PaperlayerView exact revision on arXiv

Source-linked AI summary

Investigating the limits of randomized benchmarking protocols

Jeffrey M. Epstein, Andrew W. Cross, Easwar Magesan, Jay M. Gambetta

arXiv:1308.2928v1quant-ph

TL;DR

Gate-error benchmarking is needed to compare implementations and check error thresholds, while QPT is exponentially costly and vulnerable to SPAM. The paper tests randomized benchmarking under realistic noise models and finds estimates within a factor of two in the tested cases.

  • Problem

    Gate-error benchmarking is needed to compare implementations and check error thresholds, while QPT scales exponentially and is vulnerable to SPAM.

    Source: §I · p. 1

  • Method

    The paper simulates standard and interleaved randomized benchmarking under physically relevant noise models and estimates confidence intervals by linearizing a least-squares fit.

    Source: §I · p. 1, A. Finite Sampling Effects · p. 5

  • Results

    For K ∼100 or greater, RB estimated the actual average error rate within a factor of two in all cases tested.

    Source: B. Results and Discussion · p. 9

  • Takeaways & Limitations

    These benchmarks provide a way to compare experimental implementations and assess compliance with error thresholds.

    Source: §I · p. 1

    With 1/f noise, the exponential decay model is not fully accurate, so increasing sample size beyond some Kmax does not improve estimate accuracy.

    Source: B. Results and Discussion · p. 10

Abstract

from arXiv · show

In this paper, we analyze the performance of randomized benchmarking protocols on gate sets under a variety of realistic error models that include systematic rotations, amplitude damping, leakage to higher levels, and 1/f noise. We find that, in almost all cases, benchmarking provides better than a factor-of-two estimate of average error rate, suggesting that randomized benchmarking protocols are a valuable tool for verification and validation of quantum operations. In addition, we derive new models for fidelity decay curves under certain types of non-Markovian noise models such as 1/f and leakage errors. We also show that, provided the standard error of the fidelity measurements is small, only a small number of trials are required for high confidence estimation of gate errors.

I. INTRODUCTION

Quantum process tomography is complete but exponentially costly and vulnerable to state-preparation and measurement errors; randomized benchmarking instead extracts selected error parameters from fidelity decay. The paper studies RB models and finite-sampling behavior under realistic noise, while outlining protocol assumptions and steps.

I. INTRODUCTION

  • QPT reconstructs the complete error process but scales exponentially and can be inaccurate when SPAM errors rival gate errors.The paper notes that QPT overestimated small errors by several orders of magnitude in a recent study.Source: §I · p. 1
  • RB avoids exponential scaling and SPAM sensitivity by extracting selected noise parameters, but complex noise and finite sampling may challenge its decay models and estimates.The paper motivates further study of RB where analytic support for current methods is limited.Source: §I · p. 1
  • The paper combines theoretical work on finite sampling and decay models with simulations of standard and interleaved RB under Markovian and non-Markovian single-qubit noise.The models include systematic rotations, amplitude damping, 1/f noise, and leakage to higher levels.Source: §I · p. 1

II. RANDOMIZED BENCHMARKING PROTOCOLS

  • RB selects random gate sequences, appends an inverting gate, measures survival probabilities, averages them by sequence length, and fits a fidelity-decay model.Standard RB estimates average gate fidelity over a group; interleaved RB targets an individual gate.Source: §II · p. 2
  • The protocol assumes small samples approximate the full sequence average and that gate errors remain close to their average despite possible gate, time, or history dependence.Leakage and 1/f noise are cited as examples of history-dependent errors.Source: §II · p. 2

Standard Clifford Benchmarking

Standard Clifford RB estimates average error over the Clifford group, whereas interleaved RB uses paired experiments to estimate an individual gate's error. The interleaved estimate relies on a product-twirl approximation, though bounds remain possible when it fails.

