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Robust shadow estimation

Senrui Chen, Wenjun Yu, Pei Zeng, Steven T. Flammia

arXiv:2011.09636v2quant-ph

TL;DR

Noisy near-term quantum devices reduce the accuracy of shadow-based property estimates, motivating methods that remain efficient without detailed device characterization. The paper adds a calibration stage that learns the effective noise channel and uses it for robust classical-shadow estimation. Numerical experiments show that RShadow remains faithful as noise or system size increases, while standard shadow estimation degrades.

  • Problem

    Noise in near-term quantum devices undermines shadow-estimation accuracy, while existing approaches rely on ideal operations or restrictive assumptions.

  • Method

    RShadow calibrates the effective noisy measurement channel and uses its estimate in robust classical post-processing for shadow estimation.

  • Results

    RShadow remains faithful as noise and system size increase, whereas standard shadow estimation deviates from the true value and accumulates larger errors.

  • Takeaways & Limitations

    The calibrated protocol enables sample-efficient, noise-resilient estimation of useful quantum properties under minimal experimental assumptions.

  • Takeaways & Limitations

    Rigorous guarantees assume noise that is gate-independent, time-stationary, and Markovian; gate-dependent and non-Markovian noise is supported only by numerical evidence.

Abstract

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Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory. While quantum computers promise speedups for many such tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates. Here we show how to mitigate errors in the shadow estimation protocol recently proposed by Huang, Kueng, and Preskill. By adding an experimentally friendly calibration stage to the standard shadow estimation scheme, our robust shadow estimation algorithm can obtain an unbiased estimate of the classical shadow of a quantum system and hence extract many useful properties in a sample-efficient and noise-resilient manner given only minimal assumptions on the experimental conditions. We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments.

I. INTRODUCTION

Quantum property estimation seeks efficient access to information about large quantum states, but noisy devices undermine accuracy and force difficult assumptions or costly verification. The paper introduces robust shadow estimation, adding calibration to make shadow estimation noise-resilient while retaining efficiency under minimal experimental assumptions.

  • Quantum property learning targets quantities such as fidelity, entanglement, and energy, with robustness and efficiency as central goals.
  • Noise in NISQ circuits complicates property estimation because benchmarking devices are also noisy, creating a loop that assumptions or device-independent methods only partially address.
  • Quantum tomography can require exponentially many samples, while structured alternatives rely on restrictive state assumptions; random Pauli measurements estimate all k-body observables with O(k3k log n) samples.
  • Shadow estimation can estimate many observables more efficiently than full tomography or separate measurements, but standard schemes assume perfect gates and ideal computational-basis measurements.
  • Robust shadow estimation adds experimentally friendly calibration to estimate and mitigate noise, requiring only high-fidelity preparation of |0⟩ while preserving sample efficiency.

II. PRELIMINARIES

The standard shadow protocol estimates many observable expectations by inserting an invertible measurement channel between the state and observables, then correcting measurement outcomes classically. Random unitary twirling makes the dephasing measurement channel invertible and enables parallel estimation of multiple observables.

  • Shadow estimation setup: The protocol estimates Tr(O_iρ) for many observables by inserting a prepare-and-measure superoperator whose inverse preserves the target expectation values.
  • Measurement channel: A Z-basis measurement channel is not invertible because it lacks X- and Y-basis information, so random unitary twirling is introduced to restore invertibility.
  • Measurement channel: The twirling group must produce nonzero coefficients across its irreducible representation sectors, which makes the resulting channel invertible.
  • Algorithm: Each round applies a randomly sampled unitary, measures in the computational basis, and classically evaluates ⟨⟨O_i|M−1U†|b⟩⟩ for every observable.
  • Algorithm: Median-of-means combines R = NK single-round estimators into K group means and returns their median to obtain the final estimates.

