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Efficient estimation of Pauli channels

Steven T. Flammia, Joel J. Wallman

arXiv:1907.12976v3quant-ph

TL;DR

Efficient estimation of Pauli channels is needed because these models are central to quantum error correction while biased and correlated noise can alter threshold behavior. The paper develops randomized-benchmarking-based procedures for complete channels, arbitrary error subsets, and bounded-degree correlated channels, supporting noise-tailored codes, decoders, and fault-tolerance methods.

  • Problem

    Few systematic methods efficiently estimate Pauli channels beyond the complexity of naive full channel tomography, despite their central role in quantum error correction and fault tolerance.

  • Method

    The paper develops randomized-benchmarking procedures for complete Pauli channels, arbitrary error subsets, and channels factorized over bounded-degree factor graphs.

  • Results

    The procedures provide relative-precision estimation for complete channels and arbitrary error subsets, and polynomial-time reconstruction for nice channels with bounded-degree correlations and positive marginals.

  • Takeaways & Limitations

    The methods support learning residual Pauli noise and tailoring codes, decoders, fault-tolerance schemes, and compiling to device-specific errors.

Abstract

from arXiv · show

Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here we prove several results on efficiently learning Pauli channels, and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on $n$ qubits with high probability to a relative precision $ε$ using $O\bigl(ε^{-2} n 2^n\bigr)$ measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measurement errors which, together with the relative precision, makes it especially appropriate for applications involving characterization of high-accuracy quantum gates. Next we show that the error rates for an arbitrary set of $s$ Pauli errors can be estimated to a relative precision $ε$ using $O\bigl(ε^{-4} \log s\log s/ε\bigr)$ measurements. Finally, we show that when the Pauli channel is given by a Markov field with at most $k$-local correlations, we can learn an entire $n$-qubit Pauli channel to relative precision $ε$ with only $O_k\bigl(ε^{-2} n^2 \log n \bigr)$ measurements, which is efficient in the number of qubits. These results enable a host of applications beyond just characterizing noise in a large-scale quantum system: they pave the way to tailoring quantum codes, optimizing decoders, and customizing fault tolerance procedures to suit a particular device.

I. INTRODUCTION

The paper develops efficient procedures for estimating Pauli channels across three complementary regimes: complete channels, arbitrary error subsets, and bounded-degree correlated channels. The procedures use randomized-benchmarking variants, provide relative-precision guarantees, and support noise-tailored error correction and fault tolerance.

  • Motivation: Randomized compiling motivates estimating Pauli noise because it maps general noise to a Pauli projection with the same average fidelity.The paper’s procedures target residual noise after randomized compiling and tolerate specified state-preparation and measurement errors.
  • Motivation: Pauli channels underpin much quantum error-correction and fault-tolerance analysis, while biased or correlated noise can substantially change threshold estimates.The introduction notes threshold changes exceeding a factor of 4 in some code-capacity settings.
  • Complete-channel estimation: The complete-channel procedure estimates all 4^n Pauli error probabilities using randomized Pauli gates and full n-bit measurements.Relative precision and SPAM robustness are emphasized for characterizing very low error rates.
  • Subset estimation: An arbitrary set E of s Pauli errors can be estimated with O(ε^-4 log(s) log(s/ε^2)) measurements.The method estimates a subnormalized distribution over E and uses randomized sampling.
  • Correlated-channel estimation: A bounded-degree factor graph with positive marginals permits polynomial-time reconstruction of a nice n-qubit Pauli channel.The factor graph topology is assumed known, and the reconstructed estimate is represented as a tensor network.

B. Proof techniques

The proof strategy extends randomized benchmarking from estimating one average error-rate parameter to reconstructing Pauli error information. For correlated channels, it combines marginal estimation with factor-graph reconstruction.

