19,278 research outputs found
New cubic self-dual codes of length 54, 60 and 66
We study the construction of quasi-cyclic self-dual codes, especially of
binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell
with the algebraic approach of [9]. In particular, we improve the previous
results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50
new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more
binary cubic self-dual codes with length 54, 60 and 66.Comment: 8 page
Construction of quasi-cyclic self-dual codes
There is a one-to-one correspondence between -quasi-cyclic codes over a
finite field and linear codes over a ring . Using this correspondence, we prove that every
-quasi-cyclic self-dual code of length over a finite field
can be obtained by the {\it building-up} construction, provided
that char or , is a prime , and
is a primitive element of . We determine possible weight
enumerators of a binary -quasi-cyclic self-dual code of length
(with a prime) in terms of divisibility by . We improve the result of
[3] by constructing new binary cubic (i.e., -quasi-cyclic codes of length
) optimal self-dual codes of lengths (Type I), 54 and
66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and
60. When , we obtain a new 8-quasi-cyclic self-dual code
over and a new 6-quasi-cyclic self-dual code over
. When , we find a new 4-quasi-cyclic self-dual
code over and a new 6-quasi-cyclic self-dual code
over .Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201
New extremal singly even self-dual codes of lengths and
For lengths and , we construct extremal singly even self-dual codes
with weight enumerators for which no extremal singly even self-dual codes were
previously known to exist. We also construct new inequivalent extremal
doubly even self-dual codes with covering radius meeting the
Delsarte bound.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1706.0169
Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order
Foliated fracton order is a qualitatively new kind of phase of matter. It is
similar to topological order, but with the fundamental difference that a
layered structure, referred to as a foliation, plays an essential role and
determines the mobility restrictions of the topological excitations. In this
work, we introduce a new kind of field theory to describe these phases: a
foliated field theory. We also introduce a new lattice model and
string-membrane-net condensation picture of these phases, which is analogous to
the string-net condensation picture of topological order.Comment: 22+15 pages, 8 figures; v3 added a summary of our model near the end
of the introductio
Encoding a qubit in an oscillator
Quantum error-correcting codes are constructed that embed a
finite-dimensional code space in the infinite-dimensional Hilbert space of a
system described by continuous quantum variables. These codes exploit the
noncommutative geometry of phase space to protect against errors that shift the
values of the canonical variables q and p. In the setting of quantum optics,
fault-tolerant universal quantum computation can be executed on the protected
code subspace using linear optical operations, squeezing, homodyne detection,
and photon counting; however, nonlinear mode coupling is required for the
preparation of the encoded states. Finite-dimensional versions of these codes
can be constructed that protect encoded quantum information against shifts in
the amplitude or phase of a d-state system. Continuous-variable codes can be
invoked to establish lower bounds on the quantum capacity of Gaussian quantum
channels.Comment: 22 pages, 8 figures, REVTeX, title change (qudit -> qubit) requested
by Phys. Rev. A, minor correction
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
Symmetry-protected self-correcting quantum memories
A self-correcting quantum memory can store and protect quantum information
for a time that increases without bound with the system size and without the
need for active error correction. We demonstrate that symmetry can lead to
self-correction in 3D spin-lattice models. In particular, we investigate codes
given by 2D symmetry-enriched topological (SET) phases that appear naturally on
the boundary of 3D symmetry-protected topological (SPT) phases. We find that
while conventional on-site symmetries are not sufficient to allow for
self-correction in commuting Hamiltonian models of this form, a generalized
type of symmetry known as a 1-form symmetry is enough to guarantee
self-correction. We illustrate this fact with the 3D "cluster-state" model from
the theory of quantum computing. This model is a self-correcting memory, where
information is encoded in a 2D SET-ordered phase on the boundary that is
protected by the thermally stable SPT ordering of the bulk. We also investigate
the gauge color code in this context. Finally, noting that a 1-form symmetry is
a very strong constraint, we argue that topologically ordered systems can
possess emergent 1-form symmetries, i.e., models where the symmetry appears
naturally, without needing to be enforced externally.Comment: 39 pages, 16 figures, comments welcome; v2 includes much more
explicit detail on the main example model, including boundary conditions and
implementations of logical operators through local moves; v3 published
versio
A spectral scheme for Kohn-Sham density functional theory of clusters
Starting from the observation that one of the most successful methods for
solving the Kohn-Sham equations for periodic systems -- the plane-wave method
-- is a spectral method based on eigenfunction expansion, we formulate a
spectral method designed towards solving the Kohn-Sham equations for clusters.
This allows for efficient calculation of the electronic structure of clusters
(and molecules) with high accuracy and systematic convergence properties
without the need for any artificial periodicity. The basis functions in this
method form a complete orthonormal set and are expressible in terms of
spherical harmonics and spherical Bessel functions. Computation of the occupied
eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a
combination of preconditioned block eigensolvers and Chebyshev polynomial
filter accelerated subspace iterations. Several algorithmic and computational
aspects of the method, including computation of the electrostatics terms and
parallelization are discussed. We have implemented these methods and algorithms
into an efficient and reliable package called ClusterES (Cluster Electronic
Structure). A variety of benchmark calculations employing local and non-local
pseudopotentials are carried out using our package and the results are compared
to the literature. Convergence properties of the basis set are discussed
through numerical examples. Computations involving large systems that contain
thousands of electrons are demonstrated to highlight the efficacy of our
methodology. The use of our method to study clusters with arbitrary point group
symmetries is briefly discussed.Comment: Manuscript submitted (with revisions) to Journal of Computational
Physic
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