3,143 research outputs found
New quantum algorithm for studying NP-complete problems
Ordinary approach to quantum algorithm is based on quantum Turing machine or
quantum circuits. It is known that this approach is not powerful enough to
solve NP-complete problems. In this paper we study a new approach to quantum
algorithm which is a combination of the ordinary quantum algorithm with a
chaotic dynamical system. We consider the satisfiability problem as an example
of NP-complete problems and argue that the problem, in principle, can be solved
in polynomial time by using our new quantum algorithm.Comment: 11 pages, 1 figur
Quantum Algorithm for SAT Problem and Quantum Mutual Entropy
It is von Neumann who opened the window for today's Information epoch. He
defined quantum entropy including Shannon's information more than 20 years
ahead of Shannon, and he introduced a concept what computation means
mathematically. In this paper I will report two works that we have recently
done, one of which is on quantum algorithum in generalized sense solving the
SAT problem (one of NP complete problems) and another is on quantum mutual
entropy properly describing quantum communication processes.Comment: 19 pages, Proceedings of the von Neumann Centennial Conference:
Linear Operators and Foundations of Quantum Mechanics, Budapest, Hungary,
15-20 October, 200
A stochastic limit approach to the SAT problem
We propose a new approach to solve an NP complete problem by means of
stochastic limit.Comment: 8 page
A Random Matrix Model of Adiabatic Quantum Computing
We present an analysis of the quantum adiabatic algorithm for solving hard
instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory
(RMT). We determine the global regularity of the spectral fluctuations of the
instantaneous Hamiltonians encountered during the interpolation between the
starting Hamiltonians and the ones whose ground states encode the solutions to
the computational problems of interest. At each interpolation point, we
quantify the degree of regularity of the average spectral distribution via its
Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from
chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor
spacings. We find that for hard problem instances, i.e., those having a
critical ratio of clauses to variables, the spectral fluctuations typically
become irregular across a contiguous region of the interpolation parameter,
while the spectrum is regular for easy instances. Within the hard region, RMT
may be applied to obtain a mathematical model of the probability of avoided
level crossings and concomitant failure rate of the adiabatic algorithm due to
non-adiabatic Landau-Zener type transitions. Our model predicts that if the
interpolation is performed at a uniform rate, the average failure rate of the
quantum adiabatic algorithm, when averaged over hard problem instances, scales
exponentially with increasing problem size.Comment: 9 pages, 7 figure
The role of singularities in chaotic spectroscopy
We review the status of the semiclassical trace formula with emphasis on the
particular types of singularities that occur in the Gutzwiller-Voros zeta
function for bound chaotic systems. To understand the problem better we extend
the discussion to include various classical zeta functions and we contrast
properties of axiom-A scattering systems with those of typical bound systems.
Singularities in classical zeta functions contain topological and dynamical
information, concerning e.g. anomalous diffusion, phase transitions among
generalized Lyapunov exponents, power law decay of correlations. Singularities
in semiclassical zeta functions are artifacts and enters because one neglects
some quantum effects when deriving them, typically by making saddle point
approximation when the saddle points are not enough separated. The discussion
is exemplified by the Sinai billiard where intermittent orbits associated with
neutral orbits induce a branch point in the zeta functions. This singularity is
responsible for a diverging diffusion constant in Lorentz gases with unbounded
horizon. In the semiclassical case there is interference between neutral orbits
and intermittent orbits. The Gutzwiller-Voros zeta function exhibit a branch
point because it does not take this effect into account. Another consequence is
that individual states, high up in the spectrum, cannot be resolved by
Berry-Keating technique.Comment: 22 pages LaTeX, figures available from autho
The Road to Quantum Computational Supremacy
We present an idiosyncratic view of the race for quantum computational
supremacy. Google's approach and IBM challenge are examined. An unexpected
side-effect of the race is the significant progress in designing fast classical
algorithms. Quantum supremacy, if achieved, won't make classical computing
obsolete.Comment: 15 pages, 1 figur
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