  • Randomized benchmarking: The work studies randomized benchmarking over the Pauli group rather than only the unitary or Clifford groups.This extension is needed because Pauli-channel learning involves many parameters rather than a single decay parameter.
  • Randomized benchmarking: The proof extends relative-precision estimation from one average error rate to the 4^n parameters of a Pauli channel.The method must isolate multiple exponential decays associated with the Pauli-group setting.
  • Factor-graph reconstruction: For Markov random-field channels, the procedure estimates marginal channels on factor-graph factors and rounds them into a global distribution using the Hammersley-Clifford theorem.The factor graph is assumed known and has bounded-degree correlations.

C. Comparison with Prior Work

The paper contrasts its relative-precision, SPAM-robust procedures with prior tomography methods that mainly provide additive estimates and are not inherently SPAM-robust. It also positions the results as scalable beyond earlier demonstrations and distinguishes the learning problem from conventional Markov-field settings.

  • Comparison with prior work: Results 2 and 3 give recovery guarantees for broad non-trivial quantum-channel classes with scaling efficient in n when s = poly(n) or k = O(1).The paper identifies these as the first such guarantees known in those regimes.
  • Comparison with prior work: Simulations reach 100 qubits and a method variant was implemented on a publicly accessible 14-qubit device, compared with a previous 3-qubit tomography record.The authors describe this as a qualitative shift in quantum-device characterization.
  • Comparison with prior work: Prior tomographic techniques provide additive rather than relative-precision estimates and are not robust to SPAM errors.SPAM errors create an unknown systematic error and their magnitude and complexity scale with system size.
  • Comparison with prior work: Randomized benchmarking techniques provide multiplicative precision and robustness to SPAM errors, unlike the cited tomographic approaches.Randomized benchmarking tomography can reconstruct channels, but its additive error scales as O(d/√log d).
  • Comparison with prior work: Learning a symplectic Markov field differs inherently from previously considered tasks because commuting observables cannot reveal all variables simultaneously.Some variables are observable only to selected queries, while probing one variable can hide others.

II. MATHEMATICAL PRELIMINARIES

The preliminaries encode n-qubit Pauli operators as binary strings with a symplectic commutation structure, then use stabilizer groups and coverings to organize compatible measurements. A covering can be bounded by the number of logical degrees of freedom, with an unconditional bound of 2^n + 1.

  • Pauli-group representation: The phase-quotiented n-qubit Pauli group is Abelian and represented by bit strings of length 2n.The quotient removes the group center, enabling binary labeling of Pauli operators.
  • Commutation structure: The binary symplectic form ⟨a,b⟩ = a_x · b_z + a_z · b_x mod 2 determines Pauli commutation.A stabilizer group is a subspace whose elements have zero symplectic pairing and therefore commute.
  • Commutation structure: A commutant consists of Pauli elements symplectically orthogonal to a set, while the anti-commutant is the quotient by that commutant.The double commutant returns the original group when the set is a group.
  • Stabilizer coverings: A stabilizer covering is a collection of stabilizer groups whose union contains a target error set X.The target need not itself be a group, such as errors of weight at most w.
  • Stabilizer coverings: A stabilizer covering of any X can satisfy |O| ≤ 2^n + 1, with tighter bounds determined by the logical-qubit structure of ⟨X⟩.When X is a group, mutually unbiased bases achieve the minimal covering bound described in the passage.

A. States and measurements

The paper constructs stabilizer states and syndrome measurements from commuting Pauli subgroups. States are indexed by error syndromes, while the associated measurement operators form orthogonal projectors.

  • States and measurements: For each Pauli label and stabilizer group, the construction defines a valid stabilizer state.States with labels differing by an element of the stabilizer commutant are identical and can be indexed by syndromes.
  • States and measurements: Stabilizer states are uniquely labeled by error syndromes in the anti-commutant of the stabilizer group.This quotient-like labeling removes the redundancy from commuting stabilizer elements.
  • States and measurements: The corresponding syndrome measurement is a valid measurement whose elements are orthogonal projectors.When the stabilizer group has dimension n, the state is pure and the measurement is rank-1 projective.

