The Computational Power of Symmetric Hamiltonians

TL;DR

This paper demonstrates that symmetries in Hamiltonian systems do not simplify their quantum computational complexity, which remains BQP-complete, and introduces a universal quantum interface enabling control with minimal intervention.

Contribution

It shows that symmetry does not reduce the complexity of simulating Hamiltonian dynamics and introduces a universal quantum interface for control with minimal control requirements.

Findings

Simulating symmetric Hamiltonians remains BQP-complete.

A universal quantum interface enables control of quantum systems through a fixed set of spins.

Cooling Hamiltonians with random magnetic fields solves a QMA-complete problem.

Abstract

The presence of symmetries, be they discrete or continuous, in a physical system typically leads to a reduction in the problem to be solved. Here we report that neither translational invariance nor rotational invariance reduce the computational complexity of simulating Hamiltonian dynamics; the problem is still BQP complete, and is believed to be hard on a classical computer. This is achieved by designing a system to implement a Universal Quantum Interface, a device which enables control of an entire computation through the control of a fixed number of spins, and using it as a building-block to entirely remove the need for control, except in the system initialisation. Finally, it is shown that cooling such Hamiltonians to their ground states in the presence of random magnetic fields solves a QMA-complete problem.