1,904 research outputs found
Small polygons and toric codes
We describe two different approaches to making systematic classifications of
plane lattice polygons, and recover the toric codes they generate, over small
fields, where these match or exceed the best known minimum distance. This
includes a [36,19,12]-code over F_7 whose minimum distance 12 exceeds that of
all previously known codes.Comment: 9 pages, 4 tables, 3 figure
Lattice polytopes in coding theory
In this paper we discuss combinatorial questions about lattice polytopes
motivated by recent results on minimum distance estimation for toric codes. We
also prove a new inductive bound for the minimum distance of generalized toric
codes. As an application, we give new formulas for the minimum distance of
generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 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
Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS
We study the realization of anyon-permuting symmetries of topological phases
on the lattice using tensor networks. Working on the virtual level of a
projected entangled pair state, we find matrix product operators (MPOs) that
realize all unitary topological symmetries for the toric and color codes. These
operators act as domain walls that enact the symmetry transformation on anyons
as they cross. By considering open boundary conditions for these domain wall
MPOs, we show how to introduce symmetry twists and defect lines into the state.Comment: 11 pages, 6 figures, 2 appendices, v2 published versio
Neural Decoder for Topological Codes using Pseudo-Inverse of Parity Check Matrix
Recent developments in the field of deep learning have motivated many
researchers to apply these methods to problems in quantum information. Torlai
and Melko first proposed a decoder for surface codes based on neural networks.
Since then, many other researchers have applied neural networks to study a
variety of problems in the context of decoding. An important development in
this regard was due to Varsamopoulos et al. who proposed a two-step decoder
using neural networks. Subsequent work of Maskara et al. used the same concept
for decoding for various noise models. We propose a similar two-step neural
decoder using inverse parity-check matrix for topological color codes. We show
that it outperforms the state-of-the-art performance of non-neural decoders for
independent Pauli errors noise model on a 2D hexagonal color code. Our final
decoder is independent of the noise model and achieves a threshold of .
Our result is comparable to the recent work on neural decoder for quantum error
correction by Maskara et al.. It appears that our decoder has significant
advantages with respect to training cost and complexity of the network for
higher lengths when compared to that of Maskara et al.. Our proposed method can
also be extended to arbitrary dimension and other stabilizer codes.Comment: 12 pages, 12 figures, 2 tables, submitted to the 2019 IEEE
International Symposium on Information Theor
Generalized Toric Codes Coupled to Thermal Baths
We have studied the dynamics of a generalized toric code based on qudits at
finite temperature by finding the master equation coupling the code's degrees
of freedom to a thermal bath. As a consequence, we find that for qutrits new
types of anyons and thermal processes appear that are forbidden for qubits.
These include creation, annihilation and diffusion throughout the system code.
It is possible to solve the master equation in a short-time regime and find
expressions for the decay rates as a function of the dimension of the
qudits. Although we provide an explicit proof that the system relax to the
Gibbs state for arbitrary qudits, we also prove that above a certain crossing
temperature, qutrits initial decay rate is smaller than the original case for
qubits. Surprisingly this behavior only happens with qutrits and not with other
qudits with .Comment: Revtex4 file, color figures. New Journal of Physics' versio
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