Predicting quantum channels over general product distributions

TL;DR

This paper introduces a new quantum channel prediction method that works effectively over general product distributions, overcoming previous limitations to specific invariant distributions, and employs a biased Pauli analysis technique.

Contribution

The authors develop a novel approach for predicting quantum channels over arbitrary product distributions, expanding beyond prior work limited to Clifford-invariant distributions.

Findings

Achieves accurate quantum channel prediction over most product distributions.

Introduces a biased Pauli analysis technique for quantum information.

Addresses challenges due to lack of orthogonal basis in quantum setting.

Abstract

We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $E$ and an observable $O$, we aim to learn the mapping \begin{equation*} \rho \mapsto \mathrm{Tr}(O E[\rho]) \end{equation*} to within a small error for most $\rho$ sampled from a distribution $D$. Previously, Huang, Chen, and Preskill proved a surprising result that even if $E$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/\epsilon))}$, where $\epsilon$ is the target prediction error. However, their guarantee applied only to input distributions $D$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $\rho$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $D$, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.