Oracle complexity classes and local measurements on physical Hamiltonians
Abstract
The canonical problem for the class Quantum MerlinArthur (QMA) is that of estimating ground state energies of local Hamiltonians. Perhaps surprisingly, [Ambainis, CCC 2014] showed that the related, but arguably more natural, problem of simulating local measurements on ground states of local Hamiltonians (APXSIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APXSIM is P^QMA[log]complete, for P^QMA[log] the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APXSIM is P^QMA[log]complete even when restricted to more physical Hamiltonians, obtaining as intermediate steps a variety of related complexitytheoretic results. We first give a sequence of results which together yield P^QMA[log]hardness for APXSIM on wellmotivated Hamiltonians: (1) We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. These equalities simplify the proofs of our subsequent results. (2) Next, we show that the hardness of APXSIM is preserved under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a byproduct, we obtain a full complexity classification of APXSIM, showing it is complete for P, P^NP, P^StoqMA, or P^QMA depending on the Hamiltonians employed. (3) Leveraging the above, we show that APXSIM is P^QMA[log]complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians, including physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APXSIM remains P^QMA[log]complete even for local Hamiltonians on a 1D line of 8dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.05981
 Bibcode:
 2019arXiv190905981G
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 38 pages, 3 figures