431 research outputs found
Multilayer wave functions: A recursive coupling of local excitations
Finding a succinct representation to describe the ground state of a
disordered interacting system could be very helpful in understanding the
interplay between the interactions that is manifested in a quantum phase
transition. In this work we use some elementary states to construct recursively
an ansatz of multilayer wave functions, where in each step the higher-level
wave function is represented by a superposition of the locally "excited states"
obtained from the lower-level wave function. This allows us to write the
Hamiltonian expectation in terms of some local functions of the variational
parameters, and employ an efficient message-passing algorithm to find the
optimal parameters. We obtain good estimations of the ground-state energy and
the phase transition point for the transverse Ising model with a few layers of
mean-field and symmetric tree states. The work is the first step towards the
application of local and distributed message-passing algorithms in the study of
structured variational problems in finite dimensions.Comment: 23 pages, including 3 appendices and 6 figures. A shortened version
published in EP
Sign problem in the Bethe approximation
We propose a message-passing algorithm to compute the Hamiltonian expectation
with respect to an appropriate class of trial wave functions for an interacting
system of fermions. To this end, we connect the quantum expectations to average
quantities in a classical system with both local and global interactions, which
are related to the variational parameters and use the Bethe approximation to
estimate the average energy within the replica-symmetric approximation. The
global interactions, which are needed to obtain a good estimation of the
average fermion sign, make the average energy a nonlocal function of the
variational parameters. We use some heuristic minimization algorithms to find
approximate ground states of the Hubbard model on random regular graphs and
observe significant qualitative improvements with respect to the mean-field
approximation.Comment: 19 pages, 9 figures, one figure adde
Low-temperature excitations within the Bethe approximation
We propose the variational quantum cavity method to construct a minimal
energy subspace of wave vectors that are used to obtain some upper bounds for
the energy cost of the low-temperature excitations. Given a trial wave function
we use the cavity method of statistical physics to estimate the Hamiltonian
expectation and to find the optimal variational parameters in the subspace of
wave vectors orthogonal to the lower-energy wave functions. To this end, we
write the overlap between two wave functions within the Bethe approximation
which allows us to replace the global orthogonality constraint with some local
constraints on the variational parameters. The method is applied to the
transverse Ising model and different levels of approximations are compared with
the exact numerical solutions for small systems.Comment: 14 pages, 4 figure
Bethe free-energy approximations for disordered quantum systems
Given a locally consistent set of reduced density matrices, we construct
approximate density matrices which are globally consistent with the local
density matrices we started from when the trial density matrix has a tree
structure. We employ the cavity method of statistical physics to find the
optimal density matrix representation by slowly decreasing the temperature in
an annealing algorithm, or by minimizing an approximate Bethe free energy
depending on the reduced density matrices and some cavity messages originated
from the Bethe approximation of the entropy. We obtain the classical Bethe
expression for the entropy within a naive (mean-field) approximation of the
cavity messages, which is expected to work well at high temperatures. In the
next order of the approximation, we obtain another expression for the Bethe
entropy depending only on the diagonal elements of the reduced density
matrices. In principle, we can improve the entropy approximation by considering
more accurate cavity messages in the Bethe approximation of the entropy. We
compare the annealing algorithm and the naive approximation of the Bethe
entropy with exact and approximate numerical simulations for small and large
samples of the random transverse Ising model on random regular graphs.Comment: 23 pages, 4 figures, 4 appendice
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