Zero-noise extrapolation (ZNE) is an increasingly popular technique for mitigating errors in noisy quantum computations without using additional quantum resources. We review the fundamentals of ZNE and propose several improvements to noise scaling and extrapolation, the two key components in the technique. We introduce unitary folding and parameterized noise scaling. These are digital noise scaling frameworks, i.e. one can apply them using only gate-level access common to most quantum instruction sets. We also study different extrapolation methods, including a new adaptive protocol that uses a statistical inference framework. Benchmarks of our techniques show error reductions of 18X to 24X over non-mitigated circuits and demonstrate ZNE's effectiveness at larger qubit numbers than have been tested previously. In addition to presenting new results, this work is a self-contained introduction to the practical use of ZNE by quantum programmers.
Zero-noise extrapolation (ZNE) is an increasingly popular technique for mitigating errors in noisy quantum computations without using additional quantum resources. We review the fundamentals of ZNE and propose several improvements to noise scaling and extrapolation, the two key components in the technique. We introduce unitary folding and parameterized noise scaling. These are digital noise scaling frameworks, i.e. one can apply them using only gate-level access common to most quantum instruction sets. We also study different extrapolation methods, including a new adaptive protocol that uses a statistical inference framework. Benchmarks of our techniques show error reductions of 18X to 24X over non-mitigated circuits and demonstrate ZNE's effectiveness at larger qubit numbers than have been tested previously. In addition to presenting new results, this work is a self-contained introduction to the practical use of ZNE by quantum programmers.
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.
Quantum circuit unoptimization is an algorithm that transforms a quantum circuit into a different circuit that uses more gate operations while maintaining the same unitary transformation. We demonstrate that this method can implement digital zero-noise extrapolation (ZNE), a quantum error mitigation technique. By employing quantum circuit unoptimization as a form of circuit folding, noise can be systematically amplified. The key advantages of this approach are twofold. First, its ability to generate an exponentially increasing number of distinct circuit variants as the noise level is amplified, which allows noise averaging over many circuit variants with slightly different circuit structure. Averaging over these variants can mitigate the effect of biased error propagation due to the significantly altered circuit structure from quantum circuit unoptimization, or biased noise sources on a quantum processor. Second, quantum circuit unoptimization by design resists circuit simplification back to the original unmodified circuit, making it plausible to use ZNE in contexts where circuit compiler optimization is applied server-side. We evaluate the effectiveness of quantum circuit unoptimization as a noise-scaling method for ZNE in two test cases using depolarizing noise numerical simulations: random quantum volume circuits, where the observable is the heavy output probability, and QAOA circuits for the (unweighted) maximum cut problem on random 3-regular graphs, where the observable is the cut value. We show that using quantum circuit unoptimization to perform ZNE can approximately recover signal from noisy quantum simulations.
Recent hundred-qubit processors represent a significant hardware advancement, but current limitations prevent effective quantum error correction (QEC), necessitating reliance on quantum error mitigation (QEM) to enhance result fidelity from quantum computers. Our paper introduces a noise-aware folding technique that enhances Zero-Noise Extrapolation (ZNE) by leveraging the noise characteristics of target quantum hardware to fold circuits more efficiently. Unlike traditional ZNE approaches assuming uniform error distribution, our method redistributes noise using calibration data based on hardware noise models. By employing a noise-adaptive compilation method combined with our proposed folding mechanism, we enhance the ZNE accuracy of quantum gate-based computing using superconducting quantum computers. This paper highlights the uniqueness of our method, summarizes noise accumulation, presents the scaling algorithm, and compares the reliability of our method with those of existing models using linear extrapolation model. Experimental results show that compared to existing folding methods, our approach achieved up to a 23.6% improvement on quantum computer simulators and up to a 44.8% improvement on real quantum computers, with statistical confidence intervals indicating stable performance gains across circuit scales.
Coping with noise in quantum computation poses significant challenges due to its unpredictable nature and the complexities of accurate modeling. This paper presents noise-adaptive folding, a technique that enhances zero-noise extrapolation (ZNE) through the use of adaptive scaling factors based on circuit error measurements. Furthermore, we introduce two filtering methods: one relies on measuring error strength, while the other utilizes statistical filtering to improve the extrapolation process. Comparing our approach with standard ZNE reveals that adaptive scaling factors can be optimized using either a noise model or direct error strength measurements from inverted circuits. The integration of adaptive scaling with filtering techniques leads to notable improvements in expectation-value extrapolation over standard ZNE. Our findings demonstrate that these adaptive methods effectively strengthen error mitigation against noise fluctuations, thereby enhancing the precision and reliability of quantum computations.
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 n-qubit circuit CLP-ZNE requires execution of only O(n) and at most 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=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/T2 relaxation processes, these results give evidence for the applicability of CLP-ZNE to present-day NISQ processors.
Near term quantum processors operate in a noise dominated regime, motivating error mitigation techniques that recover accurate expectation values without full fault tolerance. Zero Noise Extrapolation (ZNE) is a widely used but biased error mitigation method that lacks rigorous error bounds. Its effective application requires nontrivial technical choices, most notably the selection of noise scaling factors and extrapolation models, making ZNE sensitive to user expertise and often necessitating costly trial and error procedures. Here, we introduce Folding Free Zero Noise Extrapolation (FF-ZNE), a method that removes the need for noise factor selection by achieving effective noise amplification without circuit folding. FF-ZNE exploits isomorphic hardware layouts with distinct native noise profiles, such that executing a fixed circuit across these layouts induces controllable variations in the effective noise strength. Under a depolarizing noise model, we analytically show that the resulting extrapolation admits a fixed linear form, eliminating extrapolator choice and enabling a seamless, user independent mitigation procedure. We further propose two algorithms that identify sets of isomorphic hardware layouts on which a given circuit yields sufficiently distinct expectation values to enable reliable zero noise extrapolation. Experiments on a 133 qubit IBM Quantum device demonstrate that FF-ZNE yields mitigated expectation values with average deviations of ~6% and 4.5% for up to 50 qubit EfficientSU2 (sparse) and Hamiltonian simulation (dense) circuits, respectively. The method is thus scalable and applicable to a broad range of circuits. By eliminating noise factor and extrapolator selection, FF-ZNE transforms zero noise extrapolation from a technique requiring expert tuning into a practical, scalable, and broadly accessible error mitigation method for current quantum hardware.