Zero-Noise Extrapolation (ZNE)

Zero-noise extrapolation works by deliberately amplifying the noise in a quantum circuit through one of two main techniques: gate folding or pulse stretching. Gate folding replaces each gate G with G * G_dagger * G, which is logically equivalent but applies the physical gate three times, increasing noise exposure proportionally. Pulse stretching achieves a similar effect at the control-electronics level by slowing down the microwave pulses that drive gate operations, extending the time each qubit spends in a noisy environment. Both methods allow the experimenter to set a noise scale factor lambda, typically ranging from 1 (native noise) to 3 or 5, without changing the logical operation of the circuit.

Once expectation values have been collected at several noise scale factors, a classical post-processing step fits a curve through the data points and extrapolates to lambda = 0. Linear extrapolation is the simplest approach and works well when noise is moderate, requiring only two data points. Polynomial extrapolation fits a higher-degree curve and can capture more complex noise dependencies but requires more circuit executions. Exponential extrapolation, fitting an equation of the form A + B * exp(-C * lambda), often models depolarizing-like noise most faithfully and is the default in several software frameworks. Richardson extrapolation, borrowed from numerical analysis, provides a systematic way to cancel error terms of successive polynomial orders using a specific set of scale factors and weights.

ZNE is natively supported in Mitiq, an open-source error mitigation library from Unitary Fund that wraps any circuit-execution backend, and in Qiskit Runtime’s Estimator primitive when resilience_level=1 is set. In Qiskit Runtime the user simply sets the resilience level in the options dictionary; the service handles noise scaling and extrapolation transparently, making ZNE accessible without manual circuit folding. The technique is backend-agnostic and has been demonstrated on superconducting, trapped-ion, and photonic hardware, making it one of the most broadly applicable error mitigation strategies available for near-term devices.

The primary overhead of ZNE is shot count: running circuits at three or more noise levels multiplies total measurement time by roughly 3 to 5 times compared to an unmitigated estimate. This cost is significant but far lower than the exponential overhead of probabilistic error cancellation (PEC), which provides an unbiased estimate but requires exponentially many samples to achieve a target variance. ZNE introduces a systematic bias because the zero-noise limit is an extrapolation rather than a direct measurement, meaning it works best when the noise model is smooth and well-behaved. In practice ZNE is often paired with Pauli twirling to symmetrize the noise before extrapolation, improving the quality of the fit and reducing the residual bias in the final mitigated expectation value.