Complexity Classification of Two-Qubit Commuting Hamiltonians
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.
Quantum Computing
Sampling Problems
Commuting Hamiltonians
IQP
Gate Classification Theorems
28:1-28:33
Regular Paper
Adam
Bouland
Adam Bouland
Laura
Mancinska
Laura Mancinska
Xue
Zhang
Xue Zhang
10.4230/LIPIcs.CCC.2016.28
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode