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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, Irfan Siddiqi

arXiv:2010.00215v2quant-ph

TL;DR

Coherent control errors can make average gate-error rates poor predictors of quantum-algorithm performance. The paper uses randomized compiling to tailor these errors into stochastic Pauli noise and demonstrates improved performance for QFT and variable-depth random circuits, with predictions based on experimentally measured error rates.

  • Problem

    Coherent errors can make worst-case gate infidelity and structured-circuit impact much larger and less predictable than average error rates indicate.

  • Method

    Randomized compiling inserts and compiles random single-qubit twirling gates to convert coherent errors into stochastic Pauli noise while preserving logical circuit behavior.

  • Results

    Randomized compiling improves average TVD at all tested depths, improves more than 81% of random-input QFT circuits by an average dTV,bare/dTV,RC ≈1.9, and reaches within 2.7% of its N = 20 level after N = 10 randomizations.

  • Takeaways & Limitations

    Randomized compiling can reduce unpredictable coherent-error effects and support scalable prediction of algorithm performance from cycle-benchmarking error rates.

Abstract

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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 which 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.

I. Introduction

Quantum algorithms face both incoherent and coherent errors, but standard average gate-error metrics can substantially understate coherent errors’ worst-case and algorithmic impact. Randomized compiling is introduced as a scalable, generalizable approach to reduce and stabilize these effects while enabling performance prediction through cycle benchmarking.

  • Error mechanisms: Systematic control imperfections and crosstalk produce coherent, purity-preserving errors, unlike environmental incoherent errors that reduce state purity.Coherent single-qubit errors appear as unwanted unitary rotations.
  • Limits of average metrics: ∼10% coherent error within r(E) ∼10^-4 can produce worst-case gate infidelity around r(E) ∼10^-2.Average-case and worst-case infidelities can therefore differ by orders of magnitude.
  • Limits of average metrics: Coherent errors make structured-circuit performance difficult to predict because their impact can scale quadratically with circuit depth and interfere across an algorithm.This limits the reliability of average error rates as predictors of global algorithm performance.
  • Proposed approach: Randomized compiling reduces and stabilizes the otherwise unpredictable impact of coherent errors in QFT and variable-depth random circuits without requiring prior knowledge of the specific error model.The protocol is presented as more scalable and generalizable than Pauli-frame randomization and simple Pauli twirling.

II. Randomized Compiling Protocol

Randomized compiling inserts logically cancelling single-qubit twirling gates into easy/hard-gate cycles, producing equivalent circuits whose coherent errors are averaged into stochastic Pauli noise. This tailoring reduces depth-dependent error growth and supports algorithm evaluation using total variation distance.

  • Protocol: Randomized compiling inserts random single-qubit twirling gates between easy and hard cycles, then compiles them into new easy cycles while preserving the circuit’s unitary.The randomized circuits remain logically equivalent to the bare circuit without increasing circuit depth.
  • Protocol: Tensor products of single-qubit Paulis are typically used as twirling gates, making randomized compiling efficiently compatible with universal quantum computation.For Clifford hard gates, correction gates also remain in the Pauli set.
  • Implementation: Large numbers of randomizations are more practical on superconducting circuits than on platforms with slower gate times because each randomization must be measured.The protocol’s classical generation overhead is low, but measurement resources remain hardware-dependent.
  • Noise tailoring: Averaging N logically equivalent randomizations converts coherent errors in each nonfinal computational cycle into Pauli channels such as random phase and bit flips.Generating the randomizations requires low classical overhead and can be done before runtime.
  • Noise tailoring: Stochastic Pauli errors grow linearly with circuit depth in the small-error limit, contrasting with coherent errors that can accumulate quadratically.The tailored channel also suppresses off-diagonal error-process terms in the limit of perfect Pauli twirling.
  • Performance metric: The total variation distance measures the statistical distance between ideal and experimental bit-string distributions, with lower values indicating better algorithmic performance.For the single-qubit experiment, dTV,bare = 0.073(8) decreased to dTV,RC = 0.008(2) in the computational basis.

III. Cycle Error Reconstruction

Cycle benchmarking measures errors in parallel gate cycles and can capture errors affecting both active and idling qubits. Combined with cycle error reconstruction, it produces Pauli error maps that support targeted tuning and performance prediction under randomized compiling.

