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The complexity of estimating local physical quantities

Sevag Gharibian ( Virginia Commonwealth University (USA) )

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^QMA[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 P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following results: (1) The P^QMA[log]-completeness result of [Ambainis, CCC 2014] above requires O(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^QMA[log]-complete. (2) We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is P^QMA[log]-complete. (3) P^QMA[log] 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^QMA[log] is in PP. This improves the known containment that QMA is in PP [Kitaev, Watrous, STOC 2000]. (4) 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 P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on [Ambainis, CCC 2014] to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions.

This talk is based on joint work with Justin Yirka (Virginia Commonwealth University), available at No background in quantum complexity theory is assumed, and plenty of questions are encouraged.

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