Randomized Compiling for Scalable Quantum Computing on a Noisy Superconducting Quantum Processor

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Randomized Compiling for Scalable Quantum Computing on a Noisy Superconducting Quantum Processor

Akel Hashim, Ravi K. Naik, Alexis Morvan, Jean-Loup Ville, Bradley Mitchell, John Mark Kreikebaum, Marc Davis, Ethan Smith, Costin Iancu, Kevin P. O’Brien, Ian Hincks, Joel J. Wallman, Joseph Emerson, and Irfan Siddiqi

Phys. Rev. X 11, 041039 – Published 24 November 2021

Abstract

The successful implementation of algorithms on quantum processors relies on the accurate control of quantum bits (qubits) to perform logic gate operations. In this era of noisy intermediate-scale quantum (NISQ) computing, systematic miscalibrations, drift, and crosstalk in the control of qubits can lead to a coherent form of error that has no classical analog. Coherent errors severely limit the performance of quantum algorithms in an unpredictable manner, and mitigating their impact is necessary for realizing reliable quantum computations. Moreover, the average error rates measured by randomized benchmarking and related protocols are not sensitive to the full impact of coherent errors and therefore do not reliably predict the global performance of quantum algorithms, leaving us unprepared to validate the accuracy of future large-scale quantum computations. Randomized compiling is a protocol designed to overcome these performance limitations by converting coherent errors into stochastic noise, dramatically reducing unpredictable errors in quantum algorithms and enabling accurate predictions of algorithmic performance from error rates measured via cycle benchmarking. In this work, we demonstrate significant performance gains under randomized compiling for the four-qubit quantum Fourier transform algorithm and for random circuits of variable depth on a superconducting quantum processor. Additionally, we accurately predict algorithm performance using experimentally measured error rates. Our results demonstrate that randomized compiling can be utilized to leverage and predict the capabilities of modern-day noisy quantum processors, paving the way forward for scalable quantum computing.

Received 13 May 2021
Accepted 3 September 2021

DOI:https://doi.org/10.1103/PhysRevX.11.041039

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)

Quantum algorithms Quantum benchmarking Quantum control Quantum information with solid state qubits

Quantum Information, Science & Technology

Authors & Affiliations

Akel Hashim ^1,2,3,*,†, Ravi K. Naik^1,3,†, Alexis Morvan ^1,3, Jean-Loup Ville¹, Bradley Mitchell^1,3, John Mark Kreikebaum^1,4,‡, Marc Davis ³, Ethan Smith³, Costin Iancu³, Kevin P. O’Brien ⁵, Ian Hincks^6,7, Joel J. Wallman ^6,7,8, Joseph Emerson^6,7,8, and Irfan Siddiqi^1,3,4

¹Quantum Nanoelectronics Laboratory, Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
²Graduate Group in Applied Science and Technology, University of California at Berkeley, Berkeley, California 94720, USA
³Computational Research Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
⁴Materials Sciences Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
⁵Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142, USA
⁶Quantum Benchmark Inc., Kitchener, Ontario N2H 5G5, Canada
⁷Keysight Technologies Canada, Kanata, Ontario K2K 2W5, Canada
⁸Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

^*To whom correspondence should be addressed. ahashim@berkeley.edu
^†These authors contributed equally to this work.
^‡Present address: Google Quantum AI, Mountain View, California, USA.

Popular Summary

The successful implementation of algorithms on quantum processors relies on the accurate control of quantum bits (qubits) to perform logic gate operations. While classical bits can be 0 or 1, qubits lie in a continuous state space, making them susceptible to imperfections in their analog control signals. This leads to a coherent form of error that has no classical analog, placing limits on the scale of reliable quantum computations. Here, we show that by converting these coherent errors into stochastic noise, one can improve the performance of quantum algorithms.

