7,505 research outputs found
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.Comment: 21 page
Variational Theory and Domain Decomposition for Nonlocal Problems
In this article we present the first results on domain decomposition methods
for nonlocal operators. We present a nonlocal variational formulation for these
operators and establish the well-posedness of associated boundary value
problems, proving a nonlocal Poincar\'{e} inequality. To determine the
conditioning of the discretized operator, we prove a spectral equivalence which
leads to a mesh size independent upper bound for the condition number of the
stiffness matrix. We then introduce a nonlocal two-domain variational
formulation utilizing nonlocal transmission conditions, and prove equivalence
with the single-domain formulation. A nonlocal Schur complement is introduced.
We establish condition number bounds for the nonlocal stiffness and Schur
complement matrices. Supporting numerical experiments demonstrating the
conditioning of the nonlocal one- and two-domain problems are presented.Comment: Updated the technical part. In press in Applied Mathematics and
Computatio
Quantum Spectrum Testing
In this work, we study the problem of testing properties of the spectrum of a
mixed quantum state. Here one is given copies of a mixed state
and the goal is to distinguish whether 's
spectrum satisfies some property or is at least -far in
-distance from satisfying . This problem was promoted in
the survey of Montanaro and de Wolf under the name of testing unitarily
invariant properties of mixed states. It is the natural quantum analogue of the
classical problem of testing symmetric properties of probability distributions.
Here, the hope is for algorithms with subquadratic copy complexity in the
dimension . This is because the "empirical Young diagram (EYD) algorithm"
can estimate the spectrum of a mixed state up to -accuracy using only
copies. In this work, we show that given a
mixed state : (i) copies
are necessary and sufficient to test whether is the maximally mixed
state, i.e., has spectrum ; (ii)
copies are necessary and sufficient to test with
one-sided error whether has rank , i.e., has at most nonzero
eigenvalues; (iii) copies are necessary and
sufficient to distinguish whether is maximally mixed on an
-dimensional or an -dimensional subspace; and (iv) The EYD
algorithm requires copies to estimate the spectrum of
up to -accuracy, nearly matching the known upper bound. In
addition, we simplify part of the proof of the upper bound. Our techniques
involve the asymptotic representation theory of the symmetric group; in
particular Kerov's algebra of polynomial functions on Young diagrams.Comment: 70 pages, 6 figure
Time-parallel iterative solvers for parabolic evolution equations
We present original time-parallel algorithms for the solution of the implicit
Euler discretization of general linear parabolic evolution equations with
time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory
of parabolic problems, we show that the standard nonsymmetric time-global
system can be equivalently reformulated as an original symmetric saddle-point
system that remains inf-sup stable with respect to the same natural parabolic
norms. We then propose and analyse an efficient and readily implementable
parallel-in-time preconditioner to be used with an inexact Uzawa method. The
proposed preconditioner is non-intrusive and easy to implement in practice, and
also features the key theoretical advantages of robust spectral bounds, leading
to convergence rates that are independent of the number of time-steps, final
time, or spatial mesh sizes, and also a theoretical parallel complexity that
grows only logarithmically with respect to the number of time-steps. Numerical
experiments with large-scale parallel computations show the effectiveness of
the method, along with its good weak and strong scaling properties
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