Standard Clifford Benchmarking

  • Standard Clifford RB twirls errors to a depolarizing channel and estimates the average error rate over the full n-qubit Clifford group.The protocol's fidelity-decay model absorbs SPAM errors into fit parameters.Source: Standard Clifford Benchmarking · pp. 2–3

Interleaved Randomized Benchmarking

  • Interleaved RB compares standard and interleaved experiments to estimate an individual gate's depolarizing parameter and error rate.The gate of interest is inserted after each randomly selected Clifford gate.Source: Interleaved Randomized Benchmarking · p. 3
  • The interleaved estimate assumes the twirl of a product is approximated by the product of twirls; partial error inversion can make it underestimate error.The approximation is exact if at least one channel is depolarizing.Source: Interleaved Randomized Benchmarking · p. 3
  • Even when the product-twirl assumption fails, bounds on the interleaved gate's error can still be obtained.The assumption is therefore not required for every useful estimate.Source: Interleaved Randomized Benchmarking · p. 3

III. SIMULATION METHODS

The simulations use single-qubit Clifford operations and compare benchmarking accuracy and confidence under specified simulation settings. The Clifford set can be decomposed into physical generators or simple mathematical elements.

III. SIMULATION METHODS

  • The single-qubit Clifford group contains 24 operators, and combinations of the Pauli and exchange groups form a 12-element 2-design that fully depolarizes any quantum operation.The full Clifford group is generated from the Hadamard, Pauli, and exchange groups.Source: §III · p. 3
  • Table I lists Clifford decompositions into physical generators or mathematical elements; the operators above its line form a group and 2-design.The mean number of physical generators per Clifford is 1.875.Source: Table I · p. 4, §III · p. 4
  • Unless otherwise specified, simulations use K = 10000 sequences, maximum length Mmax = 4096, and sequence lengths doubling from 1.The fidelity-decay fits use nonlinear least squares and 90% confidence intervals.Source: §III · p. 4
  • RB accuracy measures multiplicative deviations between estimated and true average error rates, while confidence is the size of the fit's 90% interval.The accuracy definition uses logarithms to weight multiplicative deviations symmetrically.Source: §III · p. 4

IV. SRB: MARKOVIAN ERRORS

The Markovian-error analysis uses unitary, random CPTP, and amplitude-damping maps, with diamond-norm distance from an equal-error depolarizing channel used as a predictor of RB performance. Unitary channels are farthest from that channel and are treated as a worst-case test.

IV. SRB: MARKOVIAN ERRORS

  • The analysis treats Markovian errors as history-independent and considers miscalibration, amplitude damping, and fast control-field fluctuations as examples.The authors use diamond-norm distance from an equal-error depolarizing channel to predict benchmarking performance.Source: §IV · p. 4
  • Random CPTP maps are generated by interpreting matrix elements of a random unitary map as Kraus operators; random unitary and amplitude-damping maps are also compared.Maps are evaluated at matched average error rates.Source: §IV · pp. 4–5
  • Unitary channels were farthest from, and amplitude-damping channels closest to, same-error depolarizing channels; unitary errors were therefore used as an adequate worst-case SRB test.Random maps typically fell between those extremes; distance was measured with the diamond norm.Source: Figure 1 · p. 5, §IV · p. 5

A. Finite Sampling Effects

The finite-sampling analysis finds that fewer than 100 sequences can yield error-rate estimates within a factor of two, far below the Hoeffding bound’s per-point trial estimate. It derives confidence intervals by linearizing the fidelity-decay model around its least-squares solution.