The method we use is known as randomized compiling: By creating a family of quantum circuits with different gates that all perform the same operation and combining their results, coherent errors in the original circuit are averaged into stochastic noise. The difference between coherent errors and stochastic noise can be understood using a rough analogy to the signal-to-noise ratio in classical systems, such as astronomy: The signal will coherently accumulate with more measurements, whereas random and uncorrelated noise grows more slowly.

In our experiments, we show marked improvement of algorithms such as the quantum Fourier transform by reducing the error rate per gate with randomized compiling. Further, we show that randomized compiling enhances the predictability of quantum circuit performance, finding that the measured accuracy is strongly correlated with predictions based on benchmarked gate error rates. This is not the case when dominated by coherent errors, which interfere unpredictably from gate to gate.

We perform our experiments on a superconducting quantum processor, a popular platform for scalable quantum computing. However, the general principle underlying randomized compiling, and the classes of quantum algorithms amenable to this type of noise tailoring, are applicable to all qubit platforms.

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Vol. 11, Iss. 4 — October - December 2021

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Figure 1
Experimental realization of noise tailoring via randomized compiling on a superconducting quantum processor. (a) False-colored micrograph of our eight-qubit superconducting quantum processor. In this work, we use four transmon qubits [33] (green) with independent microwave control lines (blue). Two-qubit cross-resonance [34, 35, 36, 37] gates are mediated by coupling resonators (CR, purple) between nearest neighbors. The qubits are simultaneously measured via dispersive coupling [38] to independent readout resonators (RO, red) coupled to a multiplexed readout bus (MRB, cyan). (b) Randomization of a quantum circuit. The bare circuit (top), split into $K$ interleaved cycles of easy and hard gates (separated by dashed lines), (middle) is converted into a logically equivalent circuit by inserting random single-qubit twirling gates between each easy and hard cycle, inverting them in the following cycle, and (bottom) then compiling the twirling gates into a new easy-gate cycle. (c) Experimental single-qubit state-tomography results demonstrating noise tailoring. The black vector is the ideal (noiseless) final state of the qubit, but coherent errors cause an over-rotation in the measured state (blue vector). The orange vector represents the final state of the combined distribution of $N = 12$ randomizations (orange data points), which has a lower purity because of the tailored stochastic noise. RC significantly mitigates the impact of coherent errors, as indicated by a reduction in the TVD from $d_{TV, bare} = 0.170 (8)$ to $d_{TV, RC} = 0.029 (2)$ in the ${| + ⟩, | - ⟩}$ basis, from $d_{TV, bare} = 0.069 (4)$ to $d_{TV, RC} = 0.060 (1)$ in the ${| + i ⟩, | - i ⟩}$ basis, and from $d_{TV, bare} = 0.073 (8)$ to $d_{TV, RC} = 0.008 (2)$ in the ${| 0 ⟩, | 1 ⟩}$ (computational) basis. This improvement is not captured by the bare ( $F = 0.862$ ) and RC ( $F = 0.879$ ) state fidelities, which are approximately equal.
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Figure 2
Cycle error reconstruction of the tailored noise under cycle benchmarking. (a) Schematic of the process by which one- and two-body gate errors can be reconstructed using targeted CB measurements of a parallel gate cycle (e.g., CNOT between Q5 and Q6, with Q4 and Q7 idling), providing detailed information about the Pauli (Kraus) errors occurring during the cycle, as shown in panel (b). (b) Cycle error reconstruction results of four-qubit cycles containing a single CNOT gate and identity gates on the spectator qubits. The $y$ axis ( $x$ axis) labels the type of error (where the error occurs in the cycle), and the color (gradient) indicates the marginal error rate from all Pauli contributions (95% confidence interval). The first (third) row of subplots shows one-body errors acting in idling (CNOT) qubits, the second row of subplots shows correlated one- and two-body errors between idling qubits, and the last row of subplots shows correlated one- and two-body errors between an idling spectator qubit and qubits participating in the CNOT. Here, a $k$ -body error