B. Quantum channels

The channel preliminaries represent Pauli errors, channel eigenvalues, and error rates in compatible algebraic forms, using Walsh-Hadamard transforms to move between them. Pauli twirling reduces a general linear map to a Pauli channel whose rates are determined by Kraus-operator coefficients.

  • Quantum channels: A general linear map is represented through Kraus operators, with completely positive maps admitting nonnegative Pauli error rates.Effective negative rates can arise with initial correlations even though ordinary CP-map rates are nonnegative.
  • Assumptions: The analysis focuses on channels close to identity whose relevant eigenvalues lie in a small strictly positive interval [ε, 1].For completely positive channels, the other eigenvalues lie within a bounded interval determined by the channel parameters.
  • Pauli twirling: Pauli twirling projects a general channel onto a Pauli channel, with twirled error rates determined directly by the original map’s Kraus-operator coefficients.The paper derives this relationship by expanding Kraus operators in the Pauli basis.
  • Pauli-channel representation: Any Pauli channel has a superoperator representation in the Pauli basis, where Pauli error rates and channel eigenvalues provide dual descriptions.The error-rate vector p is related to Pauli eigenvalues through a Walsh-Hadamard transform.
  • Pauli-channel representation: The estimation strategy infers r and then recovers p by applying the inverse transform p = W^-1(1 − r).The transform can be evaluated using the stated algebraic lemmas.
  • Reconstruction quality: Reconstruction quality can be measured by a norm on error-rate vectors or by channel figures of merit such as average gate infidelity and diamond distance.The authors may project reconstructions onto probability or subnormalized-probability distributions, using the 2-norm when uniqueness is needed.

C. Model assumptions

The procedure assumes gate-independent, time-stationary, Markovian noise, together with sufficiently weak noise and stable SPAM parameters. These assumptions make the noisy operations analyzable through a fixed channel model and bounded signals.

  • Noise model: GTM noise is gate-independent, time-stationary, and Markovian, with each noisy Pauli implementation written as PΛ.State preparations and measurements are fixed density operators and POVMs.
  • Scope: The assumptions are standard in randomized-benchmarking analyses and may be relaxed with sufficient effort, but here are applied to tensor-product groups of single-qubit channels.The scope of the stated assumption is narrower than analyses involving general multiqubit circuits.
  • Weakness and stability: The analysis assumes noise and SPAM are sufficiently weak to support the ratio estimators used later.Weakness and stability are defined through operator-norm proximity and lower bounds on SPAM parameters.
  • Weakness and stability: The c-weak condition requires the Pauli-twirled noise map to be close to identity, equivalently keeping its Pauli eigenvalues in [1−c, 1].A sufficient condition is an identity-error probability p0 at least 1−c.

III. ESTIMATION PROCEDURE AND SAMPLE COMPLEXITY

The estimation procedure generalizes cycle benchmarking by combining stabilizer-state preparation, randomized Pauli sequences, and syndrome measurements to estimate Pauli eigenvalues under GTM noise. Its analysis converts exponentially many decay parameters into tractable estimators with explicit concentration bounds.

  • Protocol: General stabilizer-group measurements extract more information per experiment than single-Pauli measurements while encompassing several benchmarking protocols.Character benchmarking, direct randomized benchmarking, and cycle benchmarking appear as special cases.
  • Protocol: The protocol prepares a stabilizer state, applies uniformly random Pauli gates for m rounds, and records a syndrome-measurement outcome.The procedure uses noisy implementations of these primitives in experiments.
  • Signal model: Under GTM noise, the measurement distribution depends on SPAM coefficients and mth powers of the channel’s Pauli eigenvalues.This characterizes the output of one RunCB(G, H, m) call.
  • Estimation: The method avoids a direct multi-exponential fit by transforming the output into a single-exponential estimation problem.This is important because the original model contains exponentially many parameters.
  • Estimation: The V estimator averages bounded, correlated binomial variables whose covariance can be computed exactly, enabling concentration bounds for estimating A_j f_j^m.A union-bound and Hoeffding analysis supplies a conservative failure-probability bound.