A. Finite Sampling Effects

  • The Hoeffding estimate can require K = 10^8 trials per data point, while least-squares estimation uses information from all data points to estimate α and r.The bound assumes i.i.d. trials at each sequence length.Source: A. Finite Sampling Effects · p. 5
  • In numerical analysis of various time-independent Markovian error models, essentially identical results across all models led the authors to present the gate- and time-independent unitary-error case.Fig. 2a plots the size of the confidence interval C at 90% confidence for parameter α at r = 10−4 and r = 10−3.Source: A. Finite Sampling Effects · p. 5
  • For K ∼10–100, confidence intervals are on the order of underlying errors, and the estimated r converges within a factor of two of true r.The plotted accuracy reaches within a factor of two by K = 100.Source: Figure 2 · p. 5, A. Finite Sampling Effects · p. 5
  • Confidence intervals are obtained by linearizing the nonlinear fidelity-decay model around its least-squares solution, assuming normal noise with variance σ²/K and length-invariant σ.The fidelity-decay model is assumed to describe the data accurately.Source: A. Finite Sampling Effects · pp. 5–6

B. Results and Discussions

Across the tested Markovian noise models, standard randomized benchmarking estimated error rates within a factor of two of their actual averages. Amplitude damping was estimated best, while generator-dependent noise was estimated poorest.

B. Results and Discussions

  • The tested models included gate-dependent and fixed random unitaries, generator-dependent unitaries, amplitude damping, fast Gaussian noise, and slow drift.Generator-dependent errors were assigned to decomposed X and Y generators; the decomposition was non-unique, but results did not depend on its choice.Source: B. Results and Discussions · pp. 6–7
  • Slow drift used a fixed unitary whose error rate increased linearly from r/2 to 3r/2 over K sequences.The drift model otherwise followed the fixed-unitary case.Source: B. Results and Discussions · p. 7
  • Generator-dependent noise was estimated poorest; gate-to-gate error-rate variation may cause deviations from exponential fidelity decay and larger confidence intervals.The proposed explanation is qualified as possible, not established.Source: B. Results and Discussions · p. 7
  • Across all tested Markovian models, SRB estimated error rates within a factor of two of the actual average rates.Amplitude damping performed best; the figure reports estimates within 25% of r for all models except generator-dependent unitaries, which were within 50%.Source: Figure 3 · p. 7, B. Results and Discussions · p. 7

V. SRB: 1/f NOISE

The section models 1/f noise using random telegraph signals and reports randomized-benchmarking accuracy across noise conditions. Markovian-error estimates were generally more accurate than non-Markovian estimates.

V. SRB: 1/f NOISE

  • The noise is characterized by a power spectral density S(f) = A/f, with A a constant.The model is introduced for a one-qubit system subject to semiclassical phase noise.Source: §V · p. 7
  • The 1/f model sums random telegraph noise signals with different switching rates, using a switching-rate density proportional to 1/f over a frequency interval.The resulting spectrum is proportional to 1/f when the frequency lies well between the cutoffs.Source: §V · pp. 7–8
  • Markovian-error SRB accuracies were generally about an order of magnitude better than non-Markovian-error accuracies.Precision and fit confidence were not significantly different between noise types, except that SRB performed especially well with amplitude damping.Source: §V · p. 8

A. Simulated Ramsey Experiments

Simulated Ramsey experiments show that the 1/f-noise model produces Gaussian coherence decay. When T* exceeds the gate time, its corresponding gate fidelity is higher than for stochastic dephasing with the same T*.

A. Simulated Ramsey Experiments

  • Each noise realization uses 50 random-telegraph signals, with cutoffs fmin = (10NΔt)^−1 and fmax = (2Δt)^−1.Here Δt is the smallest simulation time step and N is the total number of time steps in a realization.Source: A. Simulated Ramsey Experiments · p. 8
  • Averaging 2,000 noise realizations produces coherence with Gaussian decay, σ = e^−(t/T*)^2.The simulated Ramsey experiments verify this decay for several noise powers.Source: A. Simulated Ramsey Experiments · p. 8
  • The noisy identity-gate fidelity is Fg = (2 + σ(tg))/3, with gate time tg set to 20Δt.The coherence is evaluated at the gate time.Source: A. Simulated Ramsey Experiments · p. 8
  • When T* > tg, 1/f noise yields higher gate fidelity than stochastic dephasing with the same T*.The figure compares the extracted 1/f gate fidelities with the fidelity for stochastic noise having exponential coherence decay.Source: A. Simulated Ramsey Experiments · p. 8, Figure 4 · p. 8

B. Results and Discussion

Random Clifford gates slow fidelity decay under 1/f noise, while RB estimates average error rates across the tested cases with useful accuracy. For 1/f noise, however, the exponential decay fit becomes inaccurate and further sampling eventually stops improving accuracy.