III. ROBUST SHADOW ESTIMATION

Robust shadow estimation models noisy gates and measurements through a twirled noise channel, learns that channel in a calibration stage, and uses its estimate for subsequent property estimation. The protocol assumes gate-independent, time-stationary Markovian noise and initially exact, later high-fidelity, preparation of |0⟩.

  • Noise modeling: The method learns noise as a simple stochastic model and compensates through robust classical post-processing, with twirling reducing the effective noise parameters.
  • Assumptions: The analysis assumes gate-independent, time-stationary, Markovian noise and a computational-basis preparation capability, while high-fidelity rather than exact |0⟩ preparation is later allowed.
  • Calibration: Calibration prepares |0⟩, applies a noisy sampled unitary, measures computationally, and estimates the twirled-channel coefficients with median-of-means processing.
  • RShadow protocol: RShadow first calibrates the noisy measurement channel, then supplies its estimate to the standard shadow estimator for property prediction.
  • Implementation scope: The protocol focuses on global and tensor-product single-qubit Clifford groups, for which the authors provide NoiseEst constructions and analyze correctness and efficiency.

IV. ROBUST SHADOW ESTIMATION USING GLOBAL CLIFFORD GROUP

The global-Clifford RShadow protocol calibrates the noisy measurement channel and then mitigates its effect in shadow estimation. Under suitable fidelity and low-noise conditions, it removes systematic error while retaining near-standard sample complexity.

  • Correctness guarantee: Assuming FZ ≫ 2^-n, global-Clifford RShadow estimates arbitrary observables for any quantum state with high probability after calibration.The correctness guarantee applies to any observable O and quantum state ρ under the theorem’s stated calibration conditions.
  • Global Clifford calibration: The protocol eliminates systematic shadow-estimation error by calibrating the noise channel before estimating properties.Without calibration, estimates converge to values conflating the noise map; the calibrated expression removes that map.
  • Noise scaling: For local noise of fixed strength, FZ(Λ)^-2 ≈ exp(2nξ), allowing efficient treatment of system sizes n comparable to ξ^-1.The scaling is stated for Λ = N_i=1^n Λ_i with each single-qubit channel satisfying FZ(Λ_i) ≥ 1 − ξ.
  • Sample complexity: In the low-noise regime, RShadow has nearly the same sample-complexity order as noise-free standard shadow estimation.The comparison concerns the calibration and estimation sample requirements established by the informal theorem.
  • Computational complexity: Calibration is computationally efficient via Gottesman–Knill, but efficient estimation additionally requires observables such as stabilizer states or Pauli operators.The computational advantage of the estimation stage is therefore conditional on extra observable structure.

V. ROBUST SHADOW ESTIMATION USING LOCAL CLIFFORD GROUP

The local-Clifford RShadow protocol provides an experimentally friendlier way to calibrate and mitigate errors for local observables. Its sample complexity remains comparable to noiseless shadow estimation under local, sufficiently weak noise, while general-noise bounds remain unavailable.

  • Local-Clifford protocol: Local-Clifford RShadow efficiently calibrates and mitigates errors when estimating local properties using an experimentally friendly alternative to global Clifford operations.The local Clifford group is the n-fold tensor product of single-qubit Clifford groups.
  • Calibration scope: The calibration sample requirement for all k-local observables depends on k but essentially not on the system size n, even under gate-independent global unitary noise.This scope is stated for the local-Clifford correctness theorem.
  • Limitations: Theoretical sample-complexity bounds are unavailable for the most general noise channel, although numerical results show good performance in that case.The authors note that real-time monitoring of estimator standard deviations can still suppress statistical fluctuations without such a bound.
  • Correctness guarantee: For k-local observables with bounded operator norm, the protocol estimates M arbitrary linear functions up to accuracy ε1 + ε2 with high probability.The stated guarantee covers every O_i that is k-local and satisfies ∥O_i∥∞ ≤ 1.
  • Sample complexity: Under local noise that is not too strong, local-Clifford RShadow has sample complexity similar to noiseless standard shadow estimation.The local-noise model assumes Λ = N_i=1^n Λ_i with FZ(Λ_i) ≥ 1 − ξ for small ξ.
  • Computational complexity: For k-local observables, only coefficients with |z| ≤ k need computation, and the relevant coefficients can be computed in O(n^k) time by dynamic programming.Additional observable structure, such as spatial locality, can reduce the required set further.

