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arXiv:2412.01311 · Quantum

Reduced Sampling Overhead for Probabilistic Error Cancellation by Pauli Error Propagation

Timon Scheiber, Paul Haubenwallner, Matthias Heller

Published Aug 29, 2025• 0 citations
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Abstract

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.

DOI
10.22331/q-2025-08-29-1840

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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.
Current quantum computers suffer from a level of noise that prohibits extracting useful results directly from longer computations. The figure of merit in many near-term quantum algorithms is an expectation value measured at the end of the computation, which experiences a bias in the presence of hardware noise. A systematic way to remove such bias is probabilistic error cancellation (PEC). PEC requires a full characterization of the noise and introduces a sampling overhead that increases exponentially with circuit depth, prohibiting high-depth circuits at realistic noise levels. Probabilistic error reduction (PER) is a related quantum error mitigation method that systematically reduces the sampling overhead at the cost of reintroducing bias. In combination with zero-noise extrapolation, PER can yield expectation values with an accuracy comparable to PEC. Noise reduction through PER is broadly applicable to near-term algorithms, and the automated implementation of PER is thus desirable for facilitating its widespread use. To this end, we present an automated quantum error mitigation software framework that includes noise tomography and application of PER to user-specified circuits. We provide a multi-platform Python package that implements a recently developed Pauli noise tomography (PNT) technique for learning a sparse Pauli noise model and exploits a Pauli noise scaling method to carry out PER. We also provide software tools that leverage a previously developed toolchain, employing PyGSTi for gate set tomography and providing a functionality to use the software Mitiq for PER and zero-noise extrapolation to obtain error-mitigated expectation values on a user-defined circuit.
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.
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.
Probabilistic error cancellation (PEC) is a leading quantum error mitigation method that provides an unbiased estimate, although it is known to have a large sampling overhead. In this work, we propose a new method to perform PEC, which results in a lower sampling cost than the standard way. It works by decomposing the inverse channel of each gate or each circuit layer into the identity part and the non-identity part and reorganizing the full circuit as different powers of the inverse generator. The ideal circuit becomes a linear combination of noisy circuits with different weights where shots are deterministically allocated to each circuit based on its weight. This naturally sets the achievable bias given a finite amount of shots. As the number of shots is increased, smaller bias terms can be gradually resolved and become bias-free in the limit of sufficient shots. We show the saving both analytically and numerically over the standard PEC and identify situations where it can outperform heuristic approach, such as zero-noise extrapolation, due to the well-controlled bias. We also demonstrated this method experimentally and found excellent agreement between the mitigated and the ideal values.
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.
The pursuit of practical quantum utility on near-term quantum processors is critically challenged by their inherent noise. Quantum error mitigation (QEM) techniques are leading solutions to improve computation fidelity with relatively low qubit-overhead, while full-scale quantum error correction remains a distant goal. However, QEM techniques incur substantial measurement overheads, especially when applied to families of quantum circuits parameterized by classical inputs. Focusing on zero-noise extrapolation (ZNE), a widely adopted QEM technique, here we devise the surrogate-enabled ZNE (S-ZNE), which leverages classical learning surrogates to perform ZNE entirely on the classical side. Unlike conventional ZNE, whose measurement cost scales linearly with the number of circuits, S-ZNE requires only constant measurement overhead for an entire family of quantum circuits, offering superior scalability. Theoretical analysis indicates that S-ZNE achieves accuracy comparable to conventional ZNE in many practical scenarios, and numerical experiments on up to 100-qubit ground-state energy and quantum metrology tasks confirm its effectiveness. Our approach provides a template that can be effectively extended to other quantum error mitigation protocols, opening a promising path toward scalable error mitigation.