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Mitigating measurement errors in multi-qubit experiments

Sergey Bravyi, Sarah Sheldon, Abhinav Kandala, David C. Mckay, Jay M. Gambetta

arXiv:2006.14044v2quant-ph

TL;DR

Readout errors constrain scalable multi-qubit experiments, especially when noisy outcomes are used to estimate observable means. The paper introduces tensor-product and correlated Markovian noise models, calibrates them from measurement data, and mitigates noise by inverting the measurement model. Experiments on graph states and 20-qubit Clifford states demonstrate the approach, with CTMP improving readout-noise approximation and observed overhead reaching at most 20 in the graph-state experiments.

  • Problem

    Readout errors limit multi-qubit experiments that estimate observable mean values, while existing simple models omit correlated cross-talk.

  • Method

    The paper calibrates tensor-product and CTMP readout-noise models from measurement data and mitigates outcomes by inverse-noise processing or quasi-probability sampling.

  • Results

    The methods were demonstrated on graph-state and 20-qubit Clifford-state stabilizer measurements; graph-state experiments had mitigation overhead at most e^(4·0.7) ≤ 20.

  • Takeaways & Limitations

    CTMP can model correlated cross-talk and provide scalable calibration and noise inversion for multi-qubit mean-value measurements.

Abstract

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Reducing measurement errors in multi-qubit quantum devices is critical for performing any quantum algorithm. Here we show how to mitigate measurement errors by a classical post-processing of the measured outcomes. Our techniques apply to any experiment where measurement outcomes are used for computing expected values of observables. Two error mitigation schemes are presented based on tensor product and correlated Markovian noise models. Error rates parameterizing these noise models can be extracted from the measurement calibration data using a simple formula. Error mitigation is achieved by applying the inverse noise matrix to a probability vector that represents the outcomes of a noisy measurement. The error mitigation overhead, including the the number of measurements and the cost of the classical post-processing, is exponential in $εn$, where $ε$ is the maximum error rate and $n$ is the number of qubits. We report experimental demonstration of our error mitigation methods on IBM Quantum devices using stabilizer measurements for graph states with $n\le 12$ qubits and entangled 20-qubit states generated by low-depth random Clifford circuits.

I. INTRODUCTION

Readout errors limit scalable multi-qubit experiments, motivating classical mitigation methods for mean-value estimation. The paper develops tensor-product and correlated Markovian noise models with calibration and inversion procedures, then demonstrates them on multi-qubit IBM Quantum experiments.

  • Motivation: Readout errors can dominate scalability limits in near-term quantum devices, but mitigation is targeted at experiments computing observable mean values rather than single-shot measurements.Examples include variational quantum eigensolvers, quantum machine learning, and entangled-state tomography.
  • Noise models: A noisy n-qubit measurement is modeled by a 2^n × 2^n transition matrix A, where A_y,x is the probability of observing y given true outcome x.Assuming independent qubit noise reduces A to a tensor product of 2 × 2 matrices.
  • Noise models: The correlated CTMP model represents readout noise as A = e^G, with local generators for single- and two-qubit errors and 2n^2 error rates.It addresses cross-talk omitted by the tensor-product model.
  • Calibration and mitigation: Calibration uses repeated measurements of standard basis input states, and the number of required input states scales linearly with the number of qubits n.The paper gives simple analytic formulas for extracting model parameters from calibration data.
  • Calibration and mitigation: Mitigation applies the inverse noise matrix to noisy-outcome probabilities, while quasi-probability sampling avoids explicitly constructing that inverse.The overhead scales as e^(4γpoly(n)); for independent bit-flip errors, γ = nϵ.
  • Experiments: The methods were demonstrated on graph-state stabilizer measurements with n ≤ 12 and entangled 20-qubit states from low-depth random Clifford circuits.For small systems, CTMP was compared with full and tensor-product noise models.

II. UNBIASED ERROR MITIGATION

The paper constructs an unbiased estimator of an ideal observable mean from noisy computational-basis measurements. It models readout noise classically and bounds the estimator’s sampling cost through a mitigation overhead factor.

  • Setup: The goal is to estimate Tr(ρO) within precision δ when O is diagonal in the standard basis and takes values in [−1, 1].The state ρ is produced by a noisy quantum circuit on n qubits.
  • Ideal measurements: Without measurement errors, the empirical mean is unbiased and achieves error at most δ with high probability using M = 4δ^-2 measurements.The stated success probability is at least 2/3.
  • Noise model: Noisy measurements are represented by POVM elements that, under the paper’s simplifying assumption, are diagonal in the standard basis and induce a stochastic matrix A.A_y,x gives the probability of observing y when the true outcome is x.
  • Estimator: The error-mitigated random variable ξ is an unbiased estimator of Tr(ρO) with standard deviation σξ ≤ ΓM^-1/2.The factor Γ captures the amplification introduced by error mitigation.
  • Overhead: Error mitigation increases the measurement count for precision δ by a factor of Γ^2 relative to ideal measurements.For fixed n, the cost of approximating the ideal mean with tolerance δ scales as δ^-2.

