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Scalable mitigation of measurement errors on quantum computers

Paul D. Nation, Hwajung Kang, Neereja Sundaresan, Jay M. Gambetta

arXiv:2108.12518v1quant-ph

TL;DR

Measurement mitigation is difficult to scale because conventional methods form the full assignment matrix. This paper instead works in a noisy-sample subspace, supports correlated and uncorrelated errors, and uses matrix-free iterative solves with efficient uncertainty estimates. In demonstrations, mitigation completes in seconds at 42 qubits while using dramatically less memory than full-matrix approaches.

  • Problem

    Measurement mitigation methods that use the full 2^N probability space and assignment matrix do not scale beyond a handful of qubits.

  • Method

    The method constructs a reduced assignment matrix over noisy bit-strings, computes entries from calibration data, and uses matrix-free iterative solves with uncertainty estimates.

  • Results

    At 42 qubits, M3 computes mitigated expectation values in at most 1.2 seconds, while mitigation-overhead timings are 2.4 seconds for direct and 4.5 seconds for iterative solutions.

  • Takeaways & Limitations

    The demonstrated approach enables measurement-error mitigation at scales that previous full-matrix methods make intractable, with substantially lower storage requirements.

Abstract

from arXiv · show

We present a method for mitigating measurement errors on quantum computing platforms that does not form the full assignment matrix, or its inverse, and works in a subspace defined by the noisy input bit-strings. This method accommodates both uncorrelated and correlated errors, and allows for computing accurate error bounds. Additionally, we detail a matrix-free preconditioned iterative solution method that converges in $\mathcal{O}(1)$ steps that is performant and uses orders of magnitude less memory than direct factorization. We demonstrate the validity of our method, and mitigate errors in a few seconds on numbers of qubits that would otherwise be intractable.

I. INTRODUCTION

Measurement errors are a major obstacle for near-term quantum experiments, while existing mitigation methods scale poorly because they operate in the full probability space. The paper introduces subspace reduction and matrix-free iterative techniques to make mitigation scalable and memory efficient.

  • Measurement errors play an outsized role in short-depth circuits and must be corrected for many near-term experiments.
  • Individual assignment-matrix elements can be computed from bit-string values and calibration data requiring at most O(N) calibration circuits.
  • Existing techniques use the full 2^N-dimensional probability space, limiting scalability beyond a handful of qubits.
  • The proposed reduction uses the noisy input bit-strings as a subspace because weak measurement errors mainly redistribute probability to nearby strings.
  • Preconditioned matrix-free iterative methods typically converge in O(1) steps while requiring orders of magnitude less memory than direct approaches.
  • The method provides efficient variance bounds and mitigates readout errors at qubit counts that previous methods make intractable.

II. SUBSPACE REDUCTION

The method constructs a reduced assignment matrix only over bit-strings observed in the noisy distribution, computing selected entries directly from calibration data. Hamming-distance truncation controls sparsity, while column renormalization preserves the assignment interpretation.

  • The full tensored assignment matrix is represented as a Kronecker product of N single-qubit calibration matrices.
  • Individual matrix elements are computed directly from bit-string values and calibration matrices, without forming the full assignment matrix.
  • Selecting entries with Hamming distance d(row, col) ≤ D varies the reduced matrix sparsity and enables low-distance approximations.
  • The reduced matrix contains only rows and columns corresponding to bit-strings observed in the noisy probability vector.
  • Reduced-matrix columns must be renormalized after selecting entries so that each column again sums to one.
  • Finite sampling can omit small ideal-probability elements from the noisy distribution, although the paper reports this effect is minimal for typical sample counts.

III. MATRIX-FREE SOLUTION

The method solves the reduced assignment system without explicitly forming its matrix, using matrix-free iterative methods with Jacobi preconditioning. These methods converge rapidly while substantially lowering memory requirements.

  • Matrix-free iterative methods avoid explicitly constructing the reduced assignment matrix when it becomes too costly to store.Individual matrix elements can be accessed as needed during the solve.
  • A Jacobi preconditioner exploits near-diagonal dominance to accelerate convergence.The preconditioner uses diagonal entries of the reduced assignment matrix.
  • O(1) iterations achieve an absolute tolerance of 10^-5 in practice.
  • The iterative approach dramatically reduces memory requirements for measurement-error mitigation.

IV. UNCERTAINTY ESTIMATES

The paper estimates uncertainty from mitigation overhead without constructing the inverse reduced assignment matrix. The estimate is efficient, but its value depends on the circuit and device noise properties.

