The complexity of estimating local physical quantities

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, 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-complete. (Here, P is the set of decision problems solvable in polynomial time with access to O(log n) queries to a Quantum Merlin Arthur (QMA) oracle.) In this paper, we continue the study of P, obtaining the following results. • The P-completeness result of [Ambainis, CCC 2014] above requiresO(log n)-local Hamiltonians and O(log n)-local observables. Whether this could be improved to the more physically appealing O(1)-local setting was left as an open question. We resolve this question positively by showing that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P-complete. • We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P-complete. • P is thought of as “slightly harder” than QMA. We give a formal justification of this intuition by exploiting the technique of hierarchical voting of [Beigel, Hemachandra, and Wechsung, SCT 1989] to show P ⊆ PP. This improves the known containment QMA ⊆ PP [Kitaev, Watrous, STOC 2000]. • A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem (such as P). In this vein, we identify a flaw in [Ambainis, CCC 2014] regarding a P-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a “query validation” technique, we build on [Ambainis, CCC 2014] to obtain P-hardness for estimating spectral gaps under polynomial-time Turing reductions.