Predicting quantum channels over general product distributions

TL;DR

This work addresses predicting the average behavior of outputs from unknown quantum channels under product-state input distributions, generalizing beyond Clifford-invariant inputs. It introduces a biased Pauli analysis that adapts the Pauli basis to the mean of the distribution and proves a low-degree approximation result for observables when the Pauli second-moment norm satisfies ||S||_op ≤ 1−η. Consequently, there exists an efficient learning algorithm with time/sample complexity min(2^{O(n)}/ε^2, n^{O(log(1/ε)/log(1−η))})·log(1/δ) that accurately predicts Tr(O E[ρ]) on average over D^{⊗n}. The paper also establishes lower bounds showing hardness for general concentrated distributions and the limitations of degree truncation without mean-zero assumptions, highlighting the essential role of distribution structure in quantum average-case learnability. Overall, the work broadens the practical scope of quantum process learning and introduces techniques that may apply to broader quantum-information problems.

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*} ρ\mapsto \mathrm{Tr}(O E[ρ]) \end{equation*} to within a small error for most $ρ$ 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 $n^{O(\log(1/ε))}$, where $ε$ 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 $ρ$. 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.

Theorems & Definitions (27)

• Theorem 1.1: Learning an unknown quantum channel
• Remark 1.2
• Theorem 1.3: PAC learning over a concentrated product distribution
• Remark 1.4: Predicting multiple observables
• Theorem 1.5: Hardness of learning over general concentrated distributions
• Definition 2.1: Pauli covariance and second moment matrices
• Corollary 2.5
• Lemma 3.4
• proof
• proof : Proof of \ref{['thm:classical']}