Quantum Error Mitigation Techniques

Overview

Today's quantum computers make errors in nearly every operation, and they are far too small to run full quantum error correction. Quantum error mitigation is a set of techniques that reduce the effect of noise on the final answer without correcting individual errors. They do this by running extra circuits and combining the results with classical post-processing, so that the corrected estimate of an observable is closer to its noise-free value.

The price is more measurements, often exponentially more as circuits get larger. Error mitigation is therefore a bridge technology, useful for small and medium circuits on noisy devices, and not a substitute for error correction in long computations.

The problem it solves

Getting more accurate results from today's noisy quantum hardware, without the enormous number of extra physical qubits full error correction would require.

History and background

Temme, Bravyi, and Gambetta introduced zero-noise extrapolation and probabilistic error cancellation for short-depth circuits in a 2017 Physical Review Letters paper. Li and Benjamin proposed related ideas in the same year. Endo, Benjamin, and Li made them practical for near-term applications in 2018, and in 2019 Kandala and colleagues at IBM published a Nature paper demonstrating that error mitigation extends the reach of a noisy superconducting processor.

The toolkit grew quickly. Symmetry verification, introduced by Bonet-Monroig and coauthors in 2018, discards results that violate known conservation laws. Virtual distillation, proposed by Huggins and coauthors and by Koczor in 2021, suppresses errors using multiple copies of a state. Clifford data regression, learned error mitigation, and digital zero-noise extrapolation by Giurgica-Tiron and coauthors followed. In 2023 IBM and university collaborators used zero-noise extrapolation with probabilistic error amplification in a 127-qubit utility experiment. Meanwhile, Quek and coauthors proved in 2022 that the sampling cost of any error mitigation scheme must grow exponentially with circuit depth in general.

How it works

Techniques like zero-noise extrapolation run the same circuit at deliberately varied noise levels and extrapolate back to an estimated zero-noise result, rather than directly correcting errors during computation.

Suppose a circuit outputs an expectation value, say the energy of a molecule. Noise pushes that value away from the ideal, typically towards a featureless value such as zero. If you could turn the noise down, the result would move closer to the ideal. You cannot turn it down, but you can turn it up, deliberately, and see how the answer changes.

Zero-noise extrapolation runs the same circuit with noise amplified by factors such as 1, 2, and 3, for example by stretching pulses or inserting extra pairs of gates that cancel logically. The results are plotted against the noise level and a curve is fitted and extended back to zero noise. The extrapolated point is the mitigated estimate.

Probabilistic error cancellation takes a different route. It writes the ideal operation as a weighted combination of noisy operations that the device can actually perform, with some weights negative. Sampling circuits from this combination, and weighting the results by the signs, gives an unbiased estimate of the ideal value, at the price of a sampling overhead that grows with the noise.

Step by step

1. Choose the observable to estimate and the circuit that prepares the state.
2. Characterize the noise, for example by gate set tomography, Pauli twirling, or learning a noise model, if the method requires it.
3. For zero-noise extrapolation, run the circuit at several amplified noise levels, measuring the observable many times at each level.
4. For probabilistic error cancellation, generate circuits sampled from the quasi-probability representation of the ideal gates, run them, and combine results with signed weights.
5. Fit the extrapolation curve or compute the weighted average, and estimate the statistical uncertainty.
6. Report the mitigated estimate along with an error bar that reflects the additional variance from the mitigation procedure.

A worked example

Suppose the ideal expectation value of an observable is 1.0, and the noisy device returns 0.80 at the native noise level, 0.64 at twice the noise, and 0.51 at three times the noise. A linear fit gives a slope of about minus 0.145 per unit of noise, and extrapolating to zero noise gives roughly 0.94. Using an exponential fit gives about 1.0. The mitigated answer is closer to the truth than the raw 0.80, but the choice of fit adds uncertainty.

For probabilistic error cancellation with a per-gate overhead factor of 1.05 and 100 gates, the overall sampling overhead is about 1.05 to the power 200, which is roughly 17,000. That means each measurement needs about 17,000 times as many shots for the same statistical accuracy, which shows why the method is practical only for shallow circuits.

Complexity and resources

What hardware it needs

• A noisy device with reasonably stable noise, since methods depend on a model of the noise.
• Pulse-level or gate-level control for noise amplification, in some methods.
• Many shots and fast classical post-processing.

Applications

• Improving the accuracy of VQE, QAOA, and quantum simulation results on current devices.
• Utility-scale experiments such as the 127-qubit kicked Ising experiment.
• Benchmarking and calibrating devices.
• Bridging toward early fault-tolerant machines.

Limitations and caveats

• The cost grows exponentially with circuit size, so it does not scale to long computations.
• It works for expectation values, not for sampling full output distributions.
• Extrapolation can be biased if the noise model is wrong.
• Claims of advantage with mitigation are sensitive to classical simulation improvements.

Common misconceptions

• Error mitigation is not error correction. It does not make individual runs more reliable.
• It does not give exact results. It reduces bias in an average, and adds variance.
• It will not enable large quantum algorithms such as Shor's on noisy hardware.

How it compares

• Quantum error correction scales to long circuits with polynomial overhead but needs many qubits.
• Dynamical decoupling and error suppression reduce noise at the pulse level before mitigation.
• Postselection discards bad runs, and is a special case of symmetry verification.

Real-world impact

Already widely used in practice on real NISQ hardware today (including in VQE and QAOA experiments), serving as a practical bridge until full quantum error correction becomes feasible at scale.

Why it matters

Mitigation is what makes today's noisy experiments scientifically useful. It allows devices with hundreds of qubits to produce results that can be tested against classical methods, which is the basis of the utility-scale experiments of recent years.

The theory of its limits also matters: it shows clearly that noisy devices cannot be pushed indefinitely and that error correction will be needed for large computations.

Frequently asked questions

Keep learning

• Related algorithm: Quantum Error-Correcting Codes (Shor Code, Surface Code)
• Related algorithm: Variational Quantum Eigensolver (VQE)
• Related algorithm: Quantum Approximate Optimization Algorithm (QAOA)
• Related algorithm: Random Circuit Sampling (Quantum Supremacy Experiments)

This page is written for educational purposes. Complexity statements describe the best known results at the time of review (September 30, 2026), and research in this area moves quickly, so check recent papers before relying on a specific bound.

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