Logarithmic Bias of Zero-Noise Extrapolation under 1/f Noise

Abstract

Zero-noise extrapolation estimates ideal quantum observables by amplifying noise and extrapolating the measured response. Its accuracy depends on whether the extrapolation model captures the physical effect of amplification, including correlations between repeated operations. Here we derive the response of complete multiqubit gate folding in the simultaneous presence of time-local Markovian noise and stationary finite-band 1/f noise. An ordered circuit kernel couples the observable to noise memory; exact block counting shows how folding samples correlations at increasing time separations. The resulting response contains an r^2 ln r coordinate, with a coefficient fixed by the noise spectrum and the integrated forward-inverse control response. Ordinary polynomial Richardson extrapolation leaves a calculable contribution from this coordinate; the three-node rule (1,3,5) retains 2.73 theta. We construct response-matched extrapolation, including a log-domain implementation, that cancels this contribution at the retained order. Explicit bounds separate the remaining control-shape, finite-band, higher-order, and native-offset errors. The theory applies to arbitrary finite multiqubit circuits within the stated control and noise model, and distinguishes removal of memory-induced bias from an unconditional guarantee of the ideal intercept. Single- and ten-qubit numerical examples illustrate the predicted correction and its response dependence.