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Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography
Filip B. Maciejewski, Zoltán Zimborás, Michał Oszmaniec
TL;DR
Readout noise limits measurements on near-term quantum devices, motivating methods that can correct observed statistics. The paper combines Quantum Detector Tomography with classical inversion of an estimated noise map, finding substantial improvements across several IBM experiments while identifying assumptions that constrain applicability.
Problem
Near-term quantum measurements are affected by readout noise, but practical correction requires characterizing the measurement device and its errors.
Method
The method reconstructs the device POVM through Quantum Detector Tomography and classically applies the inverse of an invertible stochastic noise map to later measurement statistics.
Results
Experiments on IBM and Rigetti processors support classical readout noise as a dominant model, while IBM five-qubit tests show substantial improvements across several one-, two-, and five-qubit tasks.
Takeaways & Limitations
Classical post-processing based on detector tomography can mitigate readout errors across multiple quantum tasks when the noise is classical and the detector is sufficiently characterized.
Takeaways & Limitations
The procedure assumes approximately perfect detector tomography and is affected by coherent errors, finite statistics, nonphysical corrected probabilities, and the difficulty of reliable two-qubit QDT.
Abstract
from arXiv · showhide
We propose a simple scheme to reduce readout errors in experiments on quantum systems with finite number of measurement outcomes. Our method relies on performing classical post-processing which is preceded by Quantum Detector Tomography, i.e., the reconstruction of a Positive-Operator Valued Measure (POVM) describing the given quantum measurement device. If the measurement device is affected only by an invertible classical noise, it is possible to correct the outcome statistics of future experiments performed on the same device. To support the practical applicability of this scheme for near-term quantum devices, we characterize measurements implemented in IBM's and Rigetti's quantum processors. We find that for these devices, based on superconducting transmon qubits, classical noise is indeed the dominant source of readout errors. Moreover, we analyze the influence of the presence of coherent errors and finite statistics on the performance of our error-mitigation procedure. Applying our scheme on the IBM's 5-qubit device, we observe a significant improvement of the results of a number of single- and two-qubit tasks including Quantum State Tomography (QST), Quantum Process Tomography (QPT), the implementation of non-projective measurements, and certain quantum algorithms (Grover's search and the Bernstein-Vazirani algorithm). Finally, we present results showing improvement for the implementation of certain probability distributions in the case of five qubits.
1 Introduction
The paper addresses readout noise in near-term quantum devices by using detector tomography to characterize measurement errors and classically correct later experimental statistics. It motivates this approach through practical device imperfections, especially in IBM and Rigetti processors.
- Motivation: Near-term quantum applications require characterization and mitigation of experimental imperfections affecting quantum measurements.The paper connects this need to quantum computation, simulation, and random-number generation on accessible quantum hardware.
- Core proposal: Quantum Detector Tomography reconstructs a device’s POVM, enabling measurement errors to be inferred and subsequent statistics to be corrected.The correction is possible when measurement noise is classical, stochastic, and invertible.
- Experimental validation: IBM and Rigetti processors are experimentally characterized to assess whether their readout noise is well described by the proposed classical model.Both architectures use superconducting transmon qubits, motivating the study of classical readout noise in these devices.
- Related work: The introduction positions the work relative to prior efforts using readout calibration or classical noise assumptions to correct assignment errors.The authors relate those procedures to their method under the stated assumptions.
2 Theoretical background
The theoretical background models quantum measurements with POVMs, explains detector tomography, and formalizes distances between measurements. It then represents classical readout noise as an invertible stochastic transformation of ideal measurement statistics.
- POVM formalism: A POVM represents an n-outcome quantum measurement through positive operator effects whose outcome probabilities follow Born’s rule.Projective measurements impose the additional relation MiMj = δi,j Mi.
- Quantum Detector Tomography: Quantum Detector Tomography reconstructs a device POVM by measuring a basis of quantum states for Hermitian operators.For a d-dimensional system, the basis is d^2-dimensional, giving the minimal number of different state preparations needed for reconstruction.
- Quantum Detector Tomography: Maximum Likelihood Estimation can enforce positivity when imperfect statistics or state preparations produce nonphysical reconstructed POVM elements.This addresses practical deviations from ideal tomographic data.
- Measurement distance: Operational distance measures the worst-case statistical distinguishability between two POVMs over all quantum states and outcome subsets.It is related to total-variation distance and has an interpretation through optimal measurement discrimination.
