The complexity of simulating local measurements on quantum systems

TL;DR

This paper investigates the complexity of simulating local measurements on quantum ground states, establishing new bounds and resolving open questions about the computational hardness of these tasks within quantum complexity classes.

Contribution

It extends the understanding of P^QMA[log] by proving hardness for single-qubit measurements on 5-local Hamiltonians and analyzing two-point correlation functions, also correcting prior proof flaws.

Findings

Simulating single-qubit measurements on 5-local Hamiltonians is P^QMA[log]-complete.

Estimating two-point correlation functions is P^QMA[log]-complete.

P^QMA[log] is contained in PP, showing it is slightly harder than QMA.

Abstract

An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity class P^QMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following lower and upper bounds. Lower bounds (hardness results): (1) The P^QMA[log]-completeness result of [Ambainis, CCC 2014] requires O(log n)-local observables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[log]-complete, resolving an open question of Ambainis. (2) We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P^QMA[log]-complete. (3) We identify a flaw in [Ambainis, CCC 2014] regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on [Ambainis, CCC 2014] to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions. Upper bounds (containment in complexity classes): P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of [Beigel, Hemachandra, Wechsung, SCT 1989] to show P^QMA[log] is in PP. This improves the containment QMA is in PP [Kitaev, Watrous, STOC 2000]. This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves a promise problem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.