Learning low-degree quantum objects

TL;DR

This paper introduces methods for efficiently learning low-degree quantum objects, such as quantum channels, unitaries, and polynomials, with query complexities independent of the system size, advancing quantum learning theory.

Contribution

The paper presents novel algorithms and bounds for learning low-degree quantum objects, including quantum channels and polynomials, using query complexities that are independent of the number of qubits.

Findings

Quantum channels and unitaries of degree d can be learned with O(1/ε^d) queries.

Classical learning of polynomials from quantum algorithms requires O((1/ε)^d log n) samples.

Degree-d polynomials can be learned via O(1/ε^d) quantum queries to block-encoding unitaries.

Abstract

We consider the problem of learning low-degree quantum objects up to $\varepsilon$-error in $\ell_2$-distance. We show the following results: $(i)$ unknown $n$-qubit degree-$d$ (in the Pauli basis) quantum channels and unitaries can be learned using $O(1/\varepsilon^d)$ queries (independent of $n$), $(ii)$ polynomials $p:\{-1,1\}^n\rightarrow [-1,1]$ arising from $d$-query quantum algorithms can be classically learned from $O((1/\varepsilon)^d\cdot \log n)$ many random examples $(x,p(x))$ (which implies learnability even for $d=O(\log n)$), and $(iii)$ degree-$d$ polynomials $p:\{-1,1\}^n\to [-1,1]$ can be learned through $O(1/\varepsilon^d)$ queries to a quantum unitary $U_p$ that block-encodes $p$. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded~polynomials.