- Classical noise model: Classical readout noise is modeled as an invertible left-stochastic map Λ that transforms ideal measurement outcomes into experimental outcomes.This makes the noisy measurement equivalent to classical post-processing of ideal statistics.
3 Scheme of mitigation of readout errors
The mitigation scheme reconstructs detector noise, then applies the inverse noise map to measured statistics. Its practical scope depends on classical invertible noise, sufficient statistics, detector characterization, and manageable multi-qubit correlations.
- 3.1 Assumptions: The scheme assumes infinite statistics, classical measurement noise, and a characterized detector described by a known noise map.Violations of these assumptions are analyzed as sources of mitigation error.
- 3.1 Assumptions: Classical readout noise is supported as a practical model because it is reported as dominant in IBM superconducting-transmon devices.The paper suggests similar architectures may exhibit comparable measurement-noise behavior.
- 3.2 Correction procedure: Applying Λ−1 to experimental statistics pexp yields the ideal statistics pideal when the noise transformation is invertible and correctly characterized.The inverse is generally not stochastic, but it reverses the classical post-processing at the statistics level.
- 3.3 Multiple qubits: For multiple qubits, uncorrelated readout errors allow separate single-qubit detector tomographies, reducing reconstruction complexity from exponential to linear in the number of qubits.Correlated detector errors instead require simultaneous multi-qubit tomography.
- 3.3 Multiple qubits: For small individual errors, the total operational distance for uncorrelated multi-qubit measurements is approximately additive across qubits.The single-qubit distance is Dop(ΛiPi, Pi) = max {pi, qi}.
4 Error analysis
The analysis examines coherent noise, finite statistics, and non-physical corrected distributions as deviations from the ideal classical-noise model. It derives bounds on the resulting error and identifies omitted tomography imperfections as a remaining limitation.
- Overview: Three deviations from the ideal model are analyzed: non-classical noise, finite experimental statistics, and projection onto a physical probability vector.Imperfect measurement tomography is explicitly omitted and deferred to future work.
- 4.1 Non-classical noise: Classical post-processing cannot fully reverse non-classical measurement noise, leaving a residual term after applying Λ−1.The residual disturbance is Λ−1 ˜∆, and the induced error is bounded using ||Λ−1||1→1 and the operational distance between measurements.
- Scope and decomposition: The classical-noise decomposition is not unique, so the paper notes that optimizing the split could reduce the upper bound on mitigation error.For arbitrary ideal measurements, the decomposition into classical and non-classical parts must also be modified.
- 4.2 Statistical errors: The total error bound combines statistical error from finite samples with error caused by the non-classical part of the noise.The statistical contribution depends on the estimation confidence and sample count, while the non-classical contribution depends on the measurement mismatch and correction matrix.
- 4.3 Non-physical probability vectors: When inversion yields negative entries, the method replaces the result with the closest proper probability vector, which can introduce additional error.The paper chooses the Euclidean distance for mathematical convenience and includes this contribution as α in the overall bound δ + α.
- 4.2 Statistical errors: For N = 8192 and Prerr = 0.01, the statistical error is ϵ ≈0.018 under the stated estimation setting.This value is used in the single-qubit error-bound analysis.
5 Summary of the method
The mitigation workflow first characterizes a noisy detector, extracts and inverts its classical readout-noise matrix, and then corrects statistics from later experiments. Corrected results are checked for physicality, and success is judged by comparing the corrected-error bound with the uncorrected bound.
- Pre-processing: Quantum Detector Tomography reconstructs the detector POVM before any subsequent experiment is corrected.The reconstructed POVM supplies the measurement description used to identify the classical noise.
- Pre-processing: Discarding off-diagonal POVM terms yields ΛP, the classical-noise model for the target projective measurement P.This step separates the classical contribution from non-classical measurement errors.
- Correction: Inverting the stochastic matrix Λ produces the correction matrix Λ−1 used to process later experimental frequencies.The correction is applied after an arbitrary experiment on the characterized device.
- Physicality check: The corrected vector is accepted when it is nonnegative and sums to one; otherwise, the method solves an optimization problem for a nearby physical vector.The resulting vector is denoted p∗exp.
- Success criterion: Mitigation is successful when δ + α is smaller than Dop(P, M) + ϵ, the corresponding uncorrected error bound.If the corrected bound is greater than or equal to the uncorrected bound, the mitigation is considered unsuccessful.
