907 research outputs found
Non-commutative Nash inequalities
A set of functional inequalities - called Nash inequalities - are introduced
and analyzed in the context of quantum Markov process mixing. The basic theory
of Nash inequalities is extended to the setting of non-commutative Lp spaces,
where their relationship to Poincare and log-Sobolev inequalities are fleshed
out. We prove Nash inequalities for a number of unital reversible semigroups
Quantum logarithmic Sobolev inequalities and rapid mixing
A family of logarithmic Sobolev inequalities on finite dimensional quantum
state spaces is introduced. The framework of non-commutative \bL_p-spaces is
reviewed and the relationship between quantum logarithmic Sobolev inequalities
and the hypercontractivity of quantum semigroups is discussed. This
relationship is central for the derivation of lower bounds for the logarithmic
Sobolev (LS) constants. Essential results for the family of inequalities are
proved, and we show an upper bound to the generalized LS constant in terms of
the spectral gap of the generator of the semigroup. These inequalities provide
a framework for the derivation of improved bounds on the convergence time of
quantum dynamical semigroups, when the LS constant and the spectral gap are of
the same order. Convergence bounds on finite dimensional state spaces are
particularly relevant for the field of quantum information theory. We provide a
number of examples, where improved bounds on the mixing time of several
semigroups are obtained; including the depolarizing semigroup and quantum
expanders.Comment: Updated manuscript, 30 pages, no figure
How fast do stabilizer Hamiltonians thermalize?
We present rigorous bounds on the thermalization time of the family of
quantum mechanical spin systems known as stabilizer Hamiltonians. The
thermalizing dynamics are modeled by a Davies master equation that arises from
a weak local coupling of the system to a large thermal bath. Two temperature
regimes are considered. First we clarify how in the low temperature regime, the
thermalization time is governed by a generalization of the energy barrier
between orthogonal ground states. When no energy barrier is present the
Hamiltonian thermalizes in a time that is at most quadratic in the system size.
Secondly, we show that above a universal critical temperature, every stabilizer
Hamiltonian relaxes to its unique thermal state in a time which scales at most
linearly in the size of the system. We provide an explicit lower bound on the
critical temperature. Finally, we discuss the implications of these result for
the problem of self-correcting quantum memories with stabilizer Hamiltonians
Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials
are given, together with methods for obtaining the coefficients in the
expansions. These approximations can be used as a standalone method of
computation of Gaussian quadratures for high enough degrees, with Gaussian
weights computed from asymptotic approximations for the orthogonal polynomials.
We provide numerical evidence showing that for degrees greater than the
asymptotic methods are enough for a double precision accuracy computation
(- digits) of the nodes and weights of the Gauss--Hermite and
Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic
Computation of the Marcum Q-function
Methods and an algorithm for computing the generalized Marcum function
() and the complementary function () are described.
These functions appear in problems of different technical and scientific areas
such as, for example, radar detection and communications, statistics and
probability theory, where they are called the non-central chi-square or the non
central gamma cumulative distribution functions.
The algorithm for computing the Marcum functions combines different methods
of evaluation in different regions: series expansions, integral
representations, asymptotic expansions, and use of three-term homogeneous
recurrence relations. A relative accuracy close to can be obtained
in the parameter region ,
, while for larger parameters the accuracy decreases (close to
for and close to for ).Comment: Accepted for publication in ACM Trans. Math. Soft
Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters
The determination of the physical entropies (R\'enyi, Shannon, Tsallis) of
high-dimensional quantum systems subject to a central potential requires the
knowledge of the asymptotics of some power and logarithmic integral functionals
of the hypergeometric orthogonal polynomials which control the wavefunctions of
the stationary states. For the -dimensional hydrogenic and oscillator-like
systems, the wavefunctions of the corresponding bound states are controlled by
the Laguerre () and Gegenbauer
() polynomials in both position and momentum
spaces, where the parameter linearly depends on . In this work we
study the asymptotic behavior as of the associated
entropy-like integral functionals of these two families of hypergeometric
polynomials
Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions
Conical functions appear in a large number of applications in physics and
engineering. In this paper we describe an extension of our module CONICAL for
the computation of conical functions. Specifically, the module includes now a
routine for computing the function , a
real-valued numerically satisfactory companion of the function for . In this way, a natural basis for solving
Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication
Quantum Metropolis Sampling
The original motivation to build a quantum computer came from Feynman who
envisaged a machine capable of simulating generic quantum mechanical systems, a
task that is believed to be intractable for classical computers. Such a machine
would have a wide range of applications in the simulation of many-body quantum
physics, including condensed matter physics, chemistry, and high energy
physics. Part of Feynman's challenge was met by Lloyd who showed how to
approximately decompose the time-evolution operator of interacting quantum
particles into a short sequence of elementary gates, suitable for operation on
a quantum computer. However, this left open the problem of how to simulate the
equilibrium and static properties of quantum systems. This requires the
preparation of ground and Gibbs states on a quantum computer. For classical
systems, this problem is solved by the ubiquitous Metropolis algorithm, a
method that basically acquired a monopoly for the simulation of interacting
particles. Here, we demonstrate how to implement a quantum version of the
Metropolis algorithm on a quantum computer. This algorithm permits to sample
directly from the eigenstates of the Hamiltonian and thus evades the sign
problem present in classical simulations. A small scale implementation of this
algorithm can already be achieved with today's technologyComment: revised versio
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