@inproceedings{8160,
abstract = {An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, Ambainis 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. The P^QMA[log]-completeness result of Ambainis requires O(log n)-local observ- ables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[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 P^QMA[log]-complete. P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of Beigel, Hemachandra, and Wechsung to show P^QMA[log] \subseteq PP. This improves the containment QMA \subseteq PP from Kitaev and Watrous. 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' prior work regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on his prior work to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions.},
author = {Gharibian, Sevag and Yirka, Justin},
booktitle = {12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)},
editor = {Wilde, Mark},
keyword = {Complexity theory, Quantum Merlin Arthur (QMA), local Hamiltonian, local measurement, spectral gap},
location = {Paris, France},
pages = {2:1--2:17},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik},
title = {{The Complexity of Simulating Local Measurements on Quantum Systems}},
doi = {10.4230/LIPIcs.TQC.2017.2},
volume = {73},
year = {2018},
}
@inproceedings{8161,
abstract = {The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, Q Sigma_3, into NEXP using the Ellipsoid Method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs "equivalent"), then QMA(2) is in the Counting Hierarchy (specifically, in P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP.},
author = {Gharibian, Sevag and Santha, Miklos and Sikora, Jamie and Sundaram, Aarthi and Yirka, Justin},
booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
editor = {Potapov, Igor and Spirakis, Paul and Worrell, James},
keyword = {Complexity Theory, Quantum Computing, Polynomial Hierarchy, Semidefinite Programming, QMA(2), Quantum Complexity},
location = {Liverpool, UK},
pages = {58:1--58:16},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik},
title = {{Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)}},
doi = {10.4230/LIPIcs.MFCS.2018.58},
volume = {117},
year = {2018},
}