- Scope: For arbitrary ideal measurements, the method requires a modified decomposition into classical and non-classical noise parts.The detailed workflow described here focuses on projective measurements.
6 Device characterization
Device characterization indicates that IBM and Rigetti readout errors are largely classical, while tomography reliability depends on state-preparation quality and calibration-specific data. The mitigation error is smaller than the underlying detector error in the tested single- and two-qubit settings.
- Calibration and limitations: Measurements from different days may be combined because public-device access prevented all experiments from occurring within one calibration period, although each correction used period-specific tomography.This calibration constraint applies to the presented experimental data.
- Experimental assumptions: IBM’s single-qubit gate errors are small enough to make Quantum Detector Tomography feasible, whereas Rigetti’s high gate infidelities significantly violate the perfect-reconstruction assumption.For IBM, randomized-benchmarking errors are of order 0.1%; Rigetti’s single-qubit gate infidelities are high.
- Calibration and limitations: Separating measurement, gate, and state-preparation errors rigorously remains difficult, limiting precise estimates of mitigation accuracy.The authors note that the scheme can nevertheless be tested directly in practical applications.
- Experimental scope: The experiments characterize all five ibmqx4 qubits and five exemplary qubits from Rigetti’s Aspen-4-16Q-A using single- and two-qubit detector tomography.The IBM device has five qubits; the Rigetti characterization covers the first five of its 16 qubits.
- Single-qubit measurements: Readout noise is significant on both platforms, but the small distance between each reconstructed POVM and its diagonal form indicates that classical noise dominates.For IBM ibmqx4, readout noise is considered predominant at the individual-qubit level for short circuits because gate fidelities are high.
- Single-qubit measurements: The upper bound on mitigation error is significantly smaller than the noisy detector’s distance from the ideal detector, suggesting that correction should be beneficial.This comparison is stated for the infinite-statistics bound, which excludes statistical errors.
- Two-qubit measurements: Two-qubit tomography likewise finds relatively small coherent errors and mitigation errors much smaller than the operational distance from the ideal detector.The comparison applies to the infinite-statistics scenario.
- Two-qubit measurements: Readout-error correlations are small for most IBM qubit pairs except q3q1 and q2q1, while the examined Rigetti pairs show no significant correlations.The exceptional IBM pairs should be mitigated using joint two-qubit Quantum Detector Tomography.
7 Applications on IBM devices
The authors apply readout-error mitigation across tomography, non-projective measurements, quantum algorithms, and five-qubit probability distributions on IBM devices. The procedure generally improves results, while correlation-aware tomography helps selectively.
- Quantum State Tomography and Quantum Process Tomography: Error mitigation reduces infidelities in every tested single-qubit state and process tomography case.The improvement depends strongly on the input state.
- Quantum State Tomography and Quantum Process Tomography: Two-qubit tomography improves, but correlated detector tomography outperforms tensor-product tomography for only one of three tested states.Nonphysical corrected probability vectors and the higher sample complexity of two-qubit QDT may explain this result.
- Implementation of non-projective measurements: Error mitigation generally improves reconstructed POVMs for non-projective measurements implemented through Naimark’s extension.Correlated and non-correlated detector tomography perform nearly identically because correlations were relatively small during these experiments.
- Quantum algorithms: Correlation-aware mitigation provides a significant advantage for Grover’s search on the highly correlated pair q2q1, with a larger relative improvement than for Bernstein-Vazirani.Both algorithms were implemented on three qubits, including one ancilla, and benchmarked by the probability of a particular outcome.
- Probability distributions: Five-qubit correction remains beneficial, but its effectiveness depends on the target distribution and whether readout correlations are modeled.For the uniform distribution, modeling correlations in q2q1 yields high improvement; the ’Mixed’ distribution benefits in both tested cases.
8 Conclusions and further research directions
The paper presents detector-tomography-based classical post-processing as a readout-error mitigation scheme for noisy quantum devices. Experiments on IBM’s five-qubit processor show substantial improvements, while scalability and coherent-error correction remain open issues.
- Conclusions: The scheme uses classical post-processing based on Quantum Detector Tomography and applies in principle when readout noise is classical and detector tomography is approximately perfect.The method is intended for noisy and imperfect quantum hardware.