is labeled by $k$ nonidentity gates acting on $n$ qubits (e.g., $k = 1$ for CNOT qubits, because these errors occur within a single gate body acting on two qubits). Curly brackets indicate error types that cannot be distinguished due to degeneracies since any local error acting on either qubit in the entangling operation will be transformed by the CNOT. All rows in which all errors are below 30% of the maximum value have been omitted for clarity. This detailed information can be used to perform targeted gate tuneup to address the most harmful errors. The residual errors in our system are broadly distributed among many pathways, so any further targeted tuneup will come with diminishing returns.
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Figure 3
Improving the quantum Fourier transform with randomized compiling. (a) Measured probability distributions for the QFT applied to $| 0000 ⟩$ , $| 000 + ⟩$ , $| + + + + ⟩$ , and a random input state [ $S U (2)_{rand}^{\otimes 4} | 0000 ⟩$ ]. (b) Bare and RC TVDs for all four-qubit QFT results, as a function of distribution uniformity of the ideal results. RC provides more improvement as the resultant distribution spans more basis states [ $d_{TV} (P_{ideal}, P_{uniform}) \to 0$ ]. Circles indicate the QFT applied to Pauli basis states ( ${| 0 ⟩$ , $| 1 ⟩$ , $| + ⟩}$ ), and triangles indicate random inputs states [ $S U (2)_{rand}^{\otimes 4} | 0000 ⟩$ ]; we use the same distinction in panels (c) and (d), but omit the description in the legend for clarity. Pearson $r$ values listed in the legend quantify the correlation strength of each data set, justifying linear fits for the RC data (transparent bands indicate the 95% confidence intervals). (c) Experimental vs simulated TVDs from two models based on the Pauli error rates in Fig. 2. The blue (orange) markers denote the bare (RC) circuits simulated with the coherent error model, and the cyan markers denote the bare circuits simulated with the Pauli model. (d) Accuracy of the two models compared to experimental results. The bare circuits simulated with the Pauli model are plotted against the experimental RC results in panel (c), which are also used to compute the model accuracy in panel (d). (e) Summary of the improvement under RC for all two-, three-, and four-qubit random input QFT results, showing good agreement between experiment (blue) and theory (gray). Simulations in which single-qubit (green) and two-qubit (pink) error rates have been scaled down by a factor of 10 suggest that RC performance increases as error rates decrease. (Error bars on the TVD [ $O (10^{- 3})$ ] for panels (b)–(d) are smaller than the markers.)
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Figure 4
Randomized compiling extends the computational reach with respect to circuit depth. (a) Bare and RC TVDs as a function of circuit depth $K$ . RC reduces the TVD on average for all circuit depths tested, allowing one to perform longer gate sequences under a fixed TVD error budget. The semitransparent blue (orange) points indicate the TVDs of the individual random circuits (the unioned data over all $N = 20$ randomizations of the corresponding bare circuits). Violin plots depict the distribution of results. The TVD error grows approximately linearly with $K$ for both bare and RC results, suggesting that the dominant reason for an improvement under RC is an overall reduction in the error for each gate cycle $K$ , rather than a suppression of the adversarial accumulation of coherent errors (although RC can additionally provide this benefit). (b) RC TVD improvement factor for the random circuits in panel (a) as a function of distribution uniformity. The average improvement is $d_{TV, bare} / d_{TV, RC} \approx 1.7$ . (c) TVD as a function of number of randomizations, with $K = 10$ fixed. The average TVD under RC converges to a value close to the 10% quantile level (dashed line) of the nonrandomized circuits for $N = 20$ . However, only $N = 10$ randomizations are needed to converge to within 2.7% of the $N = 20$ level. (d) For a fixed total error rate, RC provides a larger TVD improvement for systems with a higher fraction of coherent errors. The colored subsets listed in the legend highlight random single-qubit circuits that were performed in isolation (purple) or in parallel (blue and red). The average fraction of the total error rate due to coherent errors was quantified using measurements of RB and unitary RB under isolated or simultaneous operation.
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