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arXiv:2405.00037 · Physical Review A

Comment on “Recovering noise-free quantum observables”

Josu Etxezarreta Martínez, Olatz Sanz Larrarte, Javier Oliva del Moral, Reza Dastbasteh, R. M. Otxoa

Published Oct 21, 2024• 1 citation
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Abstract

Zero-noise extrapolation (ZNE) stands as the most widespread quantum error mitigation technique used in the recovery of noise-free expectation values of observables of interest by means of noisy intermediate-scale quantum (NISQ) machines. Recently, Otten and Gray proposed a multidimensional generalization of polynomial ZNE for systems where there is no tunable global noise source [M. Otten , ]. Specifically, the authors refer to multiqubit systems where each of the qubits experiences several noise processes with different rates, i.e., a nonidentically distributed noise model. The authors proposed a hypersurface method for mitigating such noise, which is technically correct in the sense that if the required samples are obtained, the observable can be mitigated. However, the proposed method presents an excessive experiment repetition overhead, making it impractical, at least from the perspective of quantum computing. Specifically, we discuss that in order to perform a simple mitigation task to multinomial order 3 considering 100 qubits, there, 2.5 years or over a million quantum processors running in parallel would be required. In this Comment, we show that the traditional extrapolation techniques can be applied for such a nonidentically distributed noise setting consisting of many different noise sources, implying that the measurement overhead is practical. In fact, we discuss that the previous mitigation task can be resolved in the order of minutes using a single quantum processor by means of standard ZNE. To do so, we clarify what it is meant by a tunable global noise source in the context of ZNE, a concept that we consider important to be clarified for a correct understanding of how and why these methods work. Published by the American Physical Society 2024

DOI
10.1103/physreva.110.046401

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Zero-noise extrapolation (ZNE) stands as the most widespread quantum error mitigation technique used in the recovery of noise-free expectation values of observables of interest by means of noisy intermediate-scale quantum (NISQ) machines. Recently, Otten and Gray proposed a multidimensional generalization of polynomial ZNE for systems where there is no tunable global noise source [M. Otten , ]. Specifically, the authors refer to multiqubit systems where each of the qubits experiences several noise processes with different rates, i.e., a nonidentically distributed noise model. The authors proposed a hypersurface method for mitigating such noise, which is technically correct in the sense that if the required samples are obtained, the observable can be mitigated. However, the proposed method presents an excessive experiment repetition overhead, making it impractical, at least from the perspective of quantum computing. Specifically, we discuss that in order to perform a simple mitigation task to multinomial order 3 considering 100 qubits, there, 2.5 years or over a million quantum processors running in parallel would be required. In this Comment, we show that the traditional extrapolation techniques can be applied for such a nonidentically distributed noise setting consisting of many different noise sources, implying that the measurement overhead is practical. In fact, we discuss that the previous mitigation task can be resolved in the order of minutes using a single quantum processor by means of standard ZNE. To do so, we clarify what it is meant by a tunable global noise source in the context of ZNE, a concept that we consider important to be clarified for a correct understanding of how and why these methods work. Published by the American Physical Society 2024
Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.
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.
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Quantum computers in the NISQ era are prone to noise. A range of quantum error mitigation techniques has been proposed to address this issue. Zero-noise extrapolation (ZNE) stands out as a promising one. ZNE involves increasing the noise levels in a circuit and then using extrapolation to infer the zero noise case from the noisy results obtained. This paper presents a novel ZNE approach that does not require circuit folding or noise scaling to mitigate depolarizing and/or decoherence noise. To mitigate depolarizing noise, we propose leveraging the extreme/infinite noisy case, which allows us to avoid circuit folding. Specifically, the circuit output with extreme noise becomes the maximally mixed state. We show that using circuit-reliability metrics, simple linear extrapolation can effectively mitigate depolarizing noise. With decoherence noise, different states decay into the all-zero state at a rate that depends on the number of excited states and time. Therefore, we propose a state- and latency-aware exponential extrapolation that does not involve folding or scaling. When dealing with a quantum system affected by both decoherence and depolarizing noise, we propose to use our two mitigation techniques in sequence: first applying decoherence error mitigation, followed by depolarizing error mitigation. A common limitation of ZNE schemes is that if the circuit of interest suffers from high noise, scaling-up noise levels could not provide useful data for extrapolation. We propose using circuit-cut techniques to break a large quantum circuit into smaller sub-circuits to overcome this limitation. This way, the noise levels of the sub-circuits are lower than the original circuit, and ZNE can become more effective in mitigating their noises.
Increasing the utility of currently available Noisy Intermediate-Scale Quantum (NISQ) devices requires developing efficient methods to mitigate hardware errors. In this work we propose a novel Cyclic Layout Permutations-based Zero-Noise Extrapolation (CLP-ZNE) protocol for such a task. The method leverages the inherent non-uniformity of gate errors in NISQ hardware to extrapolate the expectation value, averaged over cyclic circuit layout permutations, to the level of zero noise. In contrast to the previous layout permutation based approaches, for an nnn-qubit circuit CLP-ZNE requires execution of only O(n)O(n)O(n) and at most O(n2)O(n^2)O(n2) different circuit layouts for circuits of one-dimensional and arbitrary connectivity, respectively. When benchmarked against noise channels modeling the IBM Torino quantum computer, the method reduces a typical error in expectation values of n=12n=12n=12 qubit circuits by an order of magnitude, outperforming standard unitary folding ZNE. By demonstrating the ability to mitigate noise of real hardware specifications, including both depolarizing and T1/T2T_1/T_2T1​/T2​ relaxation processes, these results give evidence for the applicability of CLP-ZNE to present-day NISQ processors.
Errors are the primary bottleneck preventing practical quantum computing. This challenge is exacerbated in the distributed quantum computing regime, where quantum networks introduce additional communication-induced noise. While error mitigation techniques such as Zero Noise Extrapolation (ZNE) have proven effective for standalone quantum processors, their behavior in distributed architectures is not yet well understood. We investigate ZNE in this setting by comparing Global optimization (ZNE is applied prior to circuit partitioning), against Local optimization (ZNE is applied independently to each sub-circuit). Partitioning is performed on a monolithic circuit, which is then transformed into a distributed implementation by inserting noisy teleportation-based communication primitives between sub-circuits. We evaluate both approaches across varying numbers of quantum processing units (QPUs) and under heterogeneous local and network noise conditions. Our results demonstrate that Global ZNE exhibits superior scalability, achieving error reductions of up to 48%48\%48% across six QPUs. Moreover, we observe counterintuitive noise behavior, where increasing the number of QPUs improves mitigation effectiveness despite higher communication overhead. These findings highlight fundamental trade-offs in distributed quantum error mitigation and raise new questions regarding the interplay between circuit structure, partitioning strategies, and network noise.