The complexity of perfect quantum state classification

TL;DR

This paper investigates the computational complexity of perfectly classifying quantum states from a known set, introducing $k$-learnability, and characterizing when the problem is efficiently solvable or NP-hard.

Contribution

It introduces the concept of $k$-learnability for quantum state classification and analyzes its computational complexity, providing algorithms and hardness results.

Findings

Polynomial-time algorithms for fixed $k$ or fixed dimension cases.

NP-hardness of the general problem with variable $k$ and dimension.

Existence of succinct certificates for $k$-learnability in the general case.

Abstract

The problem of quantum state classification asks how accurately one can identify an unknown quantum state that is promised to be drawn from a known set of pure states. In this work, we introduce the notion of $k$-learnability, which captures the ability to identify the correct state using at most $k$ guesses, with zero error. We show that deciding whether a given family of states is $k$-learnable can be solved via semidefinite programming. When there are $n$ states, we present polynomial-time (in $n$) algorithms for determining $k$-learnability for two cases: when $k$ is a fixed constant or the dimension of the states is a fixed constant. When both $k$ and the dimension of the states are part of the input, we prove that there exist succinct certificates placing the problem in NP, and we establish NP-hardness by a reduction from the classical $k$-clique problem. Together, our findings delineate the boundary between efficiently solvable and intractable instances of quantum state classification in the perfect (zero-error) regime.