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arXiv:1612.00858 · Physical Review A

Efficient Z gates for quantum computing

David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, Jay Gambetta

Published Aug 31, 2017• 551 citations
arXivHTMLPDF

Abstract

For superconducting qubits, microwave pulses drive rotations around the Bloch sphere. The phase of these drives can be used to generate zero-duration arbitrary virtual $Z$ gates, which, combined with two ${X}_{\ensuremath{\pi}/2}$ gates, can generate any SU(2) gate. Here we show how to best utilize these virtual $Z$ gates to both improve algorithms and correct pulse errors. We perform randomized benchmarking using a Clifford set of Hadamard and $Z$ gates and show that the error per Clifford is reduced versus a set consisting of standard finite-duration $X$ and $Y$ gates. $Z$ gates can correct unitary rotation errors for weakly anharmonic qubits as an alternative to pulse-shaping techniques such as derivative removal by adiabatic gate (DRAG). We investigate leakage and show that a combination of DRAG pulse shaping to minimize leakage and $Z$ gates to correct rotation errors realizes a 13.3 ns ${X}_{\ensuremath{\pi}/2}$ gate characterized by low error $[1.95(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}]$ and low leakage $[3.1(6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}]$. Ultimately leakage is limited by the finite temperature of the qubit, but this limit is two orders of magnitude smaller than pulse errors due to decoherence.

