Complexity classification of two-qubit commuting hamiltonians

TL;DR

This paper classifies two-qubit commuting Hamiltonians, showing that most are computationally powerful and can perform tasks infeasible for classical computers unless the polynomial hierarchy collapses, with only entanglement-failing ones being classically simulable.

Contribution

It establishes a dichotomy theorem for two-qubit commuting Hamiltonians, identifying which are classically simulable and which enable universal quantum computation.

Findings

Most two-qubit commuting Hamiltonians are computationally intractable for classical simulation.

Only Hamiltonians that do not generate entanglement are classically simulable.

The proof introduces new postselection gadgets and employs Lie theory.

Abstract

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.