21,336 research outputs found
A Non-Commuting Stabilizer Formalism
We propose a non-commutative extension of the Pauli stabilizer formalism. The
aim is to describe a class of many-body quantum states which is richer than the
standard Pauli stabilizer states. In our framework, stabilizer operators are
tensor products of single-qubit operators drawn from the group , where and . We
provide techniques to efficiently compute various properties related to
bipartite entanglement, expectation values of local observables, preparation by
means of quantum circuits, parent Hamiltonians etc. We also highlight
significant differences compared to the Pauli stabilizer formalism. In
particular, we give examples of states in our formalism which cannot arise in
the Pauli stabilizer formalism, such as topological models that support
non-Abelian anyons.Comment: 52 page
Protecting unknown two-qubit entangled states by nesting Uhrig's dynamical decoupling sequences
Future quantum technologies rely heavily on good protection of quantum
entanglement against environment-induced decoherence. A recent study showed
that an extension of Uhrig's dynamical decoupling (UDD) sequence can (in
theory) lock an arbitrary but known two-qubit entangled state to the th
order using a sequence of control pulses [Mukhtar et al., Phys. Rev. A 81,
012331 (2010)]. By nesting three layers of explicitly constructed UDD
sequences, here we first consider the protection of unknown two-qubit states as
superposition of two known basis states, without making assumptions of the
system-environment coupling. It is found that the obtained decoherence
suppression can be highly sensitive to the ordering of the three UDD layers and
can be remarkably effective with the correct ordering. The detailed theoretical
results are useful for general understanding of the nature of controlled
quantum dynamics under nested UDD. As an extension of our three-layer UDD, it
is finally pointed out that a completely unknown two-qubit state can be
protected by nesting four layers of UDD sequences. This work indicates that
when UDD is applicable (e.g., when environment has a sharp frequency cut-off
and when control pulses can be taken as instantaneous pulses), dynamical
decoupling using nested UDD sequences is a powerful approach for entanglement
protection.Comment: 11 pages, 3 figures, published versio
A monomial matrix formalism to describe quantum many-body states
We propose a framework to describe and simulate a class of many-body quantum
states. We do so by considering joint eigenspaces of sets of monomial unitary
matrices, called here "M-spaces"; a unitary matrix is monomial if precisely one
entry per row and column is nonzero. We show that M-spaces encompass various
important state families, such as all Pauli stabilizer states and codes, the
AKLT model, Kitaev's (abelian and non-abelian) anyon models, group coset
states, W states and the locally maximally entanglable states. We furthermore
show how basic properties of M-spaces can transparently be understood by
manipulating their monomial stabilizer groups. In particular we derive a
unified procedure to construct an eigenbasis of any M-space, yielding an
explicit formula for each of the eigenstates. We also discuss the computational
complexity of M-spaces and show that basic problems, such as estimating local
expectation values, are NP-hard. Finally we prove that a large subclass of
M-spaces---containing in particular most of the aforementioned examples---can
be simulated efficiently classically with a unified method.Comment: 11 pages + appendice
Subspace Evasive Sets
In this work we describe an explicit, simple, construction of large subsets
of F^n, where F is a finite field, that have small intersection with every
k-dimensional affine subspace. Interest in the explicit construction of such
sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl
(2004) who showed how such constructions over the binary field can be used to
construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that,
over large finite fields (of size polynomial in n), subspace evasive sets can
be used to obtain explicit list-decodable codes with optimal rate and constant
list-size. In this work we construct subspace evasive sets over large fields
and use them to reduce the list size of folded Reed-Solomon codes form poly(n)
to a constant.Comment: 16 page
Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra
This article concerns the computational problem of counting the lattice
points inside convex polytopes, when each point must be counted with a weight
associated to it. We describe an efficient algorithm for computing the highest
degree coefficients of the weighted Ehrhart quasi-polynomial for a rational
simple polytope in varying dimension, when the weights of the lattice points
are given by a polynomial function h. Our technique is based on a refinement of
an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a
rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the unweighted case
(i.e., h = 1). In contrast to Barvinok's method, our method is local, obtains
an approximation on the level of generating functions, handles the general
weighted case, and provides the coefficients in closed form as step polynomials
of the dilation. To demonstrate the practicality of our approach we report on
computational experiments which show even our simple implementation can compete
with state of the art software.Comment: 34 pages, 2 figure
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