Randomized Benchmarking with Confidence

Joel J. Wallman; Steven T. Flammia

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Randomized Benchmarking with Confidence

Joel J. Wallman, Steven T. Flammia

arXiv:1404.6025v4Apr 24, 2014quant-ph

TL;DR

Randomized benchmarking needs rigorous evidence that its estimates remain precise across realistic noise and experimental sampling constraints. The paper analyzes sampling variance for standard and interleaved protocols, derives confidence bounds, and extends the method to time-dependent Markovian noise. It proves that estimates are generally highly precise for arbitrary Markovian noise, while identifying regimes where variance remains large and non-Markovianity can be detected.

Problem
Characterizing quantum-gate accuracy at scale is difficult because process tomography is exponentially costly and sensitive to SPAM errors, while randomized benchmarking requires rigorous sampling-precision guarantees.
Method
The paper derives variance bounds for randomized benchmarking under Markovian noise, including qubit-specific improvements, and analyzes time-dependent noise and confidence intervals.
Results
100 random sequences of 100 gates yield error below 0.9% with 99% confidence for single-qubit noise with average error rate 10^-4.
Takeaways & Limitations
Randomized benchmarking can be performed accurately with relatively few sequences and can characterize time-dependent Markovian noise while providing an indicator for non-Markovianity.
Takeaways & Limitations
Variance can remain constant or decay only arbitrarily slowly for some nonunital or unitary noise, making substantially more sequences necessary.

Abstract

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Randomized benchmarking is a promising tool for characterizing the noise in experimental implementations of quantum systems. In this paper, we prove that the estimates produced by randomized benchmarking (both standard and interleaved) for arbitrary Markovian noise sources are remarkably precise by showing that the variance due to sampling random gate sequences is small. We discuss how to choose experimental parameters, in particular the number and lengths of random sequences, in order to characterize average gate errors with rigorous confidence bounds. We also show that randomized benchmarking can be used to reliably characterize time-dependent Markovian noise (e.g., when noise is due to a magnetic field with fluctuating strength). Moreover, we identify a necessary property for time-dependent noise that is violated by some sources of non-Markovian noise, which provides a test for non-Markovianity.

I. INTRODUCTION

The paper develops a rigorous analysis of randomized benchmarking that addresses scalability, SPAM insensitivity, sampling precision, and time-dependent Markovian noise. It provides confidence bounds and tests for non-Markovianity while identifying qubit-specific and noise-dependent scope boundaries.

Full quantum process tomography is exponentially costly for general noise and is sensitive to state-preparation and measurement errors.
Randomized benchmarking efficiently provides partial characterizations such as average and worst-case gate error rates, while avoiding SPAM sensitivity.
The method extends randomized benchmarking to time-dependent average gate fidelities and uses parameters exceeding 1 as a sufficient indicator of non-Markovian noise under Markovian assumptions.
The analysis bounds sampling variance, showing that a small number of random sequences can yield high-precision estimates and rigorous confidence intervals.
100 random sequences of 100 gates give error below 0.9% with 99% confidence for single-qubit noise with average error rate 10^-4.
For qubits, improved variance bounds justify substantially fewer sequences, while special Pauli-diagonal noise permits a further improvement independent of preparations and measurements.

IV. ANALYZING DATA FROM RANDOMIZED BENCHMARKING WITH FINITE SAMPLING

This section derives confidence intervals for finite-sampling estimates in randomized benchmarking and extends the analysis to time-dependent noise. It also identifies how sequence length, noise structure, and temporal correlations affect sampling requirements and non-Markovianity tests.

