TY - CONF
AB - 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.
AU - Gharibian, Sevag
AU - Yirka, Justin
ED - Wilde, Mark
ID - 8160
KW - Complexity theory
KW - Quantum Merlin Arthur (QMA)
KW - local Hamiltonian
KW - local measurement
KW - spectral gap
T2 - 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)
TI - The Complexity of Simulating Local Measurements on Quantum Systems
VL - 73
ER -
TY - CONF
AB - 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.
AU - Gharibian, Sevag
AU - Santha, Miklos
AU - Sikora, Jamie
AU - Sundaram, Aarthi
AU - Yirka, Justin
ED - Potapov, Igor
ED - Spirakis, Paul
ED - Worrell, James
ID - 8161
KW - Complexity Theory
KW - Quantum Computing
KW - Polynomial Hierarchy
KW - Semidefinite Programming
KW - QMA(2)
KW - Quantum Complexity
T2 - 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
TI - Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)
VL - 117
ER -
TY - CONF
AB - The Boolean constraint satisfaction problem 3-SAT is arguably the canonical NP-complete problem. In contrast, 2-SAT can not only be decided in polynomial time, but in fact in deterministic linear time. In 2006, Bravyi proposed a physically motivated generalization of k-SAT to the quantum setting, defining the problem "quantum k-SAT". He showed that quantum 2-SAT is also solvable in polynomial time on a classical computer, in particular in deterministic time O(n^4), assuming unit-cost arithmetic over a field extension of the rational numbers, where n is number of variables. In this paper, we present an algorithm for quantum 2-SAT which runs in linear time, i.e. deterministic time O(n+m) for n and m the number of variables and clauses, respectively. Our approach exploits the transfer matrix techniques of Laumann et al. [QIC, 2010] used in the study of phase transitions for random quantum 2-SAT, and bears similarities with both the linear time 2-SAT algorithms of Even, Itai, and Shamir (based on backtracking) [SICOMP, 1976] and Aspvall, Plass, and Tarjan (based on strongly connected components) [IPL, 1979].
AU - de Beaudrap, Niel
AU - Gharibian, Sevag
ED - Raz, Ran
ID - 8159
KW - quantum 2-SAT
KW - transfer matrix
KW - strongly connected components
KW - limited backtracking
KW - local Hamiltonian
SN - 978-3-95977-008-8
T2 - Proceedings of the 31st Conference on Computational Complexity (CCC 2016)
TI - A Linear Time Algorithm for Quantum 2-SAT
VL - 50
ER -