III. TENSOR PRODUCT NOISE

The tensor-product model treats readout errors as independent single-qubit transitions and permits efficient mitigation for product-form observables. Its calibration parameters determine an overhead that grows exponentially with the aggregate noise strength.

  • Model: The tensor-product noise model takes A as a tensor product of 2 × 2 stochastic matrices describing independent single-qubit readout errors.The weak-noise analysis assumes ϵj, ηj ≤ 1/2 for every qubit.
  • Model: The model uses error rates ϵj and ηj for the transitions 0 →1 and 1 →0, respectively.These rates parameterize the single-qubit matrices used to construct A.
  • Mitigation: For tensor-product observables, the mitigated mean can be computed from observed bit strings using single-qubit matrix elements.Examples include products of Pauli Z operators and projectors onto computational-basis states.
  • Implementation: The classical post-processing runtime is approximately nM for product-form observables and nMτ for arbitrary observables, where τ is the cost of computing O(x).Thus the product-observable procedure is linear in qubit count and measurement count.
  • Overhead: Calibration data determines ϵj and ηj, and for small error rates Γ ≈ e^(2γ), making the mitigation overhead Γ^2 ≈ e^(4γ).The quantity γ is the paper’s noise-strength parameter.

IV. CORRELATED MARKOVIAN NOISE

The CTMP model extends tensor-product readout noise to include correlated single- and two-qubit errors, while enabling error mitigation through stochastic sampling without explicitly forming the inverse noise matrix. Its overhead scales exponentially with the CTMP noise strength, which is linear in n under short-range cross-talk assumptions.

  • CTMP noise model: The tensor-product model is recovered as a special case of CTMP when all two-qubit error rates vanish.Then the noise matrix becomes a tensor product of single-qubit stochastic matrices.
  • Error mitigation: Error-mitigated mean values are estimated by decomposing A^-1 into stochastic matrices and sampling signed contributions rather than explicitly computing the inverse.The resulting estimator is unbiased, and Hoeffding’s inequality provides a δ-accurate estimate with high probability.
  • Overhead: The CTMP mitigation overhead scales as e^4γ, where γ is the CTMP noise strength.For independent bit-flip errors, γ = nϵ, so the method is practical when γ remains a small constant.
  • Overhead: Under short-range cross-talk, γ ≤ Cϵn, yielding an exponential runtime factor e^4Cϵn; experiments are consistent with γ ∼ n.The Markov-chain simulation uses a sparse transition matrix, with average cost approximately γn^2.

V. MEASUREMENT CALIBRATION

Measurement calibration estimates tensor-product and CTMP noise parameters from repeated measurements of selected computational-basis inputs. CTMP calibration requires complete input sets that probe every two-qubit bit pattern, while structured choices distribute calibration resources across errors.

  • Calibration procedure: Calibration estimates noise-model parameters by repeatedly measuring prepared computational-basis states and counting each observed output.Using selected input states avoids calibrating all 2^n possible basis states.
  • Tensor-product calibration: Tensor-product calibration requires each qubit to appear in both computational-basis states, allowing its 0 →1 and 1 →0 error rates to be estimated from calibration fractions.The resulting rates define the tensor-product noise matrix.
  • CTMP calibration: A complete CTMP calibration set can include 0^n, 1^n, and all weight-1 bit strings, provided every pairwise input pattern is represented.Calibration uses local output transitions and averages single-qubit rates across relevant two-qubit noise matrices.
  • CTMP calibration: CTMP calibration requires every pair of qubits to realize all four input patterns 00, 01, 10, and 11 in at least one calibration state.This completeness condition ensures that every two-qubit readout error can be probed.
  • Calibration schemes: Hadamard calibration uses at most 2^n input states and probes each two-bit error on exactly 2^(p−2) states, evenly allocating resources among errors.For n = 4, the set consists of all even-weight bit strings.

VI. EXPERIMENTAL RESULTS

Experiments on IBM Quantum devices compared readout-noise models and evaluated CTMP and tensor-product mitigation on graph states and a 20-qubit random Clifford state. CTMP more accurately captured correlated readout noise, while the methods remained computationally practical at 20 qubits.