A. Confidence interval for randomized benchmarking: Finite-sampling confidence intervals quantify the uncertainty in ˆF_m while separating sequence-sampling variance from finite-measurement fluctuations.The analysis treats variance from sampling K_m random sequences; expectation-value measurement fluctuations require separate statistical analysis.
A. Confidence interval for randomized benchmarking: 4 × 10^-4 bounds the qubit variance for m ≈ 100 and r ≈ 10^-4, substantially improving previous rigorous sequence-count estimates.The bound is comparable to numerical estimates and enables confidence intervals closer to experimental standard errors.
A. Confidence interval for randomized benchmarking: 145 random sequences suffice for the stated precision bound, compared with 10^5 required by the variance-independent Hoeffding estimate.The comparison uses the bound 4mr^2 while ignoring higher-order terms.
A. Confidence interval for randomized benchmarking: Quadratic scaling with m appears necessary when mr ≪ 1, so longer sequence lengths should be averaged over more random sequences.Even Pauli-diagonal noise requires K_m to scale linearly with m to keep variance independent of sequence length.
A. Confidence interval for randomized benchmarking: Nonunital and unitary noise can produce constant or slowly decaying sequence-sampling variance, making randomized benchmarking most reliable when mr ≪ 1.The lowest-order bounds may remain approximately valid near mr ≈ 0.1.
B. Characterizing time-dependent noise: Time-dependent gate fidelity can be estimated when gate dependence is negligible and the time-dependent noise is identically distributed across experiments.The required number of random sequences will typically exceed that for time-independent noise.
B. Characterizing time-dependent noise: Negative estimated time-dependent infidelities indicate temporal correlations and therefore non-Markovian behavior when noise has no temporal correlations otherwise.The condition follows because the parameters r_t are then lower-bounded by zero.

V. MATHEMATICAL PRELIMINARIES

The paper introduces representation-theoretic and Liouville-representation tools for analyzing randomized benchmarking. These tools express states, measurements, and channels as vectors or matrices and exploit group twirling and irreducibility.

Representation theory: Randomized benchmarking composes randomly conjugated noise channels, motivating the use of group representations and channel-composition structure.The group twirl averages conjugated operators over the benchmarking group.
Representation theory: Irreducible representations decompose into invariant subspaces, with Schur bases providing a basis for that direct-sum decomposition.The trivial representation can also occur as a subrepresentation of tensor powers.
Liouville representation: POVM effects are represented as row vectors, making measurement probabilities equal to the vector pairing (E|ρ).States and measurement effects can therefore be treated on the same footing as channels.
Liouville representation: The Liouville representation expands density operators in an orthonormal operator basis and identifies them with column vectors resembling generalized Bloch vectors.The basis uses A_0 = 1/d and traceless remaining basis operators.

2. Transformations

This section restricts attention to states, measurements, and CPTP maps, then uses their Liouville matrices to represent channel transformations and randomized-benchmarking quantities.

2. Transformations: The analysis considers quantum channels that are states, measurements, or CPTP maps from D_d to D_d.Trace-reducing and dimension-changing channels are excluded because they require additional notation and are not used.
2. Transformations: A channel maps one density operator to another and acts linearly on the corresponding Liouville vector.The same symbol denotes the abstract channel and its matrix representation.
2. Transformations: A channel is unital exactly when α(E) = 0, so ∥α(E)∥ quantifies its nonunitality.The quantity α(E) is used as a measure of deviation from unitality.
2. Transformations: Randomized benchmarking uses group representations and their irreducibility to make group averaging analytically tractable and efficiently sampleable.Unitary 2-designs supply the needed irreducible structure without requiring the full unitary group.

3. Properties of channels in the Liouville representation

This section develops norm and positivity properties of Liouville-represented quantum channels. The results provide bounds and structural facts used in the paper’s randomized-benchmarking analysis.

3. Properties of channels in the Liouville representation: The spectral radius of a matrix is bounded by its spectral norm, supporting norm-based bounds for valid quantum channels.The spectral norm is the largest singular value.
3. Properties of channels in the Liouville representation: The adjoint of a completely positive map is completely positive, as shown using Kraus representations and Choi’s theorem.Kraus operators for the adjoint can be chosen as adjoints of the original Kraus operators.
3. Properties of channels in the Liouville representation: The adjoint of a unital CPTP channel is also unital and CPTP.This fact is used to establish improved norm bounds for unital channels.
3. Properties of channels in the Liouville representation: For CPTP channels, the stated norm relations hold, with equality in inequality (i) exactly for unitary channels.For unital channels, the spectral norm bound improves to ∥E∥_∞ = 1.

