eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-27
38:1
38:16
10.4230/LIPIcs.MFCS.2018.38
article
On Efficiently Solvable Cases of Quantum k-SAT
Aldi, Marco
1
de Beaudrap, Niel
2
Gharibian, Sevag
3
Saeedi, Seyran
4
Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA
Department of Computer Science, University of Oxford, UK
Department of Computer Science, University of Paderborn, Germany, and Virginia Commonwealth University, USA
Department of Computer Science, Virginia Commonwealth University, Richmond, VA, USA
The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open.
Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry; we hope these prove useful elsewhere.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol117-mfcs2018/LIPIcs.MFCS.2018.38/LIPIcs.MFCS.2018.38.pdf
search complexity
local Hamiltonian
Quantum SAT
algebraic geometry