Quantum State Isomorphism

TL;DR

This paper investigates the complexity of quantum state isomorphism, showing it is at least as hard as Graph Isomorphism and exploring related problems with quantum interactive proof systems and complexity classifications.

Contribution

It establishes the complexity of Quantum State Isomorphism, relates it to Graph Isomorphism, and analyzes related problems like StabilizerStateIsomorphism and MixedStateIsomorphism.

Findings

StateIsomorphism is as hard as Graph Isomorphism.

The complement of StateNonIsomorphism has a two-message quantum interactive proof.

MSI is QSZK-hard.

Abstract

We consider a problem we call StateIsomorphism: given two quantum states of n qubits, can one be obtained from the other by rearranging the qubit subsystems? Our main goal is to study the complexity of this problem, which is a natural quantum generalisation of the problem StringIsomorphism. We show that StateIsomorphism is at least as hard as GraphIsomorphism, and show that these problems have a similar structure by presenting evidence to suggest that StateIsomorphism is an intermediate problem for QCMA. In particular, we show that the complement of the problem, StateNonIsomorphism, has a two message quantum interactive proof system, and that this proof system can be made statistical zero-knowledge. We consider also StabilizerStateIsomorphism (SSI) and MixedStateIsomorphism (MSI), showing that the complement of SSI has a quantum interactive proof system that uses classical communication only, and that MSI is QSZK-hard.