  • Mitigation overhead M increases the standard deviation bound for an observable as σ_O ≤ M/s.Here s is the number of circuit samples.
  • The Hager-Higham algorithm estimates ||Ã^-1||_1 using linear systems involving à and Ã^T.This avoids explicitly constructing Ã^-1.
  • Direct factorization makes uncertainty estimation approximately twice as slow as mitigation alone, while iterative methods incur a 4–10x overhead.
  • The estimated overhead is often exact or nearly so, but depends on the executed circuit and device noise properties.

V. DEMONSTRATIONS

Experiments validate M3 on IBM Quantum systems and demonstrate mitigation for circuits and GHZ states at scales where competing methods become infeasible. The method reaches 42 qubits with low runtime and much lower memory use.

  • M3 was implemented with NumPy, SciPy, Cython, and Qiskit, and timing was measured on a quad-core Intel i3-10100 system with 32 GB memory.
  • Using at most 371 bit-strings, M3 was validated against Qiskit Ignis on a 27-qubit IBM Quantum Kolkata experiment.
  • Only M3 mitigated GHZ-state errors beyond 14 qubits, reaching demonstrations out to 42 qubits on IBM Quantum Brooklyn.
  • At 42 qubits, M3 computed mitigation in 1.2 seconds, or 2.4 seconds with direct overhead estimation and 4.5 seconds with iterative estimation.
  • The D = 3 Hamming approximation matched full M3 mitigation to a tolerance of approximately 10^-5.
  • At 42 qubits, storing a full p_noisy vector requires 16 TiB, whereas M3 iterative storage requires approximately 1 MiB.

1. Complete A-matrix

The complete A-matrix represents measurement-error transitions across all computational-basis bit-strings, while the tensored matrix factors single-qubit calibration data and remains diagonally dominant.

  • Complete A-matrix: The complete A-matrix is formed from calibration circuits for every computational-basis bit-string and is nominally sparse because nearby bit-strings have appreciable transition probabilities.For six qubits, this entails 32 bit-strings and 8192 executions per circuit.
  • Complete A-matrix: The complete A-matrix has strict diagonal dominance, which guarantees invertibility.
  • Tensored A-matrix: The tensored A-matrix is constructed from single-qubit calibration matrices using a tensor product.
  • Tensored A-matrix: The tensored matrix is strictly diagonally dominant and indicates that transitions between bit-strings close in Hamming distance are more likely.

3. Example 12 qubit truncated A-matrix

A truncated assignment matrix used in M3 contains only the observed bit-strings and can vary in its number of elements and amplitudes across circuit executions.

  • 3. Example 12 qubit truncated A-matrix: Each circuit execution can produce a truncated matrix whose number of elements and associated amplitudes vary.The example is one of 100 truncated matrices used in the M3 mitigation experiment.

Appendix B: Correlated errors

The method supports correlated readout errors by computing matrix elements from bit-string values and local calibration data, while experiments show correlated M3 can match the complete A-matrix where tensored mitigation over-corrects.

  • Appendix B: Correlated errors: The matrix construction method handles correlated errors when assignment-matrix elements can be obtained from bit-string values.Pairwise correlations can be incorporated using 4x4 stochastic local noise matrices.
  • Appendix B: Correlated errors: Under normal operating conditions, correlated and tensored M3 results agree with the experiment, consistent with predominantly uncorrelated readout errors.
  • Appendix B: Correlated errors: Increasing readout pulse amplitudes by approximately 2x produces correlated readout errors without appreciably changing average readout fidelity by more than 0.2%.
  • Appendix B: Correlated errors: In the induced-correlated-error experiment, correlated M3 agrees with the complete A-matrix, whereas tensored mitigation strongly over-corrects.
  • Appendix B: Correlated errors: The correlated-error formulation works for uncorrelated and pairwise correlated errors, but each matrix element requires O floating-point evaluations rather than N.

1. Experimental details

The experiments use the 65-qubit IBM Quantum Brooklyn system to prepare GHZ states by adding entangling layers to an initial Hadamard operation.

  • 1. Experimental details: Experiments are run on the 65-qubit IBM Quantum Brooklyn system using GHZ states prepared from a Hadamard gate on qubit 11.After the first six qubits are entangled, the preparation pattern increases the GHZ state by four qubits per layer.
  • 1. Experimental details: The experimental setup reports an average assignment error rate of 2.15% across the qubits used.

2. Memory requirements for storing full 42 qubit A-matrix to D = 3

For a full 42-qubit assignment matrix retaining elements through Hamming distance three, storage becomes far beyond practical supercomputer memory. The estimate totals 580.5 PiB, about 120 times Fugaku’s available memory.

  • Including elements up to Hamming distance three requires a large number of entries in each matrix column.
  • 580.5 PiB is required to store the 42-qubit matrix and its compressed-sparse-column indices.The values require 193.5 PiB, while indices add 387 PiB.
  • 120x larger than Fugaku’s 4.85 PiB memory, making explicit storage impractical at this scale.
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