825 research outputs found
Exotic topological order in fractal spin liquids
We present a large class of three-dimensional spin models that possess
topological order with stability against local perturbations, but are beyond
description of topological quantum field theory. Conventional topological spin
liquids, on a formal level, may be viewed as condensation of string-like
extended objects with discrete gauge symmetries, being at fixed points with
continuous scale symmetries. In contrast, ground states of fractal spin liquids
are condensation of highly-fluctuating fractal objects with certain algebraic
symmetries, corresponding to limit cycles under real-space renormalization
group transformations which naturally arise from discrete scale symmetries of
underlying fractal geometries. A particular class of three-dimensional models
proposed in this paper may potentially saturate quantum information storage
capacity for local spin systems.Comment: 18 pages, 10 figure
Universal entanglement signatures of foliated fracton phases
Fracton models exhibit a variety of exotic properties and lie beyond the
conventional framework of gapped topological order. In a previous work, we
generalized the notion of gapped phase to one of foliated fracton phase by
allowing the addition of layers of gapped two-dimensional resources in the
adiabatic evolution between gapped three-dimensional models. Moreover, we
showed that the X-cube model is a fixed point of one such phase. In this paper,
according to this definition, we look for universal properties of such phases
which remain invariant throughout the entire phase. We propose multi-partite
entanglement quantities, generalizing the proposal of topological entanglement
entropy designed for conventional topological phases. We present arguments for
the universality of these quantities and show that they attain non-zero
constant value in non-trivial foliated fracton phases.Comment: 17 pages, 7 figure
Secret Sharing Schemes with a large number of players from Toric Varieties
A general theory for constructing linear secret sharing schemes over a finite
field \Fq from toric varieties is introduced. The number of players can be as
large as for . We present general methods for obtaining
the reconstruction and privacy thresholds as well as conditions for
multiplication on the associated secret sharing schemes.
In particular we apply the method on certain toric surfaces. The main results
are ideal linear secret sharing schemes where the number of players can be as
large as . We determine bounds for the reconstruction and privacy
thresholds and conditions for strong multiplication using the cohomology and
the intersection theory on toric surfaces.Comment: 15 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1203.454
Toric Geometry and String Theory Descriptions of Qudit Systems
In this paper, we propose a new way to approach qudit systems using toric
geometry and related topics including the local mirror symmetry used in the
string theory compactification. We refer to such systems as (n,d) quantum
systems where and denote the number of the qudits and the basis states
respectively. Concretely, we first relate the (n,d) quantum systems to the
holomorphic sections of line bundles on n dimensional projective spaces CP^{n}
with degree n(d-1). These sections are in one-to-one correspondence with d^n
integral points on a n-dimensional simplex. Then, we explore the local mirror
map in the toric geometry language to establish a linkage between the (n,d)
quantum systems and type II D-branes placed at singularities of local
Calabi-Yau manifolds. (1,d) and (2,d) are analyzed in some details and are
found to be related to the mirror of the ALE space with the A_{d-1} singularity
and a generalized conifold respectively.Comment: 12 pages,latex, 2 figures. Accepted for publication in Journal of
Geometry and Physics, JPS(2015
Topological phases with generalized global symmetries
We present simple lattice realizations of symmetry-protected topological
(SPT) phases with -form global symmetries where charged excitations have
spatial dimensions. Specifically, we construct space-dimensional models
supported on a -colorable graph by using a family of unitary phase
gates, known as multi-qubit control- gates in quantum information community.
In our construction, charged excitations of different dimensionality may
coexist and form a short-range entangled state which is protected by symmetry
operators of different dimensionality. Non-triviality of proposed models, in a
sense of quantum circuit complexity, is confirmed by studying protected
boundary modes, gauged models and corresponding gapped domain walls. We also
comment on applications of our construction to quantum error-correcting codes,
and discuss corresponding fault-tolerant logical gates.Comment: 32 pages, 17 figures, single column (v2, corrected minor mistakes and
typos, to appear in PRB
Fast Decoders for Topological Quantum Codes
We present a family of algorithms, combining real-space renormalization
methods and belief propagation, to estimate the free energy of a topologically
ordered system in the presence of defects. Such an algorithm is needed to
preserve the quantum information stored in the ground space of a topologically
ordered system and to decode topological error-correcting codes. For a system
of linear size L, our algorithm runs in time log L compared to L^6 needed for
the minimum-weight perfect matching algorithm previously used in this context
and achieves a higher depolarizing error threshold.Comment: 4 pages, 4 figure
Topological Order and Memory Time in Marginally Self-Correcting Quantum Memory
We examine two proposals for marginally self-correcting quantum memory, the
cubic code by Haah and the welded code by Michnicki. In particular, we prove
explicitly that they are absent of topological order above zero temperature, as
their Gibbs ensembles can be prepared via a short-depth quantum circuit from
classical ensembles. Our proof technique naturally gives rise to the notion of
free energy associated with excitations. Further, we develop a framework for an
ergodic decomposition of Davies generators in CSS codes which enables formal
reduction to simpler classical memory problems. We then show that memory time
in the welded code is doubly exponential in inverse temperature via the Peierls
argument. These results introduce further connections between thermal
topological order and self-correction from the viewpoint of free energy and
quantum circuit depth.Comment: 19 pages, 18 figure
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