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Unfolding Quantum Computer Readout Noise

Benjamin Nachman, Miroslav Urbanek, Wibe A. de Jong, Christian W. Bauer

arXiv:1910.01969v2quant-phphysics.data-an

TL;DR

NISQ readout errors can bias quantum-computer results, and existing correction methods can become pathological when response matrices have substantial off-diagonal components. This paper applies high-energy-physics unfolding methods to quantum readout correction and finds that iterative Bayesian unfolding is robust to the failure modes of matrix inversion and ignis.

  • Problem

    Readout errors complicate interpretation of NISQ measurements, while current QIS correction methods can suffer pathologies that dedicated unfolding methods may avoid.

  • Method

    The paper studies high-energy-physics unfolding techniques for quantum readout errors, focusing on iterative Bayesian unfolding alongside matrix inversion and ignis.

  • Results

    Across representative tests, IBU avoids matrix-inversion and ignis oscillations; over 1000 pseudo-experiments, its prediction spread is a couple of percent smaller than ignis and both are about 10% more precise than matrix inversion.

  • Takeaways & Limitations

    HEP unfolding tools provide a robust approach for correcting quantum readout errors, and IBU is particularly effective against the demonstrated matrix-inversion failure mode.

  • Takeaways & Limitations

    Constructing the full response matrix requires exponentially many calibration circuits and may become prohibitive as the qubit count grows.

Abstract

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In the current era of noisy intermediate-scale quantum (NISQ) computers, noisy qubits can result in biased results for early quantum algorithm applications. This is a significant challenge for interpreting results from quantum computer simulations for quantum chemistry, nuclear physics, high energy physics, and other emerging scientific applications. An important class of qubit errors are readout errors. The most basic method to correct readout errors is matrix inversion, using a response matrix built from simple operations to probe the rate of transitions from known initial quantum states to readout outcomes. One challenge with inverting matrices with large off-diagonal components is that the results are sensitive to statistical fluctuations. This challenge is familiar to high energy physics, where prior-independent regularized matrix inversion techniques (`unfolding') have been developed for years to correct for acceptance and detector effects when performing differential cross section measurements. We study various unfolding methods in the context of universal gate-based quantum computers with the goal of connecting the fields of quantum information science and high energy physics and providing a reference for future work. The method known as iterative Bayesian unfolding is shown to avoid pathologies from commonly used matrix inversion and least squares methods.

I. INTRODUCTION

Readout errors are a major limitation for NISQ applications, and the paper frames their correction as histogram unfolding connected to established high-energy-physics methods. It addresses a gap in dedicated QIS studies by introducing this cross-field perspective.

  • Motivation: Readout errors arise from qubit decay during measurement and overlapping measurement distributions, motivating dedicated correction methods.The paper distinguishes readout-error mitigation from gate-error correction and describes complete quantum error correction as infeasible for current qubit counts and moderately deep circuits.
  • Connection to unfolding: High-energy physics provides established unfolding algorithms for correcting detector effects in binned differential cross-section measurements.The paper aims to connect these methods with algorithms used by the quantum-information community.
  • Connection to unfolding: Quantum readout correction is a classical histogram-unfolding problem whose bins correspond to the possible 2^nqubit computational-basis configurations.The response depends on calibrations or simulations from quantum hardware.
  • Gap and scope: The paper identifies a lack of dedicated unfolding studies for QIS applications and argues that HEP techniques can avoid pathologies in current QIS methods.Its scope covers the unfolding setup, methods, response-matrix construction, examples, uncertainties, discussion, and outlook.

II. THE UNFOLDING CHALLENGE

Unfolding relates true and measured bin counts through a response matrix, but direct inversion becomes unstable when detector or readout migrations are substantial. Statistical fluctuations can then produce unphysical and oscillatory corrected distributions.

  • Response-matrix setup: The response matrix R maps true bin counts t to measured counts m through m = Rt, with QIS matrices constructed by measuring computational-basis states.The matrix element Rij is the conditional probability Pr(m = i|t = j).
  • Instability of inversion: As off-diagonal migration grows, matrix inversion amplifies response-matrix uncertainties and can produce unphysical entries.For a two-state example, the variance scales as 1/(1 − 2ϵ) and diverges as ϵ approaches 1/2.
  • Gaussian example: Figure 2 compares direct matrix inversion, ignis, and ten-iteration IBU for the Gaussian example using a known response matrix and 10^4 measured events.The visual compares the estimated distributions against the true and measured probability mass functions.
  • Gaussian example: In the Gaussian example with ϵ = 25%, significant off-diagonal components generate oscillations and amplify statistical uncertainties despite similar true and measured distributions.Because the migration is symmetric, the optimal correction would be close to the identity.

