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arXiv:2105.08161 · Physical Review A

Perturbative readout-error mitigation for near-term quantum computers

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

Published Jun 30, 2023• 7 citations
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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 $\text{poly}[{2}^{n}/(\begin{array}{c}n\\ w\end{array})]$, 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.

DOI
10.1103/physreva.107.062426

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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, nnn. 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 qqq, we discuss a method to recover the probability of the all-zeros bit string p0{p}_{0}p0​ by sampling only a small subspace of the response matrix before inverting readout error, resulting in a relative speedup of poly[2n/(nw)]\text{poly}[{2}^{n}/(\begin{array}{c}n\\ w\end{array})]poly[2n/(nw​)], which we motivate using a simplified error model for which the approximation incurs only O(qw)O({q}^{w})O(qw) error for some integer www. We then provide a generalized technique to efficiently recover full output distributions with O(qw)O({q}^{w})O(qw) error in the perturbative limit. These approximate techniques for readout-error correction may greatly accelerate near-term quantum computing applications.
The readout error on near-term quantum devices is one of the dominant noise factors, which can be mitigated by classical postprocessing called quantum readout error mitigation (QREM). The standard QREM applies the inverse of noise calibration matrix to the outcome probability distribution using exponential computational resources to the number of measured qubits. This becomes infeasible for the current quantum devices with tens of qubits or more. Here we propose two efficient QREM methods finishing in O(ns2)O(n{s}^{2})O(ns2) time for probability distributions of nnn qubits and sss shots, which mainly aim at mitigating sparse probability distributions such that only a few states are dominant. We compare the proposed methods with several recent QREM methods in the following three cases: expectation values of the GHZ state, its fidelities, and the estimation error of maximum likelihood amplitude estimation (MLAE) algorithm with a modified Grover iterator. The two cases of the GHZ state are on real IBM quantum devices, while the third is with numerical simulation. Using the proposed method, the mitigation of the 65-qubit GHZ state takes only a few seconds, and we witness the fidelity of the 29-qubit GHZ state exceeding 0.5. The proposed methods also succeed in reducing the estimation error in the MLAE algorithm, outperforming the results by other QREM methods in general.
Mitigating errors is a significant challenge for near-term quantum computers. One of the most important sources of errors is related to the readout of the quantum state into a classical bit stream. A variety of techniques have been proposed to mitigate these errors with post-hoc corrections. We propose a complementary scheme to actively reduce readout errors on a shot-by-shot basis by encoding single qubits, immediately prior to readout, into multiqubit states. The computational resources of our technique are independent of the circuit depth and are compatible with the error rates and connectivity of many current devices. We analyze the potential of our approach using two types of error-correcting codes, and, as a proof of principle, we demonstrate an 80% improvement in readout error on the IBMQ Mumbai quantum computer.
Readout errors are a significant source of noise for near-term intermediate-scale quantum computers. Mismeasuring a qubit as $|1\ensuremath{\rangle}$ when it should be $|0\ensuremath{\rangle}$ occurs much less often than mismeasuring a qubit as $|0\ensuremath{\rangle}$ when it should have been $|1\ensuremath{\rangle}$. We make the simple observation that one can improve the readout fidelity of quantum computers by applying targeted XXX gates prior to performing a measurement. These XXX gates are placed so that the expected number of qubits in the $|1\ensuremath{\rangle}$ state is minimized. Classical postprocessing can undo the effect of the XXX gates so that the expectation value of any observable is unchanged. This protocol is designed to have no effect when readout errors are symmetric. We show that the statistical uncertainty following readout error corrections is smaller when using readout rebalancing. The statistical advantage is circuit and computer dependent and is demonstrated for the WWW state, a Grover search, and for a Gaussian state. The benefit in statistical precision is most pronounced (and nearly a factor of 2 in some cases) when states with many qubits in the excited state have high probability.
The approximate state estimation and the closely related classical shadows methods allow for the estimation of complicated observables with relatively few shots. As these methods make use of random measurements that can symmetrise the effect of readout errors, they have been shown to permit simplified approaches to readout error mitigation which require only a number of samples that scales as O(1)\mathcal{O}(1)O(1) with increasing numbers of qubits. However, these techniques require executing a different circuit at each shot, adding a typically prohibitive amount of latency that prohibits their practical application. In this manuscript we consider the approximate state estimation of readout-mitigated expectation values, and how to best implement that procedure on the Rigetti quantum computing hardware. We discuss the theoretical aspects involved, providing an explicit computation of the effect of readout error on the estimated expectation values and how to mitigate that effect. Leveraging improvements to the Rigetti control systems, we then demonstrate an efficient implementation of this approach. Not only do we find that we can suppress the effect of correlated errors and accurately mitigate the readout errors, we find that we can do so quickly, collecting and processing 10610^6106 samples in less than 1.51.51.5 minutes. This development opens the way for practical uses of methods with this type of randomisation.
Readout error models for noisy quantum devices almost universally assume that measurement noise is classical: the measurement statistics are obtained from the ideal computational-basis populations by a column-stochastic assignment matrix AAA. This description is equivalent to assuming that the effective positive-operator-valued measurement (POVM) is diagonal in the measurement basis, and therefore completely insensitive to quantum coherences. We relax this assumption and derive a fully general expression for the observed measurement probabilities under arbitrary completely positive trace-preserving (CPTP) noise preceding a computational-basis measurement. Writing the ideal post-circuit state ρ~\tildeρρ~​ in terms of its populations xxx and coherences yyy, we show that the observed probability vector zzz satisfies z=Ax+Cyz = A x + C yz=Ax+Cy, where AAA is the familiar classical assignment matrix and CCC is a coherence-response matrix constructed from the off-diagonal matrix elements of the effective POVM in the computational basis. The classical model z=Axz = A xz=Ax arises if and only if all POVM elements are diagonal; in this sense CCC quantifies accessible information about coherent readout distortions and interference between computational-basis states, all of which are invisible to models that retain only AAA. Our numerical experiments show that incorporating CCC into readout recovery can improve fidelity over classical inversion and enable selective Pauli twirling with exponentially reduced circuit overhead. This work therefore provides a natural, fully general framework for coherence-sensitive readout modeling on current and future quantum devices.
We develop a classical bit-flip correction method to mitigate measurement errors on quantum computers. This method can be applied to any operator, any number of qubits, and any realistic bit-flip probability. We first demonstrate the successful performance of this method by correcting the noisy measurements of the ground-state energy of the longitudinal Ising model. We then generalize our results to arbitrary operators and test our method both numerically and experimentally on IBM quantum hardware. As a result, our correction method reduces the measurement error on the quantum hardware by up to one order of magnitude. We finally discuss how to preprocess the method and extend it to other error sources beyond measurement errors. For local Hamiltonians, the overhead costs are polynomial in the number of qubits, even if multiqubit correlations are included.