Quantum Computational Complexity and Symmetry

TL;DR

This paper explores the complexity of testing symmetries in quantum states and channels, establishing their classification within various quantum complexity classes and connecting symmetry problems to quantum computational difficulty.

Contribution

It classifies the complexity of symmetry-testing problems in quantum information, linking them to major quantum complexity classes and establishing new completeness results.

Findings

Symmetry-testing problems are complete for BQP, QMA, QSZK, QIP(2), QIP_EB(2), and QIP.

Two Hamiltonian symmetry-testing problems are in QMA and QAM.

Open question on whether these Hamiltonian problems are complete for their classes.

Abstract

Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the difficulty of symmetry-testing problems involving a unitary representation of a group and a state or a channel that is being tested. In particular, we prove that various such symmetry-testing problems are complete for BQP, QMA, QSZK, QIP(2), QIP_EB(2), and QIP, thus spanning the prominent classes of the quantum interactive proof hierarchy and forging a non-trivial connection between symmetry and quantum computational complexity. Finally, we prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and QAM, while leaving it as an intriguing open question to determine whether these problems are complete for these classes.