C. Representing noisy channels

Noisy implementations can be represented relative to a target unitary either by pre-multiplying or post-multiplying noise. Because these representations are related by conjugation, the paper fixes a convention for analysis.

Representing noisy channels: A noisy channel E implementing a target unitary U can be written as E = Λpost(U)U or E = UΛpre(U).The two noise operators are related by Λpre(U) = U†Λpost(U)U.
Representing noisy channels: The apparent noise operator depends on whether noise is represented before or after the target unitary.This distinction is trivial for a single channel but can cause confusion when comparing channels.
Target dependence: Randomized benchmarking requires the relevant noise representation to be approximately independent of the target gate.In general, at most one of Λpost or Λpre has this property, depending on the physical implementation.
Convention: The paper uses pre-multiplying noise so the residual unaveraged noise occurs at the first time step and remains independent of sequence length.The same results can be derived for post-multiplying noise with small modifications.

D. Measures of noise

The paper distinguishes average gate fidelity from worst-case channel error, using the diamond distance as an operational worst-case measure. It then relates these quantities through Choi-matrix bounds.

Noise measures: Average gate fidelity and diamond distance are channel-level measures promoted from state fidelity and trace distance.The average gate fidelity compares a channel with a unitary, while the diamond distance compares two channels.
Diamond distance: The diamond distance is stable under larger entangled inputs and is bounded between 0 and 1 by the factor of 1/2 in its definition.The diamond norm extends to Hermiticity-preserving linear maps.
Diamond distance: The diamond distance has an operational meaning as the optimal success probability for distinguishing two unknown channels.It is also the worst-case error rate between the channels.
Choi-matrix bounds: The paper bounds the diamond norm using the trace norm of the corresponding Choi matrix.The Choi matrix is formed by applying the map to one half of a maximally entangled state.

VI. TIME-DEPENDENT GATE-INDEPENDENT ERRORS IN RANDOMIZED BENCHMARKING

For time-dependent gate-independent noise, randomized benchmarking samples a distribution of sequence outcomes whose mean and variance can be analyzed across sequence lengths. The variance behavior depends on whether the noise is unital or non-unital.

Protocol assumptions: The analysis assumes ideal gates sampled from a unitary 2-design and noise that depends only on the time step.The gate sequence is reparameterized using the cumulative products h_t.
Variance behavior: For unital but nonunitary noise, σ²_m decreases exponentially with sequence length m.This characterizes the sampling variance of the randomized benchmarking distribution.
Variance behavior: For non-unital noise, σ²_m converges to a constant determined by the strength of the non-unitality.The paper also derives upper bounds for small m.
Experimental implications: The variance bounds enable rigorous confidence intervals for estimates of average gate infidelity.The analysis is also shown to remain stable under gate-dependent perturbations in the noise.
Protocol scope: The treatment explicitly focuses on standard randomized benchmarking, while interleaved benchmarking can be handled with nearly identical modifications.For interleaved benchmarking, the noise is conjugated by the interleaved gate and absorbs its noise term.
Benchmarking distribution: Randomized benchmarking estimates the mean of the outcome distribution obtained by sampling random sequences at fixed length m.The sequence outcomes are treated as realizations of a random variable with mean F̄_m and variance σ²_m.

A. Mean of the benchmarking distribution

The mean randomized-benchmarking signal is derived for general CPTP maps with time-dependent noise and connected to experimentally relevant error quantities. In particular, it supports estimation of time-averaged average gate infidelity and bounds on worst-case error.