VI. ROBUSTNESS AGAINST STATE PREPARATION NOISE

RShadow remains robust when calibration uses imperfect state preparation rather than an ideal |0⟩ state. The resulting estimation error can be bounded under high-fidelity, time-independent preparation conditions.

  • State-preparation model: The protocol tolerates small state-preparation noise when the calibration state is prepared with high fidelity.The analysis replaces ideal |0⟩⟨0| preparation with a time-independent state ρ0 during calibration.
  • Sample requirements: The state-preparation theorems use the same calibration-sample counts as the corresponding ideal-preparation results.This is stated for both global- and product-state preparation settings.
  • Local observables: For product-state preparation, the estimation error is bounded to first order in ε and kξSP for k-local observables.Here k denotes the locality of the observable O.
  • Practical implication: The bounds allow experimentalists to choose practical sample numbers according to how accurately their devices prepare |0⟩⟨0|.The conclusion follows from bounding the effect of state-preparation noise in RShadow.

VII. NUMERICAL RESULTS

Numerical experiments compare RShadow with standard shadow estimation for GHZ fidelity, TFIM correlations, and TFIM energy under several noise settings and system sizes. Across these tests, RShadow remains accurate where standard shadow estimates become increasingly distorted as noise or system size grows.

  • GHZ-state fidelity: RShadow remains faithful as noise increases across depolarizing, amplitude-damping, and measurement-bit-flip models, while standard shadow estimates deviate from the true GHZ fidelity.The GHZ experiment uses 10 qubits, with separate calibration and estimation samples for RShadow.
  • GHZ-state fidelity: Under local X-rotation noise, standard-shadow GHZ fidelity degrades with system size, whereas RShadow remains accurate from 4 to 12 qubits.Both protocols use R = 10^5 trials for calibration and estimation, with rotation angles π/25, 2π/25, and 3π/25.
  • TFIM observables: For 50-spin TFIM ZZ correlations, RShadow generally gives more precise estimates than standard shadow under 5% measurement bit-flip noise.The experiment estimates correlations between the leftmost spin and all other spins.
  • TFIM observables: For TFIM energy, standard-shadow error increases with noise and system size, while RShadow remains precise and its absolute error stays close to zero as the system grows.The experiments use a 50-spin ground state under three noise models and then vary system size at 5% measurement bit-flip noise.
  • TFIM observables: The TFIM implementations calibrate only O(n) relevant parameters rather than all O(n^2) coefficients, preserving efficiency for large systems.Nearest-neighbor terms are calibrated for energy estimation, while first-qubit-to-other-qubit terms are calibrated for correlation estimation.
  • Additional observations: RShadow can reproduce faithful estimates under two-qubit correlated noise using a small number of benchmarking trials, although ratio estimation can produce systematic bias and statistical fluctuations.The authors note that physical-range truncation can improve accuracy and avoid nonphysical estimates.

VIII. GATE-DEPENDENT NOISE

The authors test RShadow under realistic gate-dependent pulse errors using H2 ground-state energy estimation. RShadow suppresses noise bias substantially, though pulse mis-calibration leaves residual bias at very high noise strengths.

