Leveraging randomized compiling for the quantum imaginary-time-evolution algorithm

Open Access

Leveraging randomized compiling for the quantum imaginary-time-evolution algorithm

Jean-Loup Ville, Alexis Morvan, Akel Hashim, Ravi K. Naik, Marie Lu, Bradley Mitchell, John-Mark Kreikebaum, Kevin P. O'Brien, Joel J. Wallman, Ian Hincks, Joseph Emerson, Ethan Smith, Ed Younis, Costin Iancu, David I. Santiago, and Irfan Siddiqi

Phys. Rev. Research 4, 033140 – Published 22 August 2022

Abstract

Recent progress in noisy intermediate-scale quantum (NISQ) hardware shows that quantum devices may be able to tackle complex problems even without error correction. However, coherent errors due to the increased complexity of these devices is an outstanding issue. They can accumulate through a circuit, making their impact on algorithms hard to predict and mitigate. Iterative algorithms like quantum imaginary time evolution are susceptible to these errors. This article presents the combination of both noise tailoring using randomized compiling and error mitigation with purification. We also show that cycle benchmarking gives an estimate of the reliability of the purification. We apply this method to the quantum imaginary time evolution of a transverse field Ising model and report an energy estimation error and a ground-state infidelity both below 1%. Our methodology is general and can be used for other algorithms and platforms. We show how combining noise tailoring and error mitigation will push forward the performance of NISQ devices.

Received 27 October 2021
Revised 18 April 2022
Accepted 31 May 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.033140

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Measurement-based quantum computing Quantum algorithms Quantum benchmarking Quantum computation Quantum control Quantum information processing Quantum simulation

Quantum Information, Science & TechnologyInterdisciplinary Physics

Authors & Affiliations

Jean-Loup Ville ^1,*, Alexis Morvan^1,2,†, Akel Hashim¹, Ravi K. Naik¹, Marie Lu¹, Bradley Mitchell¹, John-Mark Kreikebaum^1,3, Kevin P. O'Brien⁴, Joel J. Wallman^5,6, Ian Hincks⁶, Joseph Emerson^5,6, Ethan Smith ², Ed Younis ², Costin Iancu², David I. Santiago^1,2, and Irfan Siddiqi^1,2,3

¹Quantum Nanoelectronics Laboratory, Dept. of Physics, University of California at Berkeley, Berkeley, California 94720, USA
²Computational Research Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
³Materials Sciences Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
⁴Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
⁵Inst. for Quantum Computing and Dept. of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
⁶Quantum Benchmark Inc., Kitchener, Ontario, Canada N2H 5G5

^*jlville@berkeley.edu
^†morvan.alexis@gmail.com

Article Text

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Issue

Vol. 4, Iss. 3 — August - October 2022

Subject Areas

Quantum Information

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Images

Figure 1
Effect of randomized compiling on measured expectation values for random circuits using 6 czs. Panel (a) shows the distribution of errors on the measured expectation values $| E_{m} - E |$ for the different numbers of randomizations. Panel (b) shows the same data but sorted by measured expectation values to show the increasing agreement with a depolarization model $E_{m} = \bar{λ らむだ} E$ , with $\bar{λ らむだ}$ the mean of the Pauli decays measured by CB ( ${\bar{λ らむだ}}_{C B} = 0.980$ )—performed prior to the experiment. The standard deviation of the errors is similar for all points of same number of randomizations and is plotted in the inset. The data in panel (c) show the reduction of errors for all expectation values when using the purification formula (2). The standard deviation of the Pauli decays is $std ({λ らむだ}_{P}) = 0.003$ , as shown at the top of Fig. 3.
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Figure 2
Fidelity $F$ of random states versus the number of cz gates presented in blue points, together with the independent estimation from CB, using Eq. (7) as the dashed blue line. When these data were taken, the CB of this cz gate was ${\bar{λ らむだ}}_{C B} = 0.976$ . The length of the generalized Bloch vector $λ らむだ$ , is shown in green, together with its CB estimation in dashed green. The remaining error separating the two contributions corresponds to an angle error as shown by Eq. (5).
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Figure 3
Histograms of the Pauli decays ${λ らむだ}_{P}$ for the different cycles. The first histogram shows the 15 different ${λ らむだ}_{P}$ for the cz gate of Fig. 1. The dashed line indicates the mean and the dotted line the lower bound of Eq. (1). The two other histograms show for comparison the corresponding parameters when characterizing the different three-qubit cz hard cycles, including a remaining idling qubit. The spread of the 63 corresponding ${λ らむだ}_{P}$ is shown for the two cycles used in the following QITE experiments.
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Figure 4
QITE trajectories of relative energy error and infidelity to the ground state for the three-qubit Ising model, with $J = h = 1$ . The number of shots and of randomized compilations per point is varied. The mean and standard deviation of the 10 last points are shown by a dashed line and a colored region. The different colors correspond to with or without RC, with the total number of shots kept constant at 20 000 to allow for a fair comparison, and with or without purification.
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Figure 5
Phase diagram for the three-qubit Ising model varying the $h$ parameter. (a) Energy in units of $E / J$ , for the ground state and first-excited state. The experiments use 10 randomized circuits and 1024 shots for each value of $h$ . The reported energy corresponds to the mean of the five last points of the QITE evolution. Error bars corresponding to the standard deviation on these points are smaller than the markers. The error shown for each point corresponds to the relative error. (b) Measured magnetization of the same data, with the corresponding absolute error.
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Figure 6
Study of the distribution of the measured expectation values for randomly compiled circuits of the same unitary. (a) Dispersion of the single circuit estimator $E_{m}$ in the limit of $N ≫ 1$ so the single-shot noise is negligible. From this distribution, we extract the variance $σ しぐま$ of the results from random implementation to another one. Panel (b) presents the evolution of the variance of the randomized compiled estimator $E_{R C}$ as the number of randomizations increases. For each curve, the number of shots per random circuit is $N$ . Panel (c) presents the same variance as a function of the number $N$ of single shots per randomization. For large number of shots, the variances plateau to a value given by ${σ しぐま}^{2} / M$ as expected from (B4). For both panels (b) and (c), the dashed lines are models following the Eq. (B4) and the extracted value from the distribution of panel (a).
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