Computational Complexity of Learning Efficiently Generatable Pure States

TL;DR

This paper investigates the computational complexity of learning efficiently generatable pure quantum states, establishing links to complexity classes and cryptographic assumptions, and providing algorithms under certain conditions.

Contribution

It demonstrates that learning pure quantum states efficiently is connected to PP complexity class and cryptographic hardness assumptions, offering new insights into quantum learning complexity.

Findings

Existence of polynomial-time algorithms for learning pure states under certain conditions.

Connection between quantum state learning hardness and PP complexity class.

Equivalence between one-way state generators and average-case hardness of learning pure states.

Abstract

Understanding the computational complexity of learning efficient classical programs in various learning models has been a fundamental and important question in classical computational learning theory. In this work, we study the computational complexity of quantum state learning, which can be seen as a quantum generalization of distributional learning introduced by Kearns et.al [STOC94]. Previous works by Chung and Lin [TQC21], and B\u{a}descu and O$'$Donnell [STOC21] study the sample complexity of the quantum state learning and show that polynomial copies are sufficient if unknown quantum states are promised efficiently generatable. However, their algorithms are inefficient, and the computational complexity of this learning problem remains unresolved. In this work, we study the computational complexity of quantum state learning when the states are promised to be efficiently generatable. We show that if unknown quantum states are promised to be pure states and efficiently generateable, then there exists a quantum polynomial time algorithm $A$ and a language $L \in PP$ such that $A^L$ can learn its classical description. We also observe the connection between the hardness of learning quantum states and quantum cryptography. We show that the existence of one-way state generators with pure state outputs is equivalent to the average-case hardness of learning pure states. Additionally, we show that the existence of EFI implies the average-case hardness of learning mixed states.