  • Benchmark setup: The benchmark maps H2 to a four-qubit Hamiltonian with Bravyi–Kitaev encoding and estimates its ground-state energy against the classically computed true value.
  • Noise models: The experiments use local Clifford generators and model fixed pulse mis-calibration and random Gaussian over-rotation errors.The two error models are gate-dependent because the implemented rotations differ across generators or depend on random rotation errors.
  • Numerical results: RShadow significantly suppresses H2 ground-state-energy bias under both pulse mis-calibration and random over-rotation noise.The comparison uses 30,000 calibration samples and 10,000 estimation samples for RShadow, versus 10,000 samples for standard Shadow.
  • Random over-rotation: For random over-rotation noise, RShadow completely eliminates bias even at large noise strengths.
  • Pulse mis-calibration: For pulse mis-calibration, RShadow does not eliminate all bias at very large noise strength but still outperforms standard shadow estimation.The paper notes that noise intensity ∥∆0∥=0.1π is already fairly high in practice.
  • Interpretation: The authors suggest that approximate Pauli-channel behavior and sufficiently effective twirling may explain RShadow’s resilience, while rigorous analysis for general gate-dependent noise remains open.

IX. CONCLUDING REMARKS

The paper concludes that RShadow efficiently benchmarks and suppresses gate and measurement noise for global and local random Clifford protocols. It also identifies extensions and unresolved questions concerning higher-order properties, gate dependence, non-Markovianity, and channel estimation.

  • Conclusions: RShadow efficiently benchmarks and suppresses noise effects for both global and local random Clifford shadow protocols.
  • Future directions: The method currently focuses on linear properties, while higher-order properties such as subsystem Rényi-2 entropy and other shadow variants remain future directions.
  • Scope of guarantees: The performance guarantee allows coherent and highly correlated noise but assumes the noise is independent of the implemented random unitary gate.
  • Limitations: The paper provides numerical evidence for resilience to gate-dependent noise but leaves rigorous analysis of gate dependence and non-Markovianity for future work.
  • Extensions: The robust shadow framework can be extended in the Pauli-transfer representation to estimate matrix elements of unknown quantum channels between observables and input states.

Appendix B: Sample Complexity of RShadow with Global Clifford Group

The appendix specializes the robust shadow protocol to the n-qubit Clifford group.

  • Global Clifford group: The sample-complexity analysis considers RShadow with the n-qubit Clifford group Cl(2^n).

1. Calibration Procedure: Global

The global calibration procedure estimates the noisy measurement channel and constructs an inverse for subsequent shadow estimation. Under gate-independent noise, the estimator is unbiased and admits explicit precision, confidence, and sample-complexity guarantees.

  • Noise representation: The calibration channel combines noise from the random unitary operations and computational-basis measurement, reducing their effect to estimable channel coefficients after twirling.
  • Calibration procedure: Calibration estimates the noisy channel f_M, after which standard classical shadows use the estimated inverse channel to mitigate noise.The procedure prepares randomized states, measures in the computational basis, computes single-round estimators, and aggregates them with a median-of-means estimator.
  • Guarantees: Under gate-independent noise and perfect |0⟩ preparation, the calibrated estimator is unbiased and its variance can be bounded.
  • Estimator construction: Median-of-means aggregation uses R=KN rounds, with K groups of N single-round estimators, to control estimation error and failure probability.
  • Calibration complexity: Theorem 7 gives a calibration-round bound sufficient to achieve asymptotic error at most ε with success probability at least 1−δ, with dependence on Z-basis fidelity F_Z.
  • Estimation complexity: If F_Z is not too low and calibration uses sufficiently many rounds, RShadow estimates small-Hilbert-Schmidt-norm observables as efficiently as noiseless shadow estimation up to a small multiplicative factor.
  • Estimation guarantees: For M arbitrary linear functions with observables satisfying the stated norm condition, the protocol achieves combined accuracy ε1+ε2 with success probability at least 1−δ1−δ2.

Appendix C: Sample Complexity of RShadow with Local Clifford Group

The local-Clifford RShadow protocol calibrates the noisy measurement channel and uses its inverse for subsequent shadow estimation. Its guarantees cover k-local observables under weak, sufficiently characterized noise, with efficiency comparable to noiseless shadow estimation in relevant settings.

