Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance

TL;DR

This paper develops efficient algorithms for learning nearly sparse unitaries in quantum computing, focusing on their Pauli spectrum, with applications to various quantum circuit classes and establishing complexity bounds.

Contribution

It introduces a novel quantum learning algorithm for nearly sparse unitaries, extending sparse recovery techniques to general unitaries and analyzing their complexity.

Findings

Algorithm achieves $ ilde{O}(s^6/epsilon^4)$ query complexity.

Estimates Pauli coefficients exceeding a threshold with an extended sparse recovery method.

Proves exponential lower bounds for learning unitaries with bounded Pauli $ ext{l}_1$-norm.

Abstract

We study the problem of learning nearly $(s,\epsilon)$-sparse unitaries, meaning that the Pauli spectrum is concentrated on at most $s$ components with at most $\epsilon$ residual mass in Pauli $\ell_1$-norm. This class generalizes well-studied families, including sparse unitaries, quantum $k$-juntas, $2^k$-Pauli dimensional channels, and compositions of depth $O(\log\log n)$ circuits with near-Clifford circuits. Given query access to an unknown nearly sparse unitary $U$, our goal is to efficiently (both in time and query complexity) construct a quantum channel that is close in diamond distance to $U$. We design a learning algorithm achieving this guarantee using $\tilde{O}(s^6/\epsilon^4)$ forward queries to $U$, and running time polynomial in relevant parameters. A key contribution is an efficient quantum algorithm that, given query access to an arbitrary unknown unitary $U$, estimates all Pauli coefficients (up to a shared global phase) whose magnitude exceeds a given threshold $\theta$, extending existing sparse recovery techniques to general unitaries. We also study the broader class of unitaries with bounded Pauli $\ell_1$-norm. For that class, we prove an exponential query lower bound $\Omega(2^{n/2})$. We introduce a more relaxed accuracy metric which is the diamond distance restricted to a set of input states. Then, we show that, under this metric, unitaries with Pauli $\ell_1$-norm uniformly bounded by $L_1$ are learnable with $\tilde{O}(L_1^8/\epsilon^{16})$.