Project: Error-Mitigated Estimation

Near-term devices are too noisy to trust a raw expectation value, yet too small to error-correct. Error mitigation bridges the gap: it does not fix the qubits, but it post-processes biased measurements into a better estimate of the noiseless observable. This capstone implements the most widely used technique — zero-noise extrapolation (ZNE) — end to end on an observable.

The idea: amplify, then extrapolate

A noisy device returns a biased expectation value $E(\lambda)$ that depends on the noise strength $\lambda$ . We cannot reduce $\lambda$ below the hardware floor, but we can increase it on purpose — by gate folding, replacing each gate $G$ with $G G^\dagger G$ (logically the identity insertion, physically more noise). Measuring at several scale factors $\lambda, 2\lambda, 3\lambda, \dots$ samples the curve $E(\lambda)$ , and a fit extrapolated to $\lambda \to 0$ estimates the ideal value:

E_{\text{mitigated}} = \lim_{s \to 0} E(s).

Linear extrapolation

The simplest model assumes $E$ is approximately linear in the noise scale near the hardware floor:

E(s) = a + b\,s.

Two measurements pin down the line. With data $(s_1, e_1) = (1, 0.80)$ and $(s_2, e_2) = (2, 0.64)$ ,

b = \frac{e_2 - e_1}{s_2 - s_1} = \frac{0.64 - 0.80}{2 - 1} = -0.16, \qquad a = e_1 - b\,s_1 = 0.80 - (-0.16)(1) = 0.96.

The zero-noise estimate is the intercept $E(0) = a = 0.96$ — much closer to the ideal $\langle Z\rangle = 1$ than either raw point.

Try it

Implement zneLinear to fit the line through two scale points and return its zero-noise intercept. The grader checks you recover $0.96$ from the scales $1$ and $2$ .

Run your code to see the quantum state.