  • The 1/f-noise simulations preserve correlations across the full RB sequence, whose low-frequency components make phase-noise samples jointly dependent.Source: B. Results and Discussion · p. 9
  • Random Clifford gates dramatically slow the rapid Gaussian fidelity decay experienced by a qubit under 1/f phase noise.The comparison is between random Clifford gates and identity gates.Source: B. Results and Discussion · p. 9, Figure 5 · p. 9
  • For fixed T*2, 1/f noise has higher gate fidelity than noise with exponential Ramsey decay, potentially because Clifford twirling has a depolarizing effect.The proposed mechanism is that Clifford gates twirl the 1/f noise.Source: B. Results and Discussion · p. 9
  • Leakage estimates were nearly always within a factor of two, while 1/f-noise estimates were within 25% of r in all tested cases.The leakage types showed no significant difference except at the largest error rate, where random leakage performed better on accuracy, precision, and fit confidence.Source: Figure 6 · p. 9
  • For all tested cases, RB estimates average error rates within a factor of two when K is about 100 or greater.The reported comparison is summarized in Fig. 6.Source: B. Results and Discussion · p. 9, Figure 6 · p. 9
  • With 1/f noise, confidence intervals saturate as K increases because the exponential decay model is imperfect, so additional samples eventually stop improving accuracy.The paper identifies a Kmax beyond which increasing sample size does not yield a more accurate estimate.Source: B. Results and Discussion · p. 10

VI. SRB: LEAKAGE

Leakage can create coherences in higher levels, producing non-Markovian dynamics and collapse-and-revival behavior in the qubit subspace. The paper models the resulting fidelity decay and reports that RB captures the underlying error for both tested leakage models.

  • Leakage can build coherences in higher levels, leading to non-Markovian dynamics with multiple collapse-and-revival processes in the qubit subspace.Source: §VI · p. 10
  • The leakage analysis represents the qutrit dynamics using an orthogonal basis that separates qubit-subspace population and correlations from leakage-related terms.The basis also includes population inversion in the third level and correlations between the qubit and third level.Source: §VI · p. 10
  • Adding both ±π/2 and ±3π/2 rotations to the generating set makes the leakage map block diagonal and permits the stated fidelity-decay model.Source: §VI · p. 11
  • For unital leakage noise, the modeled fidelity decays asymptotically to 1/3; without leakage, it decays to 1/2.The model identifies the asymptotic fidelity as an indicator of the noise type.Source: §VI · p. 11
  • With random leakage noise, RB fidelity decays to 1/3 rather than 1/2, and both tested leakage models capture the underlying error.The estimated and actual errors agree by about 100 samples.Source: Figure 8 · p. 10, §VI · p. 11

VII. INTERLEAVED RB: MARKOVIAN ERRORS

For Markovian errors, IRB estimates are within a factor of two when the interleaved error is at least the average Clifford error, but become less accurate when it is much smaller. The paper also finds that averaging more unitary channels brings the average noise closer to depolarizing and notes numerical investigation of RB applications.

VII. INTERLEAVED RB: MARKOVIAN ERRORS

  • As the average includes more unitary channels, its diamond-norm distance from a depolarizing channel with the same average fidelity decreases.Source: §VII · p. 11, Figure 10 · p. 12
  • IRB estimates the interleaved error within a factor of two when rint ≥ r, but becomes less accurate as rint falls well below r and begins to diverge around rint/r = 0.1.Source: Figure 11 · p. 12, §VII · p. 12

VIII. CONCLUSION

  • The conclusion states that the paper reviewed RB protocols and numerically investigated their application.Source: §VIII · p. 12
Loading 1308.2928v1…