44,657 research outputs found
Simulating quantum computation by contracting tensor networks
The treewidth of a graph is a useful combinatorial measure of how close the
graph is to a tree. We prove that a quantum circuit with gates whose
underlying graph has treewidth can be simulated deterministically in
time, which, in particular, is polynomial in if
. Among many implications, we show efficient simulations for
log-depth circuits whose gates apply to nearby qubits only, a natural
constraint satisfied by most physical implementations. We also show that
one-way quantum computation of Raussendorf and Briegel (Physical Review
Letters, 86:5188--5191, 2001), a universal quantum computation scheme with
promising physical implementations, can be efficiently simulated by a
randomized algorithm if its quantum resource is derived from a small-treewidth
graph.Comment: 7 figure
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs II: semiclassical approach
A semiclassical approach to the universal ergodic spectral statistics in
quantum star graphs is presented for all known ten symmetry classes of quantum
systems. The approach is based on periodic orbit theory, the exact
semiclassical trace formula for star graphs and on diagrammatic techniques. The
appropriate spectral form factors are calculated upto one order beyond the
diagonal and self-dual approximations. The results are in accordance with the
corresponding random-matrix theories which supports a properly generalized
Bohigas-Giannoni-Schmit conjecture.Comment: 15 Page
Universal spectral form factor for chaotic dynamics
We consider the semiclassical limit of the spectral form factor of
fully chaotic dynamics. Starting from the Gutzwiller type double sum over
classical periodic orbits we set out to recover the universal behavior
predicted by random-matrix theory, both for dynamics with and without time
reversal invariance. For times smaller than half the Heisenberg time
, we extend the previously known -expansion to
include the cubic term. Beyond confirming random-matrix behavior of individual
spectra, the virtue of that extension is that the ``diagrammatic rules'' come
in sight which determine the families of orbit pairs responsible for all orders
of the -expansion.Comment: 4 pages, 1 figur
Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions
In this paper, we study the problem of compressed sensing using binary
measurement matrices and -norm minimization (basis pursuit) as the
recovery algorithm. We derive new upper and lower bounds on the number of
measurements to achieve robust sparse recovery with binary matrices. We
establish sufficient conditions for a column-regular binary matrix to satisfy
the robust null space property (RNSP) and show that the associated sufficient
conditions % sparsity bounds for robust sparse recovery obtained using the RNSP
are better by a factor of compared to the
sufficient conditions obtained using the restricted isometry property (RIP).
Next we derive universal \textit{lower} bounds on the number of measurements
that any binary matrix needs to have in order to satisfy the weaker sufficient
condition based on the RNSP and show that bipartite graphs of girth six are
optimal. Then we display two classes of binary matrices, namely parity check
matrices of array codes and Euler squares, which have girth six and are nearly
optimal in the sense of almost satisfying the lower bound. In principle,
randomly generated Gaussian measurement matrices are "order-optimal". So we
compare the phase transition behavior of the basis pursuit formulation using
binary array codes and Gaussian matrices and show that (i) there is essentially
no difference between the phase transition boundaries in the two cases and (ii)
the CPU time of basis pursuit with binary matrices is hundreds of times faster
than with Gaussian matrices and the storage requirements are less. Therefore it
is suggested that binary matrices are a viable alternative to Gaussian matrices
for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table
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