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.

Zero-noise extrapolation (ZNE) depends on how amplification transforms temporal noise correlations. This paper derives the observable response of complete global unitary folding under simultaneous time-local Markovian noise and stationary classical finite-band $1/f$ noise, allowing arbitrary finite multiqubit circuits with repeatable forward and inverse control blocks. An ordered Dyson expansion, retaining first Markovian order and second colored-noise order, gives an observable-dependent circuit kernel. Exact block-pair counting relates the response’s second difference to covariance integrated over delays around $rT$. Logarithmic memory then yields $E(r)=E^*+\delta+a_1r+a_2r^2+\theta r^2\ln r+R(r)$, with $\theta$ determined by the spectrum and integrated forward–inverse control response. Ordinary quadratic Richardson extrapolation on $(1,3,5)$ retains a logarithmic bias contribution $2.73\theta$; every minimal positive-node Richardson rule of degree at least two has a positive logarithmic moment. Response-matched weights cancel $r$, $r^2$, and $r^2\ln r$, using at least four nodes. The linear estimator has bias bounded by $H_{\rm nat}+\|w\|_1D_{\max}$; a log-domain implementation is also proposed. Single-qubit simulations give biases of $5.55\%$, $1.71\%$, and $-0.26\%$ for exponential, Richardson, and log-domain matched extrapolation, respectively, using different node sets. The four-coordinate model is not an exact closure: control-shape, band-edge, higher-order, and native-offset errors remain. Cancellation does not guarantee the ideal intercept or finite-budget improvement. Log-domain fitting requires nonzero fixed-sign signals and lacks the linear estimator’s certificate. Numerical illustrations are not hardware measurements; the ten-qubit model factorizes Markovian and classical-noise responses.