  • Noise-model accuracy: CTMP reduced the TVD from the full noise matrix by 2X relative to the tensor-product model for n = 6, 7 qubits.The comparison used 16 independent experiments per qubit count and identical calibration data for both models.
  • Graph-state experiments: CTMP provided much more accurate graph-state fidelity estimates than the tensor-product model, which systematically over-estimated fidelity.Graph-state experiments covered n ≤12, while the full A-matrix comparison was limited to n ≤7.
  • Graph-state experiments: Correlated two-qubit readout errors reached a few-percent rates for some qubit pairs in the 12-qubit graph-state register.The six-qubit analysis found increasing CTMP–TP differences for higher-weight stabilizers, which are more sensitive to correlated readout errors.
  • 20-qubit experiment: For the 20-qubit random depth-4 Clifford state, correlated two-qubit error rates were below 1% for all pairs, making CTMP and tensor-product estimates less different.Approximately 500 stabilizers with weights from 1 to 20 were measured using both mitigation methods.
  • Scalability: The CTMP overhead followed e4γ with γ ≈0.05n, and γ ≈1.1 was observed in the n = 20 experiment.The reported graph-state experiments had γ ≤0.7.
  • Scalability: At 20 qubits, CTMP post-processing for T = 10^6 samples took about 2 seconds on a laptop, while extracting CTMP error rates took about 1 second.The sample count affects classical post-processing but not the number of experiments.

VII. CONCLUSIONS

The paper introduces scalable readout-error mitigation methods for multi-qubit experiments, including mitigation of correlated cross-talk errors and efficient calibration and noise inversion.

  • The methods mitigate readout errors in multi-qubit experiments at scale.
  • The methods address correlated cross-talk errors.
  • The methods enable efficient calibration and noise inversion.

Appendix A: Proof of Lemma 1

The appendix bounds the variance of the estimator used for error mitigation. Under bounded observables, its standard deviation decreases as the inverse square root of the number of measurements, scaled by Γ.

  • The measurement outcomes are represented by a probability vector.
  • Substituting the probability-vector representation into the preceding expression yields the variance relation used in the proof.
  • The diagonal operator D contains the probabilities obtained from applying A to the input distribution.
  • M^-1/2: the estimator's standard deviation is at most Γ divided by the square root of the number of measurements when |O(x)| ≤1.

Appendix B: Proof of Lemma 2

The appendix characterizes when a real matrix can be decomposed into stochastic matrices and shows that the inverse of any stochastic matrix has such a decomposition.

  • A real matrix admits a decomposition into stochastic matrices if and only if all its column sums are equal.
  • The construction uses matrices with zero column sums to transform the matrix while preserving the equal-column-sum condition.
  • The transformed matrix is a scalar multiple of a stochastic matrix, establishing the decomposition for matrices with equal column sums.
  • The inverse of any stochastic matrix can be written as a linear combination of stochastic matrices because its column sums equal one.
  • For matrices with equal column sums, the minimum coefficient 1-norm equals the maximum 1-norm of the columns.
  • The proof establishes the norm equality by induction on the number of nonzero matrix elements, treating nonzero- and zero-column-sum cases.

Appendix C: Error mitigation for product noise

For tensor-product readout noise, the appendix instantiates the general stochastic-decomposition algorithm using single-qubit coefficients and Boolean maps. The resulting procedure achieves high-probability estimation with runtime scaling linearly in qubit number and measurements, up to observable-evaluation cost.

  • The product-noise algorithm assigns four coefficients to each qubit from its 0→1 and 1→0 error rates.
  • The four Boolean maps implement identity, bit flip, constant-zero, and constant-one transformations.
  • Each single-qubit transformation is sampled with probability proportional to its coefficient magnitude.
  • The algorithm accumulates signed estimator contributions and returns the rescaled estimate ξ′.
  • The single-qubit implementation is a special case of the general stochastic-matrix sampling algorithm.
  • The inverse product-noise matrix is decomposed into tensor products of stochastic matrices, with coefficient 1-norm Γ.
  • The estimator satisfies |ξ′ − ξ| ≤ δ with high probability, while runtime is approximately nMτ.

Appendix D: Experimental Hardware

Experiments used the 20-qubit ibmq johannesburg device, whose calibration data showed distinct CX, single-qubit, and readout error-rate distributions.

  • (1.75 ± 1.05) × 10^-2 was the typical mean CX-gate error rate on ibmq johannesburg.
  • (4.35 ± 1.70) × 10^-4 was the typical mean single-qubit-gate error rate.
  • (3.44 ± 1.72) × 10^-2 was the typical mean readout error rate, with detailed rates shown in Fig. 7 histograms.
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