DOI
10.1103/physreva.96.022330

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For superconducting qubits, microwave pulses drive rotations around the Bloch sphere. The phase of these drives can be used to generate zero-duration arbitrary virtual ZZZ gates, which, combined with two X\ensuremathπ/2{X}_{\ensuremath{\pi}/2}X\ensuremathπ/2​ gates, can generate any SU(2) gate. Here we show how to best utilize these virtual ZZZ gates to both improve algorithms and correct pulse errors. We perform randomized benchmarking using a Clifford set of Hadamard and ZZZ gates and show that the error per Clifford is reduced versus a set consisting of standard finite-duration XXX and YYY gates. ZZZ gates can correct unitary rotation errors for weakly anharmonic qubits as an alternative to pulse-shaping techniques such as derivative removal by adiabatic gate (DRAG). We investigate leakage and show that a combination of DRAG pulse shaping to minimize leakage and ZZZ gates to correct rotation errors realizes a 13.3 ns X\ensuremathπ/2{X}_{\ensuremath{\pi}/2}X\ensuremathπ/2​ gate characterized by low error [1.95(3)\ifmmode×\else\texttimes\fi10\ensuremath−4][1.95(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}][1.95(3)\ifmmode×\else\texttimes\fi10\ensuremath−4] and low leakage [3.1(6)\ifmmode×\else\texttimes\fi10\ensuremath−6][3.1(6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}][3.1(6)\ifmmode×\else\texttimes\fi10\ensuremath−6]. Ultimately leakage is limited by the finite temperature of the qubit, but this limit is two orders of magnitude smaller than pulse errors due to decoherence.
Improving the speed and fidelity of quantum logic gates is essential to reach quantum advantage with future quantum computers. However, fast logic gates lead to increased leakage errors in superconducting quantum processors based on qubits with low anharmonicity, such as transmons. To reduce leakage errors, we propose and experimentally demonstrate two new analytical methods, Fourier ansatz spectrum tuning derivative removal by adiabatic gate (FAST DRAG) and higher-derivative (HD) DRAG, both of which enable shaping single-qubit control pulses in the frequency domain to achieve stronger suppression of leakage transitions compared to previously demonstrated pulse shapes. Using the new methods to suppress the e f transition of a transmon qubit with an anharmonicity of − 212 MHz, we implement R X ( π / 2 ) gates achieving a leakage error below 3.0 × 10 − 5 down to a gate duration of 6.25 ns without the need for iterative closed-loop optimization. The obtained leakage error represents a 20-fold reduction in leakage compared to a conventional cosine DRAG pulse. Employing the FAST DRAG method, we further achieve an error per gate of ( 1.56 ± 0.07 ) × 10 − 4 at a 7.9-ns gate duration, outperforming conventional pulse shapes both in terms of error and gate speed. Furthermore, we study error-amplifying measurements for the characterization of temporal microwave control-pulse distortions, and demonstrate that non-Markovian coherent errors caused by such distortions may be a significant source of error for sub-10-ns single-qubit gates unless corrected using predistortion. Published by the American Physical Society 2024
High-fidelity control of superconducting qubits requires carefully shaped microwave pulses to avoid several different kinds of error at once. This article is a pedagogical bridging text aimed at upper-level undergraduate and early graduate students who have completed an introductory quantum mechanics course and a first course in quantum computing or quantum information, but who have not yet encountered the physical implementation of qubit gates. We integrate physical intuition for pulse design, analytical gate-level descriptions, and practical hardware considerations into a single, derivation-driven narrative, with explicit learning objectives. We begin with simple pulse envelopes and their spectral properties, showing how finite bandwidth produces leakage outside the computational subspace. This motivates the derivative removal by adiabatic gate (DRAG) technique, which we derive explicitly using the Magnus expansion, obtaining a clear, order-by-order account of which physical error channel appears at which order and why DRAG's cancellation is necessarily incomplete. We discuss the practical hardware realities of control pulse generation, focusing on arbitrary waveform generators (AWG), local oscillators (LO), and IQ mixing. Finally, we extend the discussion to two-qubit operation via the cross-resonance gate, and interpret how driving the control qubit at the target qubit's transition frequency necessarily produces several unwanted interaction terms alongside the desired one, and how successive generations of pulse-engineering strategies have been designed to suppress them.
Abstract Reaching high-speed, high-fidelity qubit operations requires precise control over the shape of the underlying pulses. For weakly anharmonic systems, such as superconducting transmon qubits, short gates lead to leakage to states outside of the computational subspace. Control pulses designed with open-loop optimal control may reduce such leakage. However, model inaccuracies can severely limit the usability of such pulses. We implemented a closed-loop optimization that simultaneously adapts all control parameters based on measurements of a cost function built from Clifford gates. We directly optimize the amplitude and phase of each sample point of the digitized control pulse. We thereby fully exploit the capabilities of the pulse generation electronics and create a 4.16 ns single-qubit pulse with 99.76 % fidelity and 0.044 % leakage. This is a sevenfold reduction of the leakage rate and a threefold reduction in standard errors of the best DRAG pulse we have calibrated at such short durations on the same system.
One of the main enablers in quantum computing is having qubit control that is precise and fast. However, qubits typically have multilevel structures making them prone to unwanted transitions from fast gates. This leakage out of the computational subspace is especially detrimental to algorithms as it has been observed to cause long-lived errors, such as in quantum error correction. This forces a choice between either achieving fast gates or having low leakage. Previous works focus on suppressing leakage by mitigating the first to second excited state transition, overlooking multi-photon transitions, and achieving faster gates with further reductions in leakage has remained elusive. Here, we demonstrate single qubit gates with a total leakage error consistently below 2.0×10−52.0\times10^{-5}2.0×10−5, and obtain fidelities above 99.98%99.98\%99.98% for pulse durations down to 6.8 ns for both XXX and X/2X/2X/2 gates. This is achieved by removing direct transitions beyond nearest-neighbor levels using a double recursive implementation of the Derivative Removal by Adiabatic Gate (DRAG) method, which we name the R2D method. Moreover, we find that at such short gate durations and strong driving strengths the main error source is from these higher order transitions. This is all shown in the widely-used superconducting transmon qubit, which has a weakly anharmonic level structure and suffers from higher order transitions significantly. We also introduce an approach for amplifying leakage error that can precisely quantify leakage rates below 10−610^{-6}10−6. The presented approach can be readily applied to other qubit types as well.
Recent achievements in the physical size of quantum computers based on superconducting qubits foresee reaching the next milestones of the exponentially growing number of qubits. The promises in the scalability present opportunities in integrated control electronics operating at 4K stage in the dilution refrigerator. In addition to the number of physical qubits, however, commercialization necessitates a further increase in the number of consecutive high-fidelity gate operations within a coherence time of∼100μs\sim 100\mu \mathsf{s}∼100μs. It leads to the stringent requirement on the duration of qubit-driving microwave pulse(i.e., <20ns)(\mathsf{i.e}.,\ < 20\mathsf{ns})(i.e., <20ns). The microwave pulses should only drive∣0⟩↔∣1⟩\vert 0\rangle\leftrightarrow\vert 1\rangle∣0⟩↔∣1⟩for single qubit gate operations while avoiding leakage to∣1⟩↔∣2⟩\vert 1\rangle\leftrightarrow\vert 2\rangle∣1⟩↔∣2⟩which is the major factor that degrades the gate fidelity. The frequency for∣1⟩↔∣2⟩,ω12\vert 1\rangle\leftrightarrow\vert 2\rangle, \omega_{12}∣1⟩↔∣2⟩,ω12​, is typically lower than the frequency for∣0⟩↔∣1⟩,ω01\vert 0\rangle\leftrightarrow\vert 1\rangle, \omega_{01}∣0⟩↔∣1⟩,ω01​, by2π×(160MHz−to−360MHz)2\pi\times(160\mathsf{MHz}-\mathsf{to}-360\mathsf{MHz})2π×(160MHz−to−360MHz). The frequency difference(ω01−ω12)(\omega_{01}-\omega_{12})(ω01​−ω12​), called anharmonicity, is an individual qubit characteristic and becomes the margin for spectral management. But, a short driving pulse even with Gaussian, sine, or raised-cosine shapes eventually widens the spectral width. It would range a few hundred MHz and result in a considerable spectral leakage atω12\omega_{12}ω12​. To mitigate this spectral leakage, the derivative removal by adiabatic gate (DRAG) has been adopted in the pulse shaping [1], [2]. As shown in Fig. 34.4.1, the first-order DRAG pulse shaping is realized by adding two parts as \begin{equation*}\mathsf{O}(\mathrm{t})=\mathsf{S}_{\mathsf{M}}(\mathsf{t})\cdot\mathsf{cos}(\omega_{01}\mathsf{t}+\varphi)+\mathsf{q}_{\mathsf{scale}}\mathsf{S}_{\mathsf{M}}(\mathsf{t})^{\prime}\cdot\mathsf{sin}(\omega_{01}\mathsf{t}+\varphi)\end{equation*} whereSM(t)\mathsf{S}_{\mathsf{M}}(\mathsf{t})SM​(t)is the main pulse shape andqscale\mathsf{q}_{\mathsf{scale}}qscale​is a coefficient inversely proportional to the anharmonicity. This first-order DRAG makes an asymmetrical spectrum that lowers the left frequency side and raises the right frequency side. Since theω12\omega_{12}ω12​is always on the left side ofω01\omega_{01}ω01​, the DRAG can effectively minimize the leakage.
Leakage errors occur when a quantum system leaves the two-level qubit subspace. Reducing these errors is critically important for quantum error correction to be viable. To quantify leakage errors, we use randomized benchmarking in conjunction with measurement of the leakage population. We characterize single qubit gates in a superconducting qubit, and by refining our use of derivative reduction by adiabatic gate pulse shaping along with detuning of the pulses, we obtain gate errors consistently below 10^{-3} and leakage rates at the 10^{-5} level. With the control optimized, we find that a significant portion of the remaining leakage is due to incoherent heating of the qubit.