  • Cycle benchmarking: Cycle benchmarking characterizes composite parallel gate cycles, giving performance estimates that better reflect their use in quantum circuits than isolated-gate benchmarking.It also captures coherent-error effects on idling spectator qubits.
  • Cycle error reconstruction: Cycle error reconstruction maps error types to their locations and marginal rates, including single-body and correlated spectator-qubit errors with 95% confidence intervals.The figure omits rows whose errors are all below 30% of the maximum for clarity.
  • Performance prediction: Because cycle benchmarking and randomized compiling use the same Pauli-twirled effective noise, cycle-benchmarking process infidelities can predict randomized-compiling algorithm performance.This links scalable cycle measurements to algorithm-level error estimates.
  • Cycle error reconstruction: Targeted cycle-benchmarking measurements reconstruct single- and two-body Pauli errors in cycles containing a CNOT and identity gates on spectator qubits.The reconstructed error rates are Pauli-channel coefficients obtained from Pauli-transfer-matrix eigenvalues.

IV. Quantum Fourier Transform

The four-qubit QFT shows that randomized compiling improves performance predictably for many input distributions, while its benefit depends on distribution uniformity and coherent-error structure.

  • Distribution dependence: RC provides more QFT improvement as the ideal output distribution spans more measurement-basis states.Distribution uniformity is quantified by dTV(Pideal, Puniform), which is zero for a uniform distribution and maximized for a singular distribution.
  • Distribution dependence: RC results correlate more strongly with ideal-distribution uniformity than bare results, with r = 0.95 (0.80) versus r = 0.66 (0.32) for basis (random) inputs.
  • Predictability: Experiment and simulation agree well for RC circuits modeled with coherent errors, whereas complete-model predictions are unreliable for individual bare circuits.The complete model includes coherent errors and is calibrated to reproduce experimentally measured error rates.
  • Random inputs: For random-input QFTs, more than 81% of circuits improve under RC by an average factor of dTV,bare/dTV,RC ≈1.9.The authors use random inputs as a proxy for unknown inputs encountered when QFT is used as a subroutine.
  • Predictability: When both single- and two-qubit error rates are reduced tenfold, RC improves more than 94% of simulated circuits by an average factor of dTV,bare/dTV,RC ≈3.4.

V. Random Circuits of Variable Depth

Randomized compiling improves four-qubit universal circuits across tested depths and requires relatively few randomizations, with benefits shaped by coherent-error content and eventual decoherence.

  • Depth dependence: RC reduces average TVD at every tested circuit depth, enabling longer gate sequences under a fixed TVD error budget.For both bare and RC circuits, TVD grows approximately linearly with depth, indicating reduced error contribution per gate cycle as the dominant benefit.
  • Depth dependence: RC's relative improvement decreases at longer depths as both bare and RC results converge toward a uniform distribution through decoherence.The experiments nevertheless observe average improvement at large K.
  • Depth dependence: The average RC improvement factor for the random circuits is dTV,bare/dTV,RC ≈1.7.
  • Randomization resources: At K = 10, N = 10 randomizations reach an average RC TVD within 2.7% of the N = 20 level.With N = 20, the average RC TVD is better than approximately 90% of non-randomized circuits.
  • Error composition: For fixed total error, RC yields larger relative TVD improvement as the coherent fraction increases, while still improving nearly coherence-limited single-qubit systems.The latter systems have rU(E)/r(E) ≲0.1.

VI. Outlook

The outlook presents randomized compiling as a broadly applicable strategy for suppressing coherent errors and improving error predictability in noisy quantum processors.

  • Outlook: RC is described as universal, agnostic to specific error models and hardware platforms, and applicable to complex crosstalk dynamics.
  • Outlook: Cycle-benchmarking-based error reconstruction can characterize emergent crosstalk errors and predict error rates under RC.
  • Outlook: The authors identify improved predictability as important for scalable quantum computing and for comparing experimental error rates with fault-tolerant thresholds.
  • Outlook: The methods and results are presented as relevant to NISQ applications including variational quantum algorithms, where coherent errors can leave ground-state wavefunction parameterization incorrect despite convergence.

Supplemental Material for:

The supplied passages identify the paper as a supplemental-material document on randomized compiling for scalable quantum computing on a noisy superconducting processor.

  • Document identification: The document concerns randomized compiling for scalable quantum computing on a noisy superconducting processor.
  • Authors: The listed authors include Akel Hashim, Ravi K. Naik, Alexis Morvan, and Irfan Siddiqi.

I. Experimental Setup

The experiments used a four-transmon superconducting processor with multiplexed readout and cross-resonance CNOT gates. Crosstalk was characterized and compensated using additional pulses, virtual phase gates, and refocusing pulses.

