Zero-Noise Extrapolation

Zero-noise extrapolation (ZNE) is a strikingly simple error-mitigation idea: if you cannot remove the noise, deliberately add more of it, watch how your result degrades, and extrapolate back to the noiseless point. It needs no extra qubits and no detailed noise model — only the ability to amplify the noise in a controlled way (Temme et al.).

Amplifying the noise on purpose

The starting observation is that an expectation value $\langle O\rangle$ measured on a noisy device depends on an effective noise scale factor $\lambda$ . At $\lambda = 1$ you run the circuit as written. To run at $\lambda = 3$ you scale the noise up by a factor of three — most often by unitary folding: replacing each gate $G$ with $G\,G^\dagger G$ , which is mathematically the identity-times- $G$ but physically applies three times the noisy operations. Other schemes stretch the gate pulses in time. Either way you obtain the same ideal circuit running under more noise.

You measure the observable at several scale factors, $E(\lambda_1), E(\lambda_2), \dots$ , with $\lambda_i \ge 1$ , then fit a curve and evaluate it at $\lambda = 0$ — the zero-noise limit you can never reach directly.

Linear (Richardson) extrapolation

The simplest model assumes the expectation value is approximately linear in the noise scale near small noise:

E(\lambda) \approx E_0 + a\,\lambda,

where $E_0 = E(0)$ is the mitigated value we want. With two data points $(\lambda_1, E_1)$ and $(\lambda_2, E_2)$ the slope and intercept are

a = \frac{E_2 - E_1}{\lambda_2 - \lambda_1}, \qquad E_0 = E_1 - a\,\lambda_1.

Concretely, suppose folding gives $E(1) = 0.7$ and $E(3) = 0.5$ . The slope is $a = (0.5 - 0.7)/(3 - 1) = -0.1$ , so

E_0 = 0.7 - (-0.1)(1) = 0.8.

The mitigated estimate $0.8$ lies beyond the least-noisy measurement $0.7$ , in the direction the data were trending — extrapolation, not interpolation. This two-point linear rule is the lowest-order Richardson extrapolation; with more scale factors one fits a higher-order polynomial or an exponential to capture curvature.

Assumptions and limits

ZNE rests on two assumptions worth stating plainly:

A faithful noise-scaling knob. Folding or pulse-stretching must amplify the same noise that afflicts the real circuit. If amplification introduces a different error character, the extrapolation chases the wrong curve.
A valid extrapolation model. Linear ZNE is exact only when $E(\lambda)$ really is linear in $\lambda$ ; for stronger noise the dependence curves and a low-order fit becomes biased.

Despite these caveats, ZNE has been shown to meaningfully extend the usable depth of real superconducting processors (Kandala et al.), which is why it is a default tool in near-term, pre-fault-tolerant computing.

Try it

You measured an observable at two noise scale factors: $E(\lambda{=}1) = 0.7$ and $E(\lambda{=}3) = 0.5$ . Perform a linear (Richardson) extrapolation and return the zero-noise value $E_0 = E(0)$ . The grader checks your value against the analytic result $0.8$ .

Run your code to see the quantum state.