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arXiv:2403.18793 · Quantum

Theory of quantum error mitigation for non-Clifford gates

David Layden, Bradley Mitchell, Karthik Siva

Published Feb 10, 2026• 1 citation
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Abstract

Quantum error mitigation techniques mimic noiseless quantum circuits by running several related noisy circuits and combining their outputs in particular ways. How well such techniques work is thought to depend strongly on how noisy the underlying gates are. Weakly-entangling gates, like R Z Z ( θ ) for small angles θ , can be much less noisy than entangling Clifford gates, like CNOT and CZ, and they arise naturally in circuits used to simulate quantum dynamics. However, such weakly-entangling gates are non-Clifford, and are therefore incompatible with two of the most prominent error mitigation techniques to date: probabilistic error cancellation (PEC) and the related form of zero-noise extrapolation (ZNE). This paper generalizes these techniques to non-Clifford gates, and comprises two complementary parts. The first part shows how to effectively transform any given quantum channel into (almost) any desired channel, at the cost of a sampling overhead, by adding random Pauli gates and processing the measurement outcomes. This enables us to cancel or properly amplify noise in non-Clifford gates, provided we can first characterize such gates in detail. The second part therefore introduces techniques to do so for noisy R Z Z ( θ ) gates. These techniques are robust to state preparation and measurement (SPAM) errors, and exhibit concentration and sensitivity—crucial statistical properties for many experiments. They are related to randomized benchmarking, and may also be of interest beyond the context of error mitigation. We find that while non-Clifford gates can be less noisy than related Cliffords, their noise is fundamentally more complex, which can lead to surprising and sometimes unwanted effects in error mitigation. Whether this trade-off can be broadly advantageous remains to be seen.