  • Hardware: The experiments used four transmon qubits on a superconducting processor cooled to approximately 10 mK.Control and measurement electronics are described in the experimental setup.
  • Readout: Multiplexed readout classified ground and excited states using Gaussian Mixture Models fitted to IQ-plane measurement statistics.Readout fidelities were determined separately for each qubit.
  • Crosstalk: Microwave control crosstalk was strongest between nearest neighbors but also occurred between more distant qubits.Its effects depended on qubit couplings and relative transition frequencies, producing local and entangling errors.
  • Crosstalk compensation: Crosstalk compensation canceled conditional Rabi-drive terms by adding equal-amplitude, opposite-phase pulses on affected neighboring qubits.Nonnearest-neighbor crosstalk was addressed by minimizing unwanted AC Stark shifts measured with Ramsey spectroscopy.
  • Two-qubit gates: Two-qubit CNOT gates used the cross-resonance effect, while spectator-qubit phase errors were corrected with virtual phase and refocusing gates.CNOT gates were applied serially relative to other gates, but neighboring spectators still experienced crosstalk-induced errors.

V. Randomized Benchmarking

The study used isolated and simultaneous randomized benchmarking to measure single- and two-qubit gate infidelities, and unitary randomized benchmarking to quantify coherent-error contributions.

  • Benchmarking protocols: Randomized benchmarking measured isolated and simultaneous single-qubit and isolated two-qubit gate infidelities.The reported error rates were defined using process infidelity.
  • Benchmarking protocols: Unitary randomized benchmarking quantified the fraction of total error attributable to coherent rather than stochastic errors.This separates coherent-error contributions from the broader gate-error rate.
  • Error metrics: Process infidelity supports composite error rates for tensor-product processes independently of the dimension of each process.Average gate infidelity does not have this dimension-independent property.

VI. Cycle Benchmarking and Cycle Error Reconstruction

Cycle benchmarking measured multi-qubit cycle errors through Pauli decays, while cycle error reconstruction inferred specific Pauli-error contributions. These measurements supported targeted tuneup but also exposed residual, broadly distributed errors.

  • Cycle benchmarking: Cycle benchmarking estimated local and global errors in multi-qubit gate cycles by interleaving the target cycle with random n-qubit Pauli operators.The error rate was measured as a function of sequence depth after Pauli-basis preparation and measurement.
  • Cycle benchmarking: 2.2 × 10-2 (1.4 × 10-3) was the total process infidelity for the four-qubit identity cycle.This represents the average process infidelity of arbitrary parallel Pauli gates on all four qubits.
  • Cycle error reconstruction: Cycle benchmarking reconstructs Pauli error rates from Pauli decays using their commutation relationships and linear inversion.The decay-error relation is represented by cP = W−1pP.
  • Cycle error reconstruction: Cycle error reconstruction quantified marginal Pauli-error rates by analyzing hard cycles containing one CNOT and identity gates on spectator qubits.The reconstructed error maps used axes for error type and affected qubits, with colors indicating marginal rates and 95% confidence intervals.
  • Targeted tuneup: Targeted cycle error reconstruction identified X and Y errors on qubit 6 during the (4,5) CNOT at a marginal error rate of approximately 0.10.A refocusing pulse on qubit 6 was then used to compensate these spectator errors.
  • Residual errors: After tuneup, the remaining error syndromes were more broadly distributed, limiting the usefulness of further targeted corrections.Residual features included a large Z error on qubit 5 during the (7,6) CNOT and dominant {XI, XX} correlators on most CNOT pairs.

VII. Quadratic Impact of Coherent Errors

Coherent errors preserve purity but can produce off-diagonal error-process terms that scale linearly with small rotation angle, whereas diagonal infidelity terms scale quadratically. Randomized compiling suppresses the off-diagonal terms and changes the average-error scaling from approximately θ to θ^2.

  • Coherent-error model: Coherent single-qubit errors can be modeled as unitary rotations that map pure states to pure states without decoherence.The rotation is specified by an angle, an axis, and the Pauli vector.
  • Error scaling: For small rotation angles, diagonal error-process terms scale as θ^2 while off-diagonal terms scale as θ.This difference explains why average gate infidelity can understate coherent-error impact.
  • Randomized compiling: Randomized compiling suppresses off-diagonal error-process terms through Pauli twirling.The idealized twirl converts the coherent-error contribution into a stochastic form.
  • Error scaling: r(E) ≃θ changes to r(E) ≃θ2 under randomized compiling, where r(E) is the average error rate.The change describes the quadratic reduction in the small-error limit.
  • Error metrics: Average gate and process infidelities are insensitive to off-diagonal error-process terms, unlike norm-based metrics such as TVD and diamond distance.The latter metrics can therefore reflect error suppression produced by randomized compiling.
  • Metric dependence: TVD benefits from randomized compiling only when the ideal target state is coherently spread across the measurement basis.Diamond distance remains sensitive to off-diagonal terms regardless of measurement basis.

