The complexity of simulating local measurements on quantum systems

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 PQMA[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 PQMA[log]-complete. In this paper, we continue the study of PQMA[log], obtaining the following lower and upper bounds.Lower bounds (hardness results): - The PQMA[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 PQMA[log]-complete, resolving an open question of Ambainis.- We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly PQMA[log]-complete. - We identify a flaw in [Ambainis, CCC 2014] regarding a PUQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a ``query validation'' technique, we build on [Ambainis, CCC 2014] to obtain PUQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions. Upper bounds (containment in complexity classes): - PQMA[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 PQMA[log]⊆PP. This improves the containment QMA⊆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.