129 research outputs found
Local Decoders for the 2D and 4D Toric Code
We analyze the performance of decoders for the 2D and 4D toric code which are
local by construction. The 2D decoder is a cellular automaton decoder
formulated by Harrington which explicitly has a finite speed of communication
and computation. For a model of independent and errors and faulty
syndrome measurements with identical probability we report a threshold of
for this Harrington decoder. We implement a decoder for the 4D toric
code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a
method for handling faulty syndromes we estimate a threshold of for
the same noise model as in the 2D case. We compare the performance of this
decoder with a decoder based on a 4D version of Toom's cellular automaton rule
as well as the decoding method suggested by Dennis et al.
arXiv:quant-ph/0110143 .Comment: 22 pages, 21 figures; fixed typos, updated Figures 6,7,8,
Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
We consider two-dimensional lattice models that support Ising anyonic
excitations and are coupled to a thermal bath. We propose a phenomenological
model for the resulting short-time dynamics that includes pair-creation,
hopping, braiding, and fusion of anyons. By explicitly constructing topological
quantum error-correcting codes for this class of system, we use our
thermalization model to estimate the lifetime of the quantum information stored
in the encoded spaces. To decode and correct errors in these codes, we adapt
several existing topological decoders to the non-Abelian setting. We perform
large-scale numerical simulations of these two-dimensional Ising anyon systems
and find that the thresholds of these models range between 13% to 25%. To our
knowledge, these are the first numerical threshold estimates for quantum codes
without explicit additive structure.Comment: 34 pages, 9 figures; v2 matches the journal version and corrects a
misstatement about the detailed balance condition of our Metropolis
simulations. All conclusions from v1 are unaffected by this correctio
Scalable Neural Network Decoders for Higher Dimensional Quantum Codes
Machine learning has the potential to become an important tool in quantum
error correction as it allows the decoder to adapt to the error distribution of
a quantum chip. An additional motivation for using neural networks is the fact
that they can be evaluated by dedicated hardware which is very fast and
consumes little power. Machine learning has been previously applied to decode
the surface code. However, these approaches are not scalable as the training
has to be redone for every system size which becomes increasingly difficult. In
this work the existence of local decoders for higher dimensional codes leads us
to use a low-depth convolutional neural network to locally assign a likelihood
of error on each qubit. For noiseless syndrome measurements, numerical
simulations show that the decoder has a threshold of around when
applied to the 4D toric code. When the syndrome measurements are noisy, the
decoder performs better for larger code sizes when the error probability is
low. We also give theoretical and numerical analysis to show how a
convolutional neural network is different from the 1-nearest neighbor
algorithm, which is a baseline machine learning method
Fast decoders for qudit topological codes
Qudit toric codes are a natural higher-dimensional generalization of the well-
studied qubit toric code. However, standard methods for error correction of
the qubit toric code are not applicable to them. Novel decoders are needed. In
this paper we introduce two renormalization group decoders for qudit codes and
analyse their error correction thresholds and efficiency. The first decoder is
a generalization of a 'hard-decisions' decoder due to Bravyi and Haah
(arXiv:1112.3252). We modify this decoder to overcome a percolation effect
which limits its threshold performance for many-level quantum systems. The
second decoder is a generalization of a 'soft-decisions' decoder due to Poulin
and Duclos-Cianci (2010 Phys. Rev. Lett. 104 050504), with a small cell size
to optimize the efficiency of implementation in the high dimensional case. In
each case, we estimate thresholds for the uncorrelated bit-flip error model
and provide a comparative analysis of the performance of both these approaches
to error correction of qudit toric codes
Efficient color code decoders in dimensions from toric code decoders
We introduce an efficient decoder of the color code in dimensions,
the Restriction Decoder, which uses any -dimensional toric code decoder
combined with a local lifting procedure to find a recovery operation. We prove
that the Restriction Decoder successfully corrects errors in the color code if
and only if the corresponding toric code decoding succeeds. We also numerically
estimate the Restriction Decoder threshold for the color code in two and three
dimensions against the bit-filp and phase-flip noise with perfect syndrome
extraction. We report that the 2D color code threshold on the square-octagon lattice is on a par with the toric code threshold
on the square lattice.Comment: 28 pages, 8 figure
Cellular automaton decoders for topological quantum codes with noisy measurements and beyond
We propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional toric code on the rhombic dodecahedral lattice with boundaries and prove that the resulting local decoder has a non-zero error threshold. We also numerically benchmark the performance of the decoder in the setting with measurement errors using various noise models. We find that this error correction procedure is remarkably robust against measurement errors and is also essentially insensitive to the details of the lattice and noise model. Our work constitutes a step towards finding simple and high-performance decoding strategies for a wide range of quantum low-density parity-check codes
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