Perturbative readout-error mitigation for near-term quantum computers

Evan Peters, Andy C. Y. Li, and Gabriel N. Perdue

Phys. Rev. A 107, 062426 – Published 30 June 2023

Abstract

Readout errors on near-term quantum computers can introduce significant error to the empirical probability distribution sampled from the output of a quantum circuit. These errors can be mitigated by classical postprocessing given the access of an experimental response matrix that describes the error associated with the measurement of each computational basis state. However, the resources required to characterize a complete response matrix and to compute the corrected probability distribution scale exponentially with the number of qubits, $n$ . In this work, we modify standard matrix inversion techniques using perturbative approximations with significantly reduced complexity and bounded error when the likelihood of high-order bit-flip events is strongly suppressed. Given a characteristic error rate $q$ , we discuss a method to recover the probability of the all-zeros bit string $p_{0}$ by sampling only a small subspace of the response matrix before inverting readout error, resulting in a relative speedup of $poly [2^{n} / (\begin{matrix} n \\ w \end{matrix})]$ , which we motivate using a simplified error model for which the approximation incurs only $O (q^{w})$ error for some integer $w$ . We then provide a generalized technique to efficiently recover full output distributions with $O (q^{w})$ error in the perturbative limit. These approximate techniques for readout-error correction may greatly accelerate near-term quantum computing applications.

Received 9 May 2023
Accepted 21 June 2023

DOI:https://doi.org/10.1103/PhysRevA.107.062426

Physics Subject Headings (PhySH)

Quantum algorithms Quantum error correction

Quantum Information, Science & Technology

Authors & Affiliations

Evan Peters ^1,2,3,*, Andy C. Y. Li ¹, and Gabriel N. Perdue ¹

¹Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
²Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
³Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

^*e6peters@uwaterloo.ca

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Vol. 107, Iss. 6 — July 2023

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Images

Figure 1
Performance of all-zeros readout-error mitigation given by Eq. (8) compared for different prior distributions. (a) The Gaussian prior is centered at 0 with $n$ -bit overflow for all bit strings with value less than $2^{n - 1}$ , i.e., $p_{j} \propto exp [{(x_{j} - 0.5)}^{2} / σ^{2}]$ , where $x_{j} \equiv 2^{- n} [(j + 2^{n - 1}) mod 2^{n}]$ and $σ = 0.25$ . This distribution is adversarial to recovering $p_{0}$ as it has significant support on the high-weight subspace. (b) The truncated Gaussian is given by the same distribution without overflow ( $x_{j} = j \times 2^{- n}, σ = 0.25)$ and renormalized, and (c) the uniform distribution is $p_{j} = 2^{- n}$ . In all plots, $w = 0$ is defined to correspond to the uncorrected probability $p_{0}^{'}$ . The dashed line indicates the bound of Eq. (19) derived for a relaxation-only model, which we observed was not violated even for the more general error model of Eq. (30).
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Figure 2
(a) Performance of Algorithm 1 as a function of truncation order $w$ , for fixed $q = 0.05$ . The elements of the initial distribution $p$ were each drawn from $Uniform (0, 1)$ and then normalized. We define $w = 0$ to represent the uncorrected case corresponding to $d (p, p^{'})$ . (b) The performance of the algorithm ( $n = 8$ ) diminishes with increasing $q$ , which corresponds to the series approximation for $R^{- 1}$ diverging. We discuss the resulting limitations and workarounds for this behavior in Appendix pp4. (c) Visualization of applying correction to a Gaussian distribution for $n = 8, w = 2$ with a characteristic rate $q = 0.6$ .
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Figure 3
Failure points for variable number of qubits occur around the same threshold $q_{max}$ in units of $10 \frac{q}{n}$ . We empirically observe that this failure threshold scales inversely with $n$ , signifying that our technique loses effectiveness when the characteristic error rate $q$ cannot be suppressed as additional qubits are added to the system.
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Figure 4
Our technique can be modified to overcome the limitations shown in Fig. 3 by exactly inverting $\sum_{j = 1}^{w} R_{j}$ . By doing so, the failure threshold $q_{max}$ approaches 0.5, indicating that the technique will work for arbitrary $R$ constructed according to the model we have employed.
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Figure 5
Performance of all-zeros readout-error mitigation with response matrices experimentally determined on IBM QPUs. We demonstrate the performance using three prior distributions, namely, (a) Gaussian distribution, (b) truncated Gaussian distribution, and (c) uniform distribution. The details of the distributions are explained in the caption of Fig. 1.
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Figure 6
The accuracy of our technique for recovering the all-zeros bit string is statistically indistinguishable to that of M3 for the distributions considered in Fig. 1 using $n = 8$ qubits, with $M = N = 10^{6}$ circuit repetitions used for both the calibration experiment and for sampling $p^{'}$ (see main text). The response matrix $R$ has the same characteristic error rate $q$ as in previous numerics. Error bars denote standard deviation.
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