VIII. Motivation for Noise Tailoring via Randomized Compiling

Randomized compiling is motivated by the mismatch between standard average-error metrics and coherent errors’ potentially much larger algorithmic impact. Its benefits depend on how the chosen output metric responds to off-diagonal error terms and the target-state distribution.

  • Error-metric motivation: For typical two-qubit RB rates r(E) ≃10^-2, coherent errors can contribute as much as r(E) ≃10^-1 to incorrect outcomes, unlike stochastic errors remaining at the 10^-2 scale.The contrast motivates converting coherent errors into stochastic noise for comparison with fault-tolerant thresholds.
  • Noise tailoring: Randomized compiling’s many-randomization limit provides a quadratic reduction in norm-based error metrics by tailoring coherent errors into stochastic Pauli noise.The tailored channel suppresses off-diagonal error-process terms and enables benchmarked gate fidelities to be compared with fault-tolerance thresholds for Pauli errors.
  • Metric dependence: Fidelity is insensitive to off-diagonal error-process terms, whereas TVD is generally sensitive to them except when the target state is a measurement-basis eigenstate.Consequently, RC’s improvement depends on metric linearity in the output state and sensitivity to coherent-error structure.
  • Metric dependence: For measurement-basis eigenstates, trace distance scales as r(E) ≃θ^2 and neither trace distance nor fidelity benefits from many-randomization RC.For other target states, trace distance can scale as r(E) ≃θ and benefit substantially from RC.
  • Practical interpretation: TVD improvements tighten bounds on same-basis expectation-value errors, but do not guarantee that every such expectation value becomes more accurate.If dTV(P, Ptrue) < ϵ, all local observables are accurate to within ϵ.

XI. Single-qubit State Tomography

State tomography shows that randomized compiling averages coherent-error trajectories toward the ideal state while converting the dominant error mechanism into stochastic noise. This tailoring improves alignment without necessarily improving measurements when the target state is a measurement-basis eigenstate.

  • Noise tailoring: Bare-state angle errors can make purity and fidelity differ significantly, whereas randomized-compiling results have approximately equal purity and fidelity at each depth.This pattern indicates that stochastic noise is the dominant error mechanism under randomized compiling.
  • Noise tailoring: Each randomization follows a different trajectory to the same ideal final state, so combining them averages coherent errors into a stochastic Pauli channel.The resulting tailored noise acts as a decoherence channel and reduces Bloch-vector purity.
  • Experimental design: The tomography experiment sampled K = 100 interleaved easy and hard cycles, corresponding to 201 single-qubit gates per qubit.The easy set was the Clifford set, while the hard set contained X45, Y45, and T = Z45.
  • State-tomography evidence: Randomized compiling produces a combined state approximately co-linear with the ideal state across circuit depths, while bare results separate from it.For Q5, this separation emerges at later depths as coherent errors accumulate.
  • TVD interpretation: At K = 5, bare and randomized-compiling TVD performance is approximately equal when the target state is aligned with the measurement basis.Randomized compiling is not expected to improve algorithmic performance when the target state is a measurement-basis eigenstate.

XIV. Random Circuits of Variable Depth

Variable-depth random-circuit experiments examined isolated and simultaneous single-qubit operation, including crosstalk-sensitive parallel settings. Randomized compiling reduced TVD on average at every tested depth, while simulations indicate stronger performance as total error rates decrease at fixed coherent-error fraction.

  • Circuit construction: Random bare circuits interleave easy and hard gates sampled from universal gate sets for isolated, simultaneous, and entangling operations.Single-qubit experiments use Clifford easy gates with X45, Y45, and T = Z45 hard gates.
  • Randomized-compiling workflow: The randomized-compiling workflow generates N equivalent circuits, measures each for n/N shots, and combines all results into one distribution.The total number of shots remains n, so measurement time is unchanged relative to the bare circuit.
  • Error characterization: RB and unitary RB process infidelities quantify the fraction of total error attributable to coherent errors before variable-depth experiments.The reported infidelities and standard deviations are scaled by 10^-3 and 10^-4, respectively.
  • Experimental results: Randomized compiling provides an average TVD reduction at all tested circuit depths, including isolated and parallel single-qubit circuits.Parallel operation increases coherent-error rates through crosstalk, yet the average reduction persists.
  • Simulation result: At fixed coherent-error fraction, simulations show randomized-compiling performance improves as the total error rate decreases.This may keep the benefit significant even when coherent errors represent a smaller share of the total error budget.
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