Error Mitigation for Short-Depth Quantum Circuits
Open accessAbstract
Two schemes are presented that mitigate the effect of errors and decoherence in short-depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate estimates of expectation values to become relevant. Decoherence and gate errors lead to wrong estimates of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and do not require additional qubit resources, so to be as practically relevant in current experiments as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasiprobability distribution.
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Two variants of polynomial extrapolation are linear extrapolation (d=1) and Richardson extrapolation (d=m-1).
“Consequently, we can obtain two other variants: the linear extrapolation ( d = 1 ) and the Richardson extrapolation ( d = m -1 ) (Temme et al. , 2017).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
The protocol on p. 2 uses n+1 noise-scaled runs for an nth-order Richardson estimator, consistent with an order of m−1 when m runs are used. However, the paper does not identify linear extrapolation as a separate variant or state the claimed distinction between linear and Richardson extrapolation. (LLM, conf=0.79)
Experimental — accuracy not yet measuredA fundamental drawback of both zero-noise extrapolation and probabilistic error cancellation or quasi-probability methods is that one needs to know the precise physical noise model in advance, which is itself difficult.
“More importantly, the fundamental drawback of both ZNE and probabilistic error cancellation (PEC) (Temme et al. , 2017), or quasi-probability method (which is discussed next) is that one needs to know the precise physical noise model in advance, which in itself is a difficult problem.”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
The paper says probabilistic error cancellation requires full characterization of noisy operations, which it describes as challenging (Conclusions, p. 4). But it explicitly says zero-noise extrapolation makes no assumption about the noise model beyond weak, time-constant noise (Introduction, p. 1). Thus the claimed drawback applies to PEC, not both methods. (LLM, conf=0.90)
Experimental — accuracy not yet measuredZero-noise extrapolation (ZNE) was independently proposed by Li and Benjamin (2017) and Temme et al. (2017).
“(Li and Benjamin, 2017) and (Temme et al. , 2017) independently proposed the Richardson extrapolation QEMor zero-noise extrapolation (ZNE) technique.”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
Probabilistic error cancellation or quasiprobability decomposition can estimate the expectation value of an observable by sampling from a set of erroneous circuits.
“The central theme of probabilistic error cancellation (PEC) or quasiprobability decomposition introduced by Ref. (Temme et al. , 2017) is that one can estimate the expectation value of an observable by sampling from a set of erroneous circuits”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
Noise-scaling can be accomplished in three ways: time-scaling, circuit folding, and gate folding.
“Noise-scaling can be accomplished in three ways. Firstly, in Ref. (Temme et al. , 2017) it was proposed to use a time-scaling approach to take λ > 1 , which means that the time-dependent driving Hamiltonian ˆ K ( t ) is now rescaled by 1 λ ˆ K ( t / λ ) .”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
The mean value of the probabilistic error cancellation outcome centers around the ideal one with a larger variance due to C.
“As a consequence, the mean value of the PEC outcome centers around the ideal one with a larger variance due to C”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
Given specific error models and full knowledge of them, the quasiprobabilities q_l can be efficiently derived.
“Given specific error models (assuming the experimentalist has full knowledge about them), the real numbers q l , which represent quasiprobabilities, can be efficiently derived.”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
Probabilistic error cancellation relies on correct knowledge of the error model L_tot.
“The above PEC method relies on the correct knowledge of error model L ( l ) tot as is apparent from Eq. (57).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Temme et al., 2017
Computing the expectation value on an approximate restricted Boltzmann machine instead of directly from measurements substantially reduces the sampling variance for a given number of shots at the cost of introducing a small, positive bias.
“However, by computing the expectation value on an approximate RBM instead of directly from measurements the sampling variance for a given number of shots is substantially reduced at the cost of introducing a small, positive bias [227].”
From Variational quantum algorithms (2021) · cites this paper as [227]
The probabilistic error-cancellation discussion (p. 3) and Numerical simulations (p. 12) describe estimators based on noisy-circuit measurements, not an approximate restricted Boltzmann machine. Searches for RBM-based expectation values, reduced sampling variance, and positive bias found no passage addressing the claimed trade-off. (LLM, conf=0.76)
Experimental — accuracy not yet measuredFrom the covariance matrix it is possible to obtain the corresponding W matrix, which is later decomposed into 2N(N-1) matchgates acting on neighboring qubits and N local rotations, using the Hoffmann algorithm.
“From the covariance matrix it is possible to obtain the corresponding W matrix, which is later decomposed into 2 N ( N -1) matchgates acting on neighboring qubits and N local rotations, using the Hoffmann algorithm 337 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [337]
Error mitigation generally requires exponential resource overhead with circuit depth.
“it was shown that error mitigation generally requires exponential resource overhead with circuit depth [32-34].”
From Quantum Dynamic Programming (2025) · cites this paper as [33]
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Wang, W.-T., He, X.-G., Kao, H.-C., & Chang, C.-R. (2024) · Chinese Journal of Physics · doi:10.1016/j.cjph.2024.05.006 · Cited by 1 · see citing works
- Authors
- Temme, K., Bravyi, S., & Gambetta, J. M.
- Affiliations
- IBM Research - Thomas J. Watson Research Center (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2017-11-03
- Volume
- 119
- Issue
- 18
- Pages
- 180509-180509
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2562526363
- Cited by
- 1,379
- References
- 24
- Citations by year
- 2026: 217 · 2025: 281 · 2024: 237 · 2023: 207 · 2022: 152 · 2021: 144 · 2020: 85 · 2019: 39 · 2018: 15 · 2017: 2
- FWCI
- 22.99
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- IBM Research Frontiers Institute
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- error mitigation, decoherence, gate errors, zero-noise extrapolation, Richardson extrapolation, quasiprobability distributions, expectation value estimation, noisy quantum devices, quantum simulation
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