IV. DATA FITTING AND ERROR ANALYSIS

The fitting strategy uses exponentially increasing sequence lengths and endpoint ratios to estimate Pauli eigenvalues multiplicatively. Under weak-noise and stable-SPAM assumptions, it provides explicit precision, failure-probability, and sequence-length controls.

  • Ratio fitting: The ratio estimator uses sequence lengths that increase exponentially and retains endpoint data to estimate decay ratios.Discarding intermediate data simplifies analysis because endpoints are more sensitive to the decay.
  • Guarantees: Pointwise multiplicative precision follows when the noise is 1/2-weak and the SPAM parameters are 1/2-stable.The assumptions ensure the relevant signals remain sufficiently bounded for the ratio estimator.
  • Assumptions: Removing the 1/2-stability assumption is possible, but then sample-complexity statements depend explicitly on the SPAM parameters.The paper retains stability to keep the complexity expression transparent.
  • Guarantees: The estimator achieves relative error O(ε) with failure probability controlled by a union bound over sequence lengths and estimated eigenvalues.The proof combines concentration bounds with the ratio construction and Taylor expansion.
  • Sequence lengths: The required maximum sequence length scales as Θ(1/r), where r is the relevant infidelity, and the number of tested lengths grows logarithmically with that maximum.The sequence-length set is κ = {2^i} through the largest required endpoint.

V. RECONSTRUCTING ERROR RATES FOR A GROUP

Estimated Pauli eigenvalues can be transformed into error-rate distributions for a subgroup, with guarantees inherited from the eigenvalue estimator. The reconstruction is generally coarse-grained because indistinguishable Pauli errors produce identical measurement signatures.

  • Reconstruction: Estimating the Pauli eigenvalues of a group G enables reconstruction of a marginal error-rate distribution over that group.The reconstruction uses the linear relation between eigenvalues and Pauli error probabilities.
  • Identifiability: Pauli errors differing by an element of the commutant cannot be distinguished from estimated eigenvalues alone, so only a coarse-grained distribution is identifiable.The indistinguishable probabilities are grouped into marginals over quotient classes.
  • Decoding application: When G contains a code’s stabilizers and logical operators, the resulting marginals give syndrome-conditioned logical-error distributions usable for maximum-likelihood decoding.This connects the reconstruction directly to decoder construction for the device’s noise.
  • Guarantees: Under the proposition’s conditions, the reconstructed distribution inherits O(ε)-relative accuracy with explicit round and measurement bounds.The estimator projects onto the probability simplex to maintain a valid distribution.
  • Full-group case: For the full Pauli group, the method estimates the Pauli projection of a channel with pointwise precision O(ε)(1−p0).Here p0 denotes the probability of no error.

VI. RECONSTRUCTING A SUBSET OF ERRORS

The section develops methods to estimate selected Pauli error probabilities and search for dominant or anomalous errors, avoiding reconstruction of the full exponential distribution. Its tree-based heuristic is useful for identifying sparse sets, but provable approximation guarantees remain unresolved.

  • Estimating selected errors: The procedure estimates probabilities for every error in an arbitrary set E with pointwise relative precision using the Ratio subroutine.The estimator uses randomized sampling and stabilizer coverings to reconstruct the selected probabilities.
  • Estimating selected errors: O(ε^-4 log(s) log(s/ε^2)) measurements estimate a subnormalized distribution over any s Pauli errors.The result applies to any selected set E of size s, with success probability at least 1 − δ under the theorem’s sampling construction.
  • Tree-based search: The search heuristic uses Select to identify error indices satisfying a chosen condition, then combines selected sets through a tree reconstruction procedure.The procedure iterates over logarithmically many stages and uses Choose, Cover, and Ratio as supporting subroutines.
  • Tree-based search: Choosing the s largest probabilities makes TreeReconstruction a heuristic for finding a good s-sparse approximation of the Pauli channel.The paper does not prove that this set is an optimal or near-optimal s-sparse approximation.
  • Limitations: Provable approximation ratios and potentially faster sparse-reconstruction methods are left for future work.The authors note that sparse Fourier or Hadamard techniques might improve runtime, but adaptation to the Pauli group remains open.
  • Applications: The method can target errors that do not fit a background model, including correlated errors beyond independent single-qubit noise.The motivating application is to identify unknown couplings for device re-engineering or compensating pulses.

