Zero-Noise Extrapolation: NISQ Error Mitigation

Noise Model

Per-gate depolarizing rate p 1.5%

Circuit depth G (gates) 60

Shots per noise scale N 150

Extrapolation

Fit model

Readouts

Ideal ⟨O⟩ (c→0, exact)

0.809

Raw noisy (c=1)

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ZNE estimate (c→0)

—

Error reduction

—

Decay base (1−p)^G

—

Decay exponent γ

—

How it works

A circuit of G gates, each depolarizing the qubit with probability p, survives coherently with probability (1−p)^G. If nothing depolarized, the measurement still reports the ideal expectation value; otherwise the qubit is maximally mixed and contributes 0. So the noisy expectation value at "noise scale" c (the circuit stretched to c·G gates via unitary folding — insert G(c−1) extra gate·gate⁻¹ pairs) is:

⟨O(c)⟩ = ⟨O⟩_ideal · (1 − p)^(c·G) = ⟨O⟩_ideal · b^c
b = (1 − p)^G   (measured decay base at c=1)

Zero-noise extrapolation runs the same circuit at several folded scales (c = 1, 2, 3 — never c = 0, that hardware doesn't exist) and fits a curve back through c = 0 to estimate the noiseless value, trading extra circuit executions for a bias-corrected result:

Linear (2 pts):       y0 = 2y1 − y2
Richardson (3 pts):   y0 = 3y1 − 3y2 + y3
Exponential:          ln|y_c| ≈ ln|a| − k·c  (least squares)
                       y0 = a · sign(y1)

Shots per scale N — each lane samples N independent ±1 measurement outcomes from the Born-rule probability P(+1) = (1+⟨O(c)⟩)/2, so the plotted lane mean carries real statistical scatter, not just the theoretical curve.
New Shots — redraws the random outcomes at the current noise parameters, showing how much the extrapolated estimate itself fluctuates run to run.
Real-world relevance: this is the core technique behind Qiskit Runtime's and Mitiq's error mitigation, and the kind of protocol quantum-hardware consortia standardize so results from different labs' noisy processors stay comparable.

+1 outcome −1 outcome Ideal ⟨O⟩ (target) Measured lane means Extrapolated c→0

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