to economize on the power requirement of two-qubit gates, make them robust to drift in experimental parameters, and shorten the gate times. Applicable to all quantum computer architectures whose two-qubit gates rely on phase-space closure, we present here a new gate-optimizing principle according to which negligible amounts of gate fidelity are traded for substantial savings in power, which, in turn, can be traded for substantial increases in gate speed and/or qubit connectivity. As a concrete example, we illustrate the method by constructing optimal pulses for entangling gates on a pair of ions within a trapped-ion chain, one of the leading quantum computing architectures. Our method is direct, noniterative, and linear, and, in some parameter regimes, constructs gate-steering pulses requiring up to an order of magnitude less power than the standard method. Additionally, our method provides increased robustness to mode drift. We verify the new trade-off principle experimentally on our trapped-ion quantum computer.
Efficiently entangling pairs of qubits is essential to fully harness the power of quantum computing. Here, we devise an exact protocol that simultaneously entangles arbitrary pairs of qubits on a trapped-ion quantum computer. The protocol requires classical computational resources polynomial in the system size, and very little overhead in the quantum control compared to a single-pair case. We demonstrate an exponential improvement in both classical and quantum resources over the current state of the art. We implement the protocol on a software-defined trapped-ion quantum computer, where we reconfigure the quantum computer architecture on demand. Our protocol may also be extended to a wide variety of other quantum computing platforms.
Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. However, one of the pressing challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the M\o{}lmer-S\o{}rensen protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. We present an analytical study of AM, using a Fourier series expansion to maintain the generality of the laser amplitude's functional form. We then apply this general AM method to gate-timing errors by imposing conditions on these Fourier coefficients, producing trade-offs between the laser power and fidelity at a fixed gate time. The conditions derived here are linear and can be used, in principle, to achieve arbitrarily high orders of insensitivity to gate-timing errors. Numerical optimization is then employed to identify the minimum-power pulse satisfying these constraints. Our central result is that the leading-order dependence on gate-timing errors is improved from $O(\mathrm{\ensuremath{\Delta}}{t}^{2})$ to $O(\mathrm{\ensuremath{\Delta}}{t}^{6})$ with the addition of one linear constraint on the Fourier coefficients and to $O(\mathrm{\ensuremath{\Delta}}{t}^{10})$ with two linear constraints without a significant increase in the average laser power. The increase approaches zero as more Fourier coefficients are included. In further studies, this protocol can be applied to other error sources and used in conjunction with other error-mitigation techniques to improve two-qubit gates.
Almost every quantum circuit is built with two-qubit gates in the current stage, which are crucial to the quantum computing in any platform. The entangling gates based on Mølmer-Sørensen schemes are widely exploited in the trapped-ion system, with the utilization of the collective motional modes of ions and two laser-controlled internal states, which are served as qubits. The key to realize high-fidelity and robust gates is the minimization of the entanglement between the qubits and the motional modes under various sources of errors after the gate operation. In this work, we propose an efficient numerical method to search high-quality solutions for phase-modulated pulses. Instead of directly optimizing a cost function, which contains the fidelity and the robustness of the gates, we convert the problem to the combination of linear algebra and the solution to quadratic equations. Once a solution with the gate fidelity of 1 is found, the laser power can be further reduced while searching on the manifold where the fidelity remains 1. Our method largely overcomes the problem of the convergence and is shown to be effective up to 60 ions, which suffices the need of the gate design in current trapped-ion experiments.
A global race towards developing a gate-based, universal quantum computer that one day promises to unlock the never before seen computational power has begun and the biggest challenge in achieving this goal arguably is the quality implementation of a two-qubit gate. In a trapped-ion quantum computer, one of the leading quantum computational platforms, a two-qubit gate is typically implemented by modulating the individual addressing beams that illuminate the two target ions, which, together with others, form a linear chain. The required modulation, expectedly so, becomes increasingly more complex, especially as the quantum computer becomes larger and runs faster, complicating the control hardware design. Here, we develop a simple method to essentially remove the pulse-modulation complexity at the cost of engineering the normal modes of the ion chain. We demonstrate that the required mode engineering is possible for a three ion chain, even with a trapped-ion quantum computational system built and optimized for a completely different mode of operations. This indicates that a system, if manufactured to target specifically for the mode-engineering based two-qubit gates, would readily be able to implement the gates without significant additional effort.
High-fidelity and parallel realization in scalable platforms of the two-qubit entangling gates fundamental to universal quantum computing constitutes one of the largest challenges in implementing fault-tolerant quantum computation. Integrated optical addressing of trapped-ion qubits offers routes to scaling the high-fidelity optical control demonstrated to date in small systems. Here we show that in addition to scaling, capabilities practically enabled by integrated optics can substantially alleviate laser powers required for both light-shift (LS) and Molmer-Sorensen (MS) geometric phase gates acting on long-lived ground-state qubit encodings in a broad range of ion species. In the proposed gate schemes utilizing carrier nulling via ion positioning at phase-stable standing-wave (SW) nodes, our calculations suggest that suppressed spontaneous photon scattering at the SW node allows for gate drives operating at smaller Raman detunings, resulting in approximately an order-of-magnitude reduction in power requirement (and significantly larger in certain parameter regimes) for gates of a given duration and scattering-limited fidelity as compared to conventional running wave (RW)-based approaches. The SW schemes have the additional benefit of simultaneously eliminating undesired coherent couplings that typically limit gate speeds. Our work quantifies power requirements for multiple ion species and enhancements to be expected from carrier-nulled configurations practically enabled by integrated delivery, and informs experiments and systems for realization of fast and power-efficient laser-based entangling gates in scalable platforms.