Quantum 3-SAT Is QMA1-Complete

Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum k-SAT problem, each constraint is specified by a k-local projector and is satisfied by any state in its nullspace. Bravyi showed that quantum 2-SAT can be solved efficiently on a classical computer and that quantum k-SAT with k ≥ 4 is QMA1-complete [4]. Quantum 3-SAT was known to be contained in QMA1 [4], but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA1-hard, and therefore complete for this complexity class.

[1]  Peter Selinger,et al.  Exact synthesis of multi-qubit Clifford+T circuits , 2012, ArXiv.

[2]  Christopher R. Laumann,et al.  Random quantum satisfiability , 2010, Quantum Inf. Comput..

[3]  Daniel Nagaj,et al.  Criticality without frustration for quantum spin-1 chains. , 2012, Physical review letters.

[4]  Chris Marriott,et al.  Quantum Arthur–Merlin games , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[5]  Bojan Mohar,et al.  Eigenvalues, diameter, and mean distance in graphs , 1991, Graphs Comb..

[6]  Julia Kempe,et al.  The Complexity of the Local Hamiltonian Problem , 2004, FSTTCS.

[7]  Zhengfeng Ji,et al.  complete characterization of the ground-space structure of two-body frustration-free hamiltonians for qubits , 2010, 1010.2480.

[8]  New construction for a QMA complete three-local Hamiltonian , 2007, quant-ph/0612113.

[9]  Sergey Bravyi,et al.  Efficient algorithm for a quantum analogue of 2-SAT , 2006, quant-ph/0602108.

[10]  Yong Zhang,et al.  Fast amplification of QMA , 2009, Quantum Inf. Comput..

[11]  Daniel Nagaj.  Fast universal quantum computation with railroad-switch local Hamiltonians , 2009, 0908.4219.

[12]  Andris Ambainis,et al.  A quantum lovász local lemma , 2009, STOC '10.

[13]  T. Osborne.  Hamiltonian complexity , 2011, Reports on progress in physics. Physical Society.

[14]  Roderich Moessner,et al.  Random quantum satisfiabiilty , 2010 .

[15]  Matthew Coudron,et al.  Unfrustration Condition and Degeneracy of Qudits on Trees , 2012 .

[16]  Richard Phillips Feynman,et al.  Quantum mechanical computers , 1984, Feynman Lectures on Computation.

[17]  Hirotada Kobayashi,et al.  Achieving perfect completeness in classical-witness quantum merlin-arthur proof systems , 2011, Quantum Inf. Comput..

[18]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[19]  Oded Regev,et al.  Quantum SAT for a Qutrit-Cinquit Pair Is QMA1-Complete , 2008, ICALP.

[20]  Scott Aaronson,et al.  On perfect completeness for QMA , 2008, Quantum Inf. Comput..

[21]  Stathis Zachos,et al.  Probabalistic Quantifiers vs. Distrustful Adversaries , 1987, FSTTCS.

[22]  Alexander Russell,et al.  Bounds on the Quantum Satisfiability Threshold , 2009, ICS.

[23]  Christopher R. Laumann,et al.  On product, generic and random generic quantum satisfiability , 2009, ArXiv.

[24]  François Le Gall,et al.  Stronger methods of making quantum interactive proofs perfectly complete , 2012, ITCS '13.

[25]  Edward Farhi,et al.  Unfrustrated qudit chains and their ground states , 2010, 1001.1006.