DOI
10.22331/q-2026-02-10-2003

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Quantum error mitigation techniques mimic noiseless quantum circuits by running several related noisy circuits and combining their outputs in particular ways. How well such techniques work is thought to depend strongly on how noisy the underlying gates are. Weakly-entangling gates, like R Z Z ( θ ) for small angles θ , can be much less noisy than entangling Clifford gates, like CNOT and CZ, and they arise naturally in circuits used to simulate quantum dynamics. However, such weakly-entangling gates are non-Clifford, and are therefore incompatible with two of the most prominent error mitigation techniques to date: probabilistic error cancellation (PEC) and the related form of zero-noise extrapolation (ZNE). This paper generalizes these techniques to non-Clifford gates, and comprises two complementary parts. The first part shows how to effectively transform any given quantum channel into (almost) any desired channel, at the cost of a sampling overhead, by adding random Pauli gates and processing the measurement outcomes. This enables us to cancel or properly amplify noise in non-Clifford gates, provided we can first characterize such gates in detail. The second part therefore introduces techniques to do so for noisy R Z Z ( θ ) gates. These techniques are robust to state preparation and measurement (SPAM) errors, and exhibit concentration and sensitivity—crucial statistical properties for many experiments. They are related to randomized benchmarking, and may also be of interest beyond the context of error mitigation. We find that while non-Clifford gates can be less noisy than related Cliffords, their noise is fundamentally more complex, which can lead to surprising and sometimes unwanted effects in error mitigation. Whether this trade-off can be broadly advantageous remains to be seen.
Quantum error mitigation is regarded as a possible path to near-term quantum utility. The methods under the quantum error mitigation umbrella term, such as probabilistic error cancellation (PEC), zero-noise extrapolation (ZNE) or Clifford data regression (CDR) are able to significantly reduce the error for the estimation of expectation values, although at an exponentially scaling cost, i.e., in the sampling overhead. In this work, we present a method to reduce the sampling overhead of PEC through Pauli error propagation combined with classical preprocessing. Our findings indicate that this method significantly reduces sampling overheads for Clifford circuits, leveraging the well-defined interaction between the Clifford group and Pauli noise. Additionally, we show that the method is applicable to non-Clifford circuits, though with more limited effectiveness, primarily constrained by the number of non-Clifford gates present in the circuit. We further provide examples of Clifford sub-circuits commonly encountered in relevant calculations, such as resource state generation in measurement-based quantum computing.
We propose a general framework for quantum error mitigation that combines and generalizes two techniques: probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE). Similar to PEC, the proposed method represents ideal operations as linear combinations of noisy operations that are implementable on hardware. However, instead of assuming a fixed level of hardware noise, we extend the set of implementable operations by noise scaling. By construction, this method encompasses both PEC and ZNE as particular cases and allows us to investigate a larger set of hybrid techniques. For example, gate extrapolation can be used to implement PEC without requiring knowledge of the device's noise model, e.g., avoiding gate-set tomography. Alternatively, probabilistic error reduction can be used to estimate expectation values at intermediate virtual noise strengths (below the hardware level), leading to partially mitigated results at a lower sampling cost. Moreover, multiple results obtained with different noise-reduction factors can be further postprocessed with ZNE to better approximate the zero-noise limit.
Classical shadows enable us to learn many properties of a quantum state ρ with very few measurements. However, near-term and early fault-tolerant quantum computers will only be able to prepare noisy quantum states ρ and it is thus a considerable challenge to efficiently learn properties of an ideal, noise-free state ρ id . We consider error mitigation techniques, such as probabilistic error cancelation (PEC), zero noise extrapolation (ZNE), and symmetry verification (SV), which have been developed for mitigating errors in single expected value measurements and generalize them for mitigating errors in classical shadows. We find that PEC is the most natural candidate and thus develop a thorough theoretical framework for PEC shadows with the following rigorous theoretical guarantees: PEC shadows are an unbiased estimator for the ideal quantum state ρ id ; the sample complexity for simultaneously predicting many linear properties of ρ id is identical to that of the conventional shadows approach up to a multiplicative factor, which is the sample overhead due to error mitigation. Due to efficient postprocessing of shadows, this overhead does not depend directly on the number of qubits but rather grows exponentially with the number of noisy gates. The broad set of tools introduced in this work may be instrumental in exploiting near-term and early fault-tolerant quantum computers: we demonstrate in detailed numerical simulations a range of practical applications of quantum computers that will significantly benefit from our techniques. Published by the American Physical Society 2024
Overcoming experimental errors is a central challenge in quantum science. This challenge is particularly acute in large and complex quantum circuits, where quantum information spreads into exponentially many paths, foiling any direct attempt to understand errors' impact. In this work, we present a novel approach to understanding and mitigating experimental errors, based on a surprisingly compact representation of a quantum circuit's information dynamics, the reactivity function. The shape of a circuit's reactivity function governs its noise response, and its amenability to different error mitigation methods. From this viewpoint, we introduce two complementary such methods. The first, Pauli-path zero-noise extrapolation, fits a parameterized model to a circuit's reactivity function and uses it to accurately extrapolate to zero noise. The second, tunable error cancellation, filters a circuit's reactivity function to cancel noise on all information up to a threshold locality. Crucially, both methods feature a tuning parameter that allows one to systematically improve their accuracy until convergence. Across three circuit families from published experiments, our methods overcome conventional zero-noise extrapolation biases as large as 30%; at ∼\sim∼10−210^{-2}10−2 root-mean-square error, they reduce sampling cost relative to probabilistic error cancellation by up to a factor of 10310^3103.
Until fault-tolerance becomes implementable at scale, quantum computing will heavily rely on noise mitigation techniques. While methods such as zero noise extrapolation with probabilistic error amplification (ZNE-PEA) and probabilistic error cancellation (PEC) have been successfully tested on hardware recently, their scalability to larger circuits may be limited. Here, we introduce the tensor-network error mitigation (TEM) algorithm, which acts in post-processing to correct the noise-induced errors in estimations of physical observables. The method consists of the construction of a tensor network representing the inverse of the global noise channel affecting the state of the quantum processor, and the consequent application of the map to informationally complete measurement outcomes obtained from the noisy state. TEM does therefore not require additional quantum operations other than the implementation of informationally complete POVMs, which can be achieved through randomised local measurements. The key advantage of TEM is that the measurement overhead is quadratically smaller than in PEC. We test TEM extensively in numerical simulations in different regimes. We find that TEM can be applied to circuits of twice the depth compared to what is achievable with PEC under realistic conditions with sparse Pauli-Lindblad noise, such as those in [E. van den Berg et al., Nat. Phys. (2023)]. By using Clifford circuits, we explore the capabilities of the method in wider and deeper circuits with lower noise levels. We find that in the case of 100 qubits and depth 100, both PEC and ZNE fail to produce accurate results by using ∼105\sim 10^5∼105 shots, while TEM succeeds.
Using near-term quantum computers to achieve a quantum advantage requires efficient strategies to improve the performance of the noisy quantum devices presently available. We develop and experimentally validate two efficient error mitigation protocols named ``Noiseless Output Extrapolation" and ``Pauli Error Cancellation" that can drastically enhance the performance of quantum circuits composed of noisy cycles of gates. By combining popular mitigation strategies such as probabilistic error cancellation and noise amplification with efficient noise reconstruction methods, our protocols can mitigate a wide range of noise processes that do not satisfy the assumptions underlying existing mitigation protocols, including non-local and gate-dependent processes. We test our protocols on a four-qubit superconducting processor at the Advanced Quantum Testbed. We observe significant improvements in the performance of both structured and random circuits, with up to 86% improvement in variation distance over the unmitigated outputs. Our experiments demonstrate the effectiveness of our protocols, as well as their practicality for current hardware platforms.