- Conclusions: Experiments on IBM’s five-qubit processor show substantial improvements across single-, two-, and five-qubit tasks and algorithms.The study compares mitigation with and without accounting for correlations in readout errors.
- Conclusions: Accounting for readout correlations is crucial for Grover’s algorithm and five-qubit probability distributions, motivating further correlation studies.The conclusion identifies correlation-aware mitigation as a future research direction.
- Further research directions: Full detector tomography becomes difficult for many-qubit systems because the problem size grows exponentially.The authors suggest applicability may remain practical for algorithms using polynomially many few-qubit measurements.
- Further research directions: Physical correction of coherent readout errors is an open research question, although uncorrelated cases may permit unitary-based correction under specified conditions.The proposed procedure focuses on classical post-processing rather than established physical correction of coherent errors.
- Further research directions: Confidence intervals for quantum detector tomography and scalable tomography for larger, irregularly connected systems are also unresolved.The paper also reports publicly available Python code implementing the method.
NOTE ADDED
A note added after manuscript completion situates the scheme alongside analogous readout-error mitigation work and a recent Qiskit update.
- NOTE ADDED: The authors identify a recent analogous readout-error mitigation scheme and a Qiskit implementation relying on a similar procedure.These observations were added after the manuscript was completed.
A Proofs of technical statements
The supplied passage contains only a repository reference and does not report technical proofs.
- Proofs of technical statements: The passage provides a GitHub repository link but no proof or technical statement.No further proof content is available in this passage.
B State dependence of correction
The mitigation criterion compares post-processed and unprocessed error bounds, but its success depends on the measured quantum state. Numerical tests show high success fractions for the studied experimental data, especially for single-qubit and uncorrelated two-qubit detectors.
- Criterion for successful mitigation: The mitigation criterion compares an upper bound on post-processed error with the corresponding worst-case unmitigated error bound.The comparison uses total-variation distances to ideal statistics and remains useful when the ideal distribution is unavailable experimentally.
- Numerical procedure: The numerical procedure generates L Haar-random pure states, computes noisy and ideal probability vectors, and samples N outcomes from the noisy distribution.The resulting empirical statistics are tested against the success criterion for each state.
- Single-qubit results: For single-qubit detectors, successful corrections cross 50% near δ + α ≈ Dop(M, P) + ϵ.This behavior supports the rule used to identify successful mitigation cases.
- Single-qubit results: 88%–99% of corrections succeed for actual single-qubit experimental data, suggesting usefulness in most single-qubit cases.The reported fraction is based on the studied IBM data and the tested detector-noise models.
- Two-qubit results: For two-qubit tensor-product detectors, successful corrections cross 50% near the regime where Eq. (47) becomes unsatisfied.The simulations vary off-diagonal POVM terms while retaining diagonal terms from experimental data.
- Two-qubit results: 98.86%–99.99% of corrections succeed for actual uncorrelated two-qubit data, suggesting success for generic quantum states in those cases.These results apply to the tested tensor-product POVMs and uncorrelated qubit pairs.
C Additional experimental data
Additional material documents the experimental reconstructions and extends the numerical analysis to two-qubit detector models. The two-qubit analysis uses repeated sampling and separates qubit pairs for clarity.
- Additional experimental data: The appendix provides explicit matrices for exemplary POVMs reconstructed in the single-qubit tomographies of IBM’s ibmqx4 and Rigetti’s Aspen-4-16Q-A.For each POVM, the second effect follows as the complement to identity.
- Additional experimental data: Figure 13 studies successful-mitigation fraction against the same noise-ratio quantity used in Figure 12.The two-qubit data use L = 10000 Haar-random states, N = 8192 samples per probability vector, and Prerr = 0.01.
- Additional experimental data: Figure 13 separates results for different qubit pairs, while the tested POVMs are formed using tensor products of single-qubit POVMs.The separation is for visual clarity.
C.2 Dates of the experiments
The paper records execution dates for the reported experiments, with specific dates given for Rigetti characterization and tomography and a separate table for IBM experiments.
- C.2 Dates of the experiments: Rigetti fidelities were checked on June 30, 2019, and Rigetti tomographic reconstructions were performed on May 30, 2019.These dates are reported for the device characterization and tomography experiments.
- C.2 Dates of the experiments: Table 6 contains execution dates for experiments performed on IBM’s ibmqx4.The table notes that all experiments were conducted in 2019.