III. UNFOLDING METHODS

The paper contrasts matrix inversion and the ignis constrained solution with iterative Bayesian unfolding (IBU), which regularizes correction through finite iteration. IBU preserves non-negative probability outputs and avoids the oscillatory behavior seen in the other approaches in the representative example.

  • Matrix inversion and ignis: The ignis method finds a physically valid vector close to the matrix-inversion result, but it equals that result whenever the inversion is already non-negative and therefore inherits some pathologies.Its optimization enforces preservation of the measured L1 norm.
  • IBU: IBU avoids fitting and matrix inversion by iteratively updating a truth spectrum from a prior, response matrix, and measured distribution.The method is also known as Richardson–Lucy deconvolution.
  • IBU: For non-negative measured counts, IBU produces a non-negative, unit-measure probability result, with the prior and iteration count specified in advance.The number of iterations depends on the desired precision, the initial prior, and the importance of off-diagonal response components.
  • IBU: Finite iteration acts as regularization: relatively small iteration counts can damp oscillations that emerge as IBU approaches convergence.The paper notes that fewer than approximately O(10) iterations are typically needed in practice.
  • Representative comparison: With ten iterations, IBU avoids the rapid oscillations seen in matrix inversion and ignis while remaining non-negative in the Gaussian comparison.The IBU result approaches matrix inversion only after very large iteration counts.

IV. CONSTRUCTING THE RESPONSE MATRIX

The response matrix is calibrated from basis-state measurements, revealing substantial qubit- and hardware-dependent readout variation. Per-qubit error models capture much of the structure, but full-matrix calibration scales exponentially with qubit count.

  • Response-matrix construction: The response matrix is measured by preparing all 2^nqubit computational-basis states with X gates and immediately measuring them.Each matrix element relates true states to measured outcomes, and the matrix is built from calibration circuits.
  • Hardware response: A 5-qubit Johannesburg matrix has substantial off-diagonal migration, including same-qubit flips of about 5–7%, and its entries drift with calibration.Higher-connectivity machines have been observed to exhibit more readout noise.
  • Universality tests: The Johannesburg global fit gives p0→1 ≈3.2% and p1→0 ≈7.5%, while independent per-qubit fits vary by about 50% and 60%, respectively.The first spread applies to p0→1 and the second to p1→0.
  • Universality tests: Transition probabilities also depend on neighboring-qubit states, although per-qubit readout errors capture most salient response-matrix features on this hardware.The authors caution that this sufficiency is hardware dependent and may weaken for higher-connectivity computers.
  • Calibration cost: Constructing the full response matrix requires exponentially many calibration resources, which can become untenable for nqubit ≫1.The study suggests that per-qubit or polynomially sampled approaches may be possible, but leaves them for future work.

V. REPRESENTATIVE RESULTS

The representative tests compare matrix inversion, Ignis, and iterative Bayesian unfolding on Gaussian distributions with pathological and realistic response matrices. Iterative Bayesian unfolding avoids the oscillatory pathology of direct inversion and is slightly more precise in repeated realistic simulations.

  • Benchmark setup: The Gaussian benchmark represents a harmonic-oscillator ground-state distribution mapped onto computational-basis states, with σ = 3.5 in the reported results.The study uses this distribution as a quantum-mechanics-motivated test case for readout errors.
  • Pathological response matrix: With the pathological response matrix, 10-iteration IBU nearly matches the truth distribution, whereas matrix inversion and Ignis show large oscillations.The inversion result is non-negative in this test, making the Ignis result nearly identical; IBU approaches matrix inversion only after about a thousand iterations.
  • Realistic response matrix: For the realistic Johannesburg response matrix, all three methods qualitatively reproduce the truth distribution despite large migrations.The measured-spectrum asymmetry arises from unequal 0→1 and 1→0 migration probabilities induced by the qubit-state mapping.
  • Quantitative comparison: Across 1000 pseudo-experiments with the Johannesburg matrix, IBU predictions are a couple of percent less spread than Ignis and both are about 10% more precise than matrix inversion.The comparison includes irreducible Poisson noise, which the unfolding methods are not expected to correct.

VI. REGULARIZATION AND UNCERTAINTIES

This section evaluates how IBU uncertainty changes with iteration count and describes statistical, response-matrix, and non-closure uncertainties. In the example, total uncertainty is minimized at two iterations, with three iterations nearly equivalent.

