Learnability of general Pauli-sparse and nearly Pauli-sparse unitaries

Determine whether there exist general, query-efficient algorithms for learning n-qubit unitaries that are s-sparse in the Pauli basis, and more broadly nearly (s, ε)-sparse (i.e., the Pauli spectrum is concentrated on at most s components with at most ε residual ℓ1-mass), from black-box forward queries to the unitary, together with precise sample complexity bounds.

Background

The paper surveys prior results showing that certain structured subclasses of Pauli-sparse unitaries—such as quantum k-juntas (a special case of 4^k-sparse unitaries) and unitaries supported on Pauli subgroups of size 2^k—are learnable with sample complexity scaling as O(4^k/ε) and Θ(2^k/ε), respectively.

In contrast, before this work, the efficient learnability of general Pauli-sparse and nearly Pauli-sparse unitaries had not been established. The authors highlight this gap as a motivating open question and then propose algorithms addressing aspects of the problem under their learning framework.