The Complexity of Simulating Local Measurements on Quantum Systems

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 results. The PQMA[log]-completeness result of [Ambainis, CCC 2014] requires O(logn)-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. 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]. A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem. In this vein, 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. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes