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 classPQMA[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 isPQMA[log]-complete. In this paper, we continue the study ofPQMA[log], obtaining the following lower and upper bounds.Lower bounds (hardness results): - ThePQMA[log]-completeness result of [Ambainis, CCC 2014] requiresO(log⁡n)-local observables and Hamiltonians. We show that simulating even asingle qubitmeasurement on ground states of5-local Hamiltonians isPQMA[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 similarlyPQMA[log]-complete. - We identify a flaw in [Ambainis, CCC 2014] regarding aPUQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a ``query validation'' technique, we build on [Ambainis, CCC 2014] to obtainPUQMA[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 showPQMA[log]⊆PP. This improves the containmentQMA⊆PP[Kitaev, Watrous, STOC 2000]. This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves apromiseproblem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.