Mean signal: The paper derives the mean of the randomized-benchmarking distribution for general CPTP maps with time-dependent noise.The derivation targets SPAM error, average time-dependent gate fidelity, and worst-case noise error.
SPAM characterization: The parameters E_0ρ_0 and E⃗ρ characterize the quality of state preparation and measurement procedures.The state ρ is redefined to include a noise term, and Tr Eρ = E_0ρ_0 + E⃗ρ.
Average error: The parameters determining the mean are connected to the average gate infidelity, which characterizes the average error rate.This links the randomized-benchmarking decay parameters to an operational noise measure.
Time-dependent estimation: Randomized benchmarking estimates the product of the time-dependent parameters f_t, enabling average gate infidelity estimation over arbitrary time intervals.The interval-averaged estimate is described in Section IV B.
Worst-case error: Average gate infidelity supplies upper and lower bounds on half the diamond distance from the identity channel.The diamond distance represents the worst-case error introduced by using Λ instead of the identity channel.
Worst-case error: The Choi-matrix relation converts average gate fidelity into bounds on the diamond norm through trace-norm inequalities.The proof uses the maximally entangled state and the relation between its fidelity and average gate infidelity.

B. Upper bounds on the variance

This section derives upper bounds on randomized-benchmarking variance from sequence sampling, with substantially sharper results for qubits and Pauli-diagonal noise.

The analysis bounds variance from sampling random sequences for time- and gate-independent noise in d-level systems.
The sampling standard error can scale as 0.1K^-1/2 when mr ≈10^-2, so Theorem 10 alone does not justify many single-qubit experimental sequence lengths.
For qubits, representation-theoretic structure and a bound on nonunital noise substantially reduce the required number of sampled sequences.The justified choice is approximately K_m ≈145 rather than K_m ≈7 × 10^4 from earlier estimates.
For Pauli-diagonal unital noise, the variance bound improves by a factor of m and becomes independent of state preparation and measurement.
The variance depends on the chosen 2-design basis, with order mr^2 in a diagonalizing basis versus order m^2r^2 in other bases.

VII. ASYMPTOTIC VARIANCE OF RANDOMIZED BENCHMARKING

The section studies long-sequence behavior of randomized-benchmarking variance using n-contractive channels and establishes exponential decay toward a nonunitality-dependent constant.

For n-contractive channels, the variance due to random-sequence sampling decays exponentially with sequence length.
The variance converges exponentially to zero for unital noise and to a positive constant proportional to nonunitality for nonunital noise.
No general bound on the decay rate is available without further assumptions, because eigenvalues can approach 1 arbitrarily closely.
A channel is n-contractive when the averaged n-fold channel has at most one eigenvalue of modulus 1.
For unital, nonunitary channels, 2-contractivity holds exactly for unitary 2-designs.

VIII. STABILITY UNDER GATE-DEPENDENT PERTURBATIONS

This section shows that randomized-benchmarking variance remains stable under sufficiently small gate-dependent noise perturbations, treated perturbatively around gate-independent noise.

Gate-dependent noise is modeled perturbatively around time-dependent, gate-independent channels with bounded perturbations.
The variance changes by a bounded correction under gate-dependent perturbations.
The correction is at most δ0 when the gate-dependent noise satisfies the theorem’s stated condition.
The bound is obtained by expressing sequence probabilities as averages over all gate sequences and controlling perturbations with norm inequalities.
The resulting estimate is acknowledged to be improvable, including by introducing a factor of at least the average infidelity r.

IX. CONCLUSION

The paper concludes that randomized benchmarking is accurate with few sequences for arbitrary Markovian noise, extends to time-dependent and interleaved settings, and exposes boundaries for higher-dimensional systems.

Randomized benchmarking can reliably characterize time-dependent Markovian noise and provide an indicator of non-Markovianity over long timescales.
For qubits, the variance is at most 4 × 10^-4 at currently achievable noise levels, supporting approximately 145 random sequences.
The number of sequences K_m should scale with sequence length m to keep variance independent of m, especially for near-unitary noise in the regime mr ≪1.
The unexpectedly good accuracy follows from the small variance of random-sequence sampling, rather than only from fitting data across sequence lengths.
Interleaved benchmarking is essentially as accurate as standard benchmarking when noise is Markovian and approximately gate-independent.
Higher-dimensional systems and multiple qubits require many more sequences for confidence levels currently justified rigorously by the weaker qudit bound.