  • Protocol: The protocol uses experimentally simpler local Clifford operations, applying random single-qubit Cliffords to calibrated states before computational-basis measurements.The local Clifford group is an n-fold product of single-qubit Clifford groups, making its unitaries easier to implement experimentally.
  • Calibration: Calibration estimates Pauli fidelities of the twirled noisy channel, after which the inverse estimated channel is supplied to standard shadow estimation.Only fidelities relevant to the target observables need to be computed; for k-local properties, strings with weight at most k suffice.
  • Estimator guarantees: The single-round Pauli-fidelity estimators are unbiased, with variance bounded by the corresponding Pauli-weight-dependent quantity.In the noiseless case, the Pauli fidelity is f_z = 3^−|z|.
  • Sample complexity: Theorem 9 guarantees that sufficiently many calibration samples enable estimation of any k-local observable with accuracy ε and success probability at least 1 − δ.The bound applies to arbitrary n-qubit states and k-local observables under the theorem’s stated conditions.
  • Noise conditions: If the noise has Z-basis average fidelity at least 1 − c, then every calibration factor is at least 1 − 2c, enabling efficient mitigation for small k and weak noise.For separable noise with adequate per-qubit Z-basis fidelity, local-Clifford RShadow matches noiseless shadow estimation up to a small multiplicative factor.
  • Limitations: The analysis does not bound the relevant norm for the most general noise channel, while computationally estimating all Pauli fidelities is unaffordable.The practical protocol therefore focuses on fidelities relevant to local or nearby-qubit properties.

2. Robustness of RShadow with Local Clifford Group

This section analyzes RShadow when the calibration state is prepared noisily. Under a product-state preparation model, the resulting estimation error remains controlled to first order in the measurement and state-preparation errors.

  • Assumptions: The robustness analysis assumes no cross-talk between qubits and a product calibration state with suitable single-qubit state-preparation fidelity.These assumptions define the local state-preparation noise model used for the result.
  • Analysis: The calibration estimators remain analyzable under state-preparation noise because the relevant Haar-integrated second-moment expression is independent of the specific single-qubit prepared states.The analysis can therefore replace the noisy single-qubit states by ideal |0⟩ states when evaluating that expression.
  • Robustness guarantee: With product-form calibration states and local state-preparation noise, the final estimate for any k-local observable is accurate up to first order in ε and kξ_SP.The stated bound is derived using the calibration estimator variance and the Pauli-fidelity lower bound.

Appendix E: More numerical results

The appendix studies local-Clifford RShadow on GHZ correlation estimation under coherent and correlated noise, and describes gate decompositions and gate-dependent noise models used in simulations.

  • GHZ correlations: Numerical experiments estimate 5-qubit GHZ 2-point ZZ correlations under single-qubit X rotations and two-qubit XX cross-talk, whose noiseless target value is 1.The reported correlations pair the fifth qubit with each other qubit and average the values.
  • GHZ correlations: The appendix evaluates whether local-Clifford RShadow remains sample-efficient against qubitwise-correlated noise and reports an affirmative numerical answer.The tested noise models are coherent single-qubit rotations and adjacent-qubit XX rotations.
  • Gate decomposition: Each single-qubit Clifford can be decomposed into up to three rotations that map the Cartesian +Z and +X directions to its stabilizer and destablizer directions.The Hadamard gate is given as an example of this rotational decomposition.
  • Gate-dependent noise: Pulse mis-calibration and random over-rotation produce gate-dependent noise because their effective channels depend on the generator or Pauli direction.For pulse mis-calibration, the gate-dependent effect appears only in higher-order terms; random over-rotation induces dephasing in a Pauli-dependent eigenbasis.
  • Pulse mis-calibration: Additional simulations sample ten directions of the calibration error and compare RShadow with standard Shadow using separate calibration and estimation budgets.The sampled error has magnitude 0.1π, with RShadow using 30,000 calibration and 10,000 estimation samples, while standard Shadow uses 10,000 samples.
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