VII. BOUNDED DEGREE GRAPHICAL MODELS

The paper models Pauli error probabilities as Gibbs random fields on bounded-degree factor graphs, using local marginals and canonical factor potentials to reconstruct the global distribution. Under stated regularity assumptions, this yields an efficiently constructible estimator, while normalization and consistency introduce practical limitations.

  • Graphical-model structure: A factor graph represents dependencies between Pauli variables through factors, with bounded-degree structure limiting each variable's and factor's participation.The example has at most four factors per variable and at most four variables per factor.
  • Graphical-model structure: The local Markov property makes variables conditionally independent of variables outside their Markov blankets.The closure of a set is the set together with its Markov blanket, providing the local context used in estimation.
  • Canonical reconstruction: The reconstruction estimates factor potentials rather than general partition functions, whose computation is generally believed to be hard.Normalization can still be handled in special cases, such as tree factor graphs or when p(0) is sufficiently large for sampling-based estimation.
  • Canonical reconstruction: Canonical factor potentials are determined by marginal probabilities on factors and their Markov-blanket closures.When each closure has constant size, each log factor potential depends on only a small amount of data.
  • Estimator limitations: Independent estimates of raw factors add a constant sample-complexity overhead but may produce mutually inconsistent empirical marginals.A covering-set approach could avoid the overhead but might bias the estimate and complicate the analysis.
  • Estimator guarantees: The estimator can be constructed in polynomial time in n when factor sizes are O(1), apart from the normalizing factor.The result assumes a known factor graph with bounded factor and degree parameters and additional channel regularity conditions.
  • Estimator limitations: Marginal-based bounds can scale as √n, but applying them theoretically is difficult because empirical marginals require estimating or bounding the reconstructed distribution's partition function.Monte Carlo sampling from factor potentials is suggested heuristically for empirical marginals.

VIII. CONCLUSION

The paper concludes that its methods enable high-precision Pauli-channel learning with fewer resources than previous approaches and can support noise characterization for large-scale devices. It also identifies open questions involving lower bounds, broader noise models, reuse of sequences, higher-dimensional systems, and structure learning.

  • Conclusion: The methods learn Pauli channels with high precision using fewer resources than previous methods and may apply to residual noise after randomized compiling.The authors present this as broadly applicable to large-scale quantum-device characterization.
  • Conclusion: Reconstructed noise descriptions could guide engineering toward the most relevant residual errors and support tailored codes, decoders, fault-tolerance schemes, and error-aware compiling.These are envisioned applications rather than demonstrated outcomes in the supplied passage.
  • Open questions: The paper does not prove lower bounds and leaves open whether procedures with better asymptotic scaling exist.This is identified as a direction for future work.
  • Open questions: Future work includes robustness to gate-dependent noise, interleaved-gate estimation, and comparison with circuit-level performance protocols.The passage specifically mentions accreditation-based assessments as a comparison point.
  • Open questions: The convergence guarantees are proved in an idealized single-shot regime, and extensions to reused sequences, d-level systems, and compressed sensing remain open.The paper also uses the qubit Pauli group exclusively in the presented treatment.
  • Open questions: Structure learning for bounded-degree Pauli Markov fields and sample-complexity improvements from parallel samples are proposed as further extensions.The authors expect bounded-degree structure learning to be efficient in the number of qubits.
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