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arXiv:2502.08735

CV4Quantum: Reducing the Sampling Overhead in Probabilistic Error Cancellation Using Control Variates

Prasanth Shyamsundar, Wern Yeen Yeong

Published Feb 12, 2025• 1 citation
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Abstract

Quasiprobabilistic decompositions (QPDs) play a key role in maximizing the utility of near-term quantum hardware. For example, Probabilistic Error Cancellation (PEC) (an error mitigation technique) and circuit cutting (which enables large quantum computations to be performed on quantum hardware with a limited number of qubits) both involve QPDs. Computations based on QPDs typically incur large sampling overheads that grow exponentially, e.g., with the number of error-terms mitigated or the number of circuit-cuts employed, limiting their practical feasibility. In this work, we adapt the control variates variance reduction technique from the statistics literature in order to reduce the sampling overhead in QPD-based computations. We demonstrate our method, dubbed CV4Quantum, using simulation experiments that mimic a realistic PEC scenario. In more than 50% of the PEC-based estimations performed in the study, we observed a more than 50% reduction in the number of samples needed to achieve a given precision when using our technique. We discuss how future research on constructing good control variates can lead to even stronger sampling overhead reduction.

DOI
10.48550/arxiv.2502.08735

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Quasiprobabilistic decompositions (QPDs) play a key role in maximizing the utility of near-term quantum hardware. For example, Probabilistic Error Cancellation (PEC) (an error mitigation technique) and circuit cutting (which enables large quantum computations to be performed on quantum hardware with a limited number of qubits) both involve QPDs. Computations based on QPDs typically incur large sampling overheads that grow exponentially, e.g., with the number of error-terms mitigated or the number of circuit-cuts employed, limiting their practical feasibility. In this work, we adapt the control variates variance reduction technique from the statistics literature in order to reduce the sampling overhead in QPD-based computations. We demonstrate our method, dubbed CV4Quantum, using simulation experiments that mimic a realistic PEC scenario. In more than 50% of the PEC-based estimations performed in the study, we observed a more than 50% reduction in the number of samples needed to achieve a given precision when using our technique. We discuss how future research on constructing good control variates can lead to even stronger sampling overhead reduction.
Quasiprobabilistic decompositions (QPDs) play a key role in maximizing the utility of near-term quantum hardware. For example, Probabilistic Error Cancellation (PEC) (an error mitigation technique) and circuit cutting (which enables large quantum computations to be performed on quantum hardware with a limited number of qubits) both involve QPDs. Computations based on QPDs typically incur large sampling overheads that grow exponentially with the number of error-terms mitigated or number of circuit-cuts employed, limiting their practical feasibility. In this work, we adapt the control variates variance reduction technique from the statistics literature in order to reduce the sampling overhead in QPD-based computations. We demonstrate our method using simulation experiments that mimic a realistic PEC scenario. In our experiments, we observed a more than 50% reduction in the number of samples needed to achieve a given precision, in more than 50% of the PEC-based estimations performed in the study when using our approach. We discuss how future research on constructing good control variates can lead to even stronger sampling overhead reduction.
Quasi-probability decompositions (QPDs) have proven essential in many quantum algorithms and protocols -- one replaces a ``difficult'' quantum circuit with an ensemble of ``easier'' circuit variants whose weighted outcomes reproduce any target observable. This, however, inevitably yields an increased configuration variance beyond Born-rule shot noise. We develop a broad framework for accounting for and reducing this variance and prove that stratified sampling -- under ideal proportional allocation -- results in an unbiased estimator with a variance that is never worse than naïve sampling (with equality only in degenerate cases). Furthermore, we provide a classical dynamic programme to enable stratification on arbitrary product-form QPDs. Numerical simulations of typical QPDs, such as Probabilistic Error Cancellation (PEC) and Probabilistic Angle Interpolation (PAI), demonstrate constant-factor reductions in overall variance (up to ∼60\sim 60∼60--80%80\%80% in an oracle model) and robust ∼10%\sim 10\%∼10% savings in the pessimistic single-shot regime. Our results can be applied immediately to reduce the net sampling cost of practically relevant QPDs that are commonly used in near term and early fault-tolerant algorithms without requiring additional quantum resources.
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.
