2,294 research outputs found
Classical and quantum satisfiability
We present the linear algebraic definition of QSAT and propose a direct
logical characterization of such a definition. We then prove that this logical
version of QSAT is not an extension of classical satisfiability problem (SAT).
This shows that QSAT does not allow a direct comparison between the complexity
classes NP and QMA, for which SAT and QSAT are respectively complete.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Classical-Quantum Mixing in the Random 2-Satisfiability Problem
Classical satisfiability (SAT) and quantum satisfiability (QSAT) are complete
problems for the complexity classes NP and QMA which are believed to be
intractable for classical and quantum computers, respectively. Statistical
ensembles of instances of these problems have been studied previously in an
attempt to elucidate their typical, as opposed to worst case, behavior. In this
paper we introduce a new statistical ensemble that interpolates between
classical and quantum. For the simplest 2-SAT/2-QSAT ensemble we find the exact
boundary that separates SAT and UNSAT instances. We do so by establishing
coincident lower and upper bounds, in the limit of large instances, on the
extent of the UNSAT and SAT regions, respectively.Comment: Updated reference
Clustering in Hilbert space of a quantum optimization problem
The solution space of many classical optimization problems breaks up into
clusters which are extensively distant from one another in the Hamming metric.
Here, we show that an analogous quantum clustering phenomenon takes place in
the ground state subspace of a certain quantum optimization problem. This
involves extending the notion of clustering to Hilbert space, where the
classical Hamming distance is not immediately useful. Quantum clusters
correspond to macroscopically distinct subspaces of the full quantum ground
state space which grow with the system size. We explicitly demonstrate that
such clusters arise in the solution space of random quantum satisfiability
(3-QSAT) at its satisfiability transition. We estimate both the number of these
clusters and their internal entropy. The former are given by the number of
hardcore dimer coverings of the core of the interaction graph, while the latter
is related to the underconstrained degrees of freedom not touched by the
dimers. We additionally provide new numerical evidence suggesting that the
3-QSAT satisfiability transition may coincide with the product satisfiability
transition, which would imply the absence of an intermediate entangled
satisfiable phase.Comment: 11 pages, 6 figure
On product, generic and random generic quantum satisfiability
We report a cluster of results on k-QSAT, the problem of quantum
satisfiability for k-qubit projectors which generalizes classical
satisfiability with k-bit clauses to the quantum setting. First we define the
NP-complete problem of product satisfiability and give a geometrical criterion
for deciding when a QSAT interaction graph is product satisfiable with positive
probability. We show that the same criterion suffices to establish quantum
satisfiability for all projectors. Second, we apply these results to the random
graph ensemble with generic projectors and obtain improved lower bounds on the
location of the SAT--unSAT transition. Third, we present numerical results on
random, generic satisfiability which provide estimates for the location of the
transition for k=3 and k=4 and mild evidence for the existence of a phase which
is satisfiable by entangled states alone.Comment: 9 pages, 5 figures, 1 table. Updated to more closely match published
version. New proof in appendi
Classical and Quantum Annealing in the Median of Three Satisfiability
We determine the classical and quantum complexities of a specific ensemble of
three-satisfiability problems with a unique satisfying assignment for up to
N=100 and N=80 variables, respectively. In the classical limit we employ
generalized ensemble techniques and measure the time that a Markovian Monte
Carlo process spends in searching classical ground states. In the quantum limit
we determine the maximum finite correlation length along a quantum adiabatic
trajectory determined by the linear sweep of the adiabatic control parameter in
the Hamiltonian composed of the problem Hamiltonian and the constant transverse
field Hamiltonian. In the median of our ensemble both complexities diverge
exponentially with the number of variables. Hence, standard, conventional
adiabatic quantum computation fails to reduce the computational complexity to
polynomial. Moreover, the growth-rate constant in the quantum limit is 3.8
times as large as the one in the classical limit, making classical fluctuations
more beneficial than quantum fluctuations in ground-state searches
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