  • Regularization choice: IBU bias is lowest at three iterations, but the true bias cannot guide iteration selection because the truth is unknown before measurement.The authors instead motivate choosing iterations by minimizing the total expected uncertainty before unblinding.
  • Uncertainty sources: The uncertainty study considers statistical uncertainty in measured counts, statistical and systematic uncertainty in the response matrix, and method non-closure uncertainty.Non-closure estimates potential unfolding bias, while response-matrix uncertainty reflects calibration impurity and can include modeled gate noise.
  • Examples: For the realistic IBM Q Johannesburg example, Gaussian measurements use a calibrated response matrix and IBU with 100 iterations, with edge deviations partly driven by low-count fluctuations.The figure uses one million shots to construct the response matrix and 10^4 shots for the truth and measured distributions.
  • Examples: Repeated pseudo-experiments compare true and unfolded counts across 25 states, with irreducible Poisson noise retained in the truth values.The setup repeats the Gaussian measurement 1000 times and evaluates distributions of true-minus-predicted counts.
  • Regularization choice: Two iterations minimize the total uncertainty, while the difference between two and three iterations is less than 1%.The total combines individual uncertainty sources in quadrature, excluding gate noise from the measurement simulation.

VII. DISCUSSION

The discussion concludes that HEP unfolding methods provide useful readout-error corrections for quantum computers, with IBU robust to matrix-inversion pathologies. It also identifies scalability and application-specific regularization as important remaining challenges.

  • Discussion: IBU is robust to a matrix-inversion failure mode, while ignis and Bayesian methods are preferred when readout errors are sufficiently small because they remain non-negative.Ignis is described as a special case of TUnfold, which adds covariance-based precision improvements and regularization.
  • Discussion: Realistic response-matrix results show that readout corrections can be significant and must be accounted for in measurements on near-term hardware.This conclusion is drawn from the IBM Q Johannesburg example.
  • Limitations: Constructing the full response matrix requires exponential resources in the number of qubits, which can become prohibitive for large systems.Per-qubit transition probabilities may suffice on hardware with few connections, while other reductions remain future work.
  • Limitations: The iteration metric used in the uncertainty study averages uncertainty across all bins, but applications may instead prioritize an expectation value or an extreme-state uncertainty.The authors emphasize that regularization criteria are application specific.

VIII. CONCLUSIONS AND OUTLOOK

Unfolding tools from high energy physics provide a more robust approach to correcting quantum readout errors. For IBU, early stopping regularizes the solution by damping oscillations that emerge with many iterations.

  • High energy physics unfolding tools are now available to the quantum information science community for more robust readout-error correction.The paper connects the two fields to improve robustness against resolution effects.
  • Increasing IBU iterations causes oscillations resembling those of other algorithms, while early stopping can damp them as a regularization.The oscillations are known to form as the iteration count approaches infinity.

Appendix B: Alternative Configurations of the IBM

Readout-error behavior depends on qubit connectivity, so the study evaluates four IBM Q Johannesburg configurations. Each configuration uses a row of connected qubits with additional inter-row links.

  • Four IBM Q Johannesburg configurations test how connectivity affects quantum readout errors.The configurations correspond to four rows of the machine's qubits.
  • Within each row, every qubit connects to its two neighbors, while edge and middle qubits also have specified inter-row connections.The first and last qubits connect to adjacent rows, with extra connections for middle qubits in the middle rows.

Appendix C: Uncertainties when using IBM Q Response Matrix

The IBM Q response-matrix example is used for the uncertainty analyses corresponding to earlier figures. These analyses examine the example introduced in Fig. 7.

  • The uncertainty analyses in Figs. 13 and 14 use the IBM Q response matrix from the Fig. 7 example.They are the counterparts of Figs. 9 and 10 for that IBM Q-based example.

Appendix D: Alternative to Harmonic Oscillator Ground State

For the spiky W-state distribution, matrix inversion produces large fluctuations because most true probabilities are exactly zero. In contrast, ignis and IBU remain positive, while the accompanying figures examine iteration dependence, universality, bias, uncertainty, and mean squared error.

  • Matrix inversion shows large fluctuations for the spiky W distribution because its many exactly zero probabilities are not specially constrained.The method can therefore produce negative results, unlike positivity-preserving alternatives described in the passage.
  • The appendix includes figures testing IBU iteration dependence, machine-configuration universality, iteration-dependent bias, uncertainty sources, and mean squared error.These figures collectively vary iterations and evaluate agreement with the target distribution.
  • The W-state test compares the noisy measured distribution with the theoretical W distribution using an IBM Q Johannesburg response matrix.The measurement uses 1000 state-measurement shots and 500 × 2^nqubits response-matrix shots, with IBU run for 100 iterations.
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