Probabilistic error cancellation (PEC) is a promising error mitigation technique that reduces the error rate without auxiliary quantum bits. However, PEC has two problems that need to be resolved: 1) there is no good PEC technique for parameterized gates and 2) sampling overhead (SO) grows exponentially with the number of PEC mitigated gates. We first propose a parameterized gate PEC (PGPEC) that mitigates the error without fully characterizing the gates, as the original PEC requires. The result shows that the number of gates requiring characterization for a thousand random circuits can be reduced by 97% or more. We next propose two novel approaches to solving the second problem. We propose a macro gate PEC (MGPEC) technique that aggregates multiple gates as a single macro gate to reduce the exponent of the SO. MGPEC reduces the SO by 49% on the QFT7 under the IBMQ noise model, which simulates real operation conditions of quantum circuits. We propose a design diversity PEC (DDPEC) technique to reduce the exponential basis of the SO. The results show that our DDPEC with design diversity check reduces overall SO by 13% on the QFT7 circuit under the IBM Q noise model. Combining the DDPEC with the MGPEC, we can reduce overall SO by 73%.
Quantum error mitigation is essential for extracting trustworthy results from noisy intermediate-scale quantum (NISQ) processors. Yet, current approaches face a core scalability bottleneck: unbiased methods such as probabilistic error cancellation (PEC) incur exponential sampling overhead, while approximate techniques like zero-noise extrapolation trade accuracy for efficiency. We introduce and experimentally demonstrate MoSAIC (Modular Spatio-temporal Aggregation for Inverted Channels), a scalable quantum error mitigation framework that preserves the unbiasedness of PEC while dramatically reducing sampling costs. MoSAIC partitions a circuit into noise-aligned blocks, learns an effective block noise model using classical variational optimization, and applies quasi-probabilistic inversion once per block instead of after every layer. This blockwise aggregation reduces both sampling overhead and circuit-depth overhead, enabling mitigation far beyond the operating regime of standard PEC. We also experimentally validate MoSAIC on IBM's 156-qubit Heron processors, performing the largest PEC-based mitigation demonstration on hardware to date. As a physically meaningful benchmark, we prepare the critical one-dimensional transverse-field Ising (TFIM) ground state for system sizes up to 50 qubits. We show that MoSAIC can achieve at least 1 to 2 orders of magnitude better accuracy than standard PEC under identical sampling budgets. This enables MoSAIC to recover accurate observables for larger system sizes, even when standard PEC fails due to its prohibitive sampling overhead. We also present CUDA-Q accelerated simulations to validate performance trends under a range of different noise models. These results demonstrate that MoSAIC is not only theoretically scalable but also practically deployable for high-accuracy, large-scale quantum experiments on today's quantum hardware.
Current quantum computers are plagued by prohibitive amounts of noise, which complicates the experimental realization of useful quantum algorithms that outperform classical computers.Quantum error mitigation techniques could constitute a potential avenue to demonstrate this feat without the need for fault-tolerant quantum error correction.One method in this family of mitigation techniques is the quasiprobability method introduced by Temme et al. [1].It simulates a noise-free quantum computer with a noisy one, with the caveat of only producing the correct expected values of measurement observables.The cost of a quasiprobability simulation manifests as a sampling overhead which scales exponentially in the number of error-mitigated gates in the circuit.In this thesis we aim to reduce the exponential basis of that overhead, which in turn allows the application of the quasiprobability method to deeper quantum circuits.A central result is the introduction of a novel scheme, which we call Stinespring algorithm, that aims to choose the quasiprobability decomposition in a noise-aware manner.Along the way, we introduce a generalization of the quasiprobability method, which we denote approximate quasiprobability method, that allows for a tradeoff between an approximation error and the sampling overhead.This method is already interesting on its own and we present a few potential applications.Finally, we present some ideas how quantum error correction can benefit from the quasiprobability method.This final part is separate to the other topics of the master thesis and gives a rough overview of a new research project that started towards the end of the thesis and that we will continue to pursue in the future.i Overview of ContributionsWe summarize the novel contributions from this thesis:• Many details in the presentation of Chapter 2 have not been stated in such precision in previous literature.However, we estimate that many of these contributions are evident for someone knowledgeable in the domain.• The channel difference decomposition (Section 2.6).• The approximate QPD and its applications (Chapter 3).• The Stinespring algorithm (Chapter 4).