63 research outputs found
Efficient Decoding of Topological Color Codes
Color codes are a class of topological quantum codes with a high error
threshold and large set of transversal encoded gates, and are thus suitable for
fault tolerant quantum computation in two-dimensional architectures. Recently,
computationally efficient decoders for the color codes were proposed. We
describe an alternate efficient iterative decoder for topological color codes,
and apply it to the color code on hexagonal lattice embedded on a torus. In
numerical simulations, we find an error threshold of 7.8% for independent
dephasing and spin flip errors.Comment: 7 pages, LaTe
New Protocols and Lower Bound for Quantum Secret Sharing with Graph States
We introduce a new family of quantum secret sharing protocols with limited
quantum resources which extends the protocols proposed by Markham and Sanders
and by Broadbent, Chouha, and Tapp. Parametrized by a graph G and a subset of
its vertices A, the protocol consists in: (i) encoding the quantum secret into
the corresponding graph state by acting on the qubits in A; (ii) use a
classical encoding to ensure the existence of a threshold. These new protocols
realize ((k,n)) quantum secret sharing i.e., any set of at least k players
among n can reconstruct the quantum secret, whereas any set of less than k
players has no information about the secret. In the particular case where the
secret is encoded on all the qubits, we explore the values of k for which there
exists a graph such that the corresponding protocol realizes a ((k,n)) secret
sharing. We show that for any threshold k> n-n^{0.68} there exists a graph
allowing a ((k,n)) protocol. On the other hand, we prove that for any k<
79n/156 there is no graph G allowing a ((k,n)) protocol. As a consequence there
exists n_0 such that the protocols introduced by Markham and Sanders admit no
threshold k when the secret is encoded on all the qubits and n>n_0
Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes
We construct a new family of quantum MDS codes from classical generalized
Reed-Solomon codes and derive the necessary and sufficient condition under
which these quantum codes exist. We also give code bounds and show how to
construct them analytically. We find that existing quantum MDS codes can be
unified under these codes in the sense that when a quantum MDS code exists,
then a quantum code of this type with the same parameters also exists. Thus as
far as is known at present, they are the most important family of quantum MDS
codes.Comment: 9 pages, no figure
Quantum Stabilizer Codes Embedding Qubits Into Qudits
We study, by means of the stabilizer formalism, a quantum error correcting
code which is alternative to the standard block codes since it embeds a qubit
into a qudit. The code exploits the non-commutative geometry of discrete phase
space to protect the qubit against both amplitude and phase errors. The
performance of such code is evaluated on Weyl channels by means of the
entanglement fidelity as function of the error probability. A comparison with
standard block codes, like five and seven qubit stabilizer codes, shows its
superiority.Comment: 15 pages, 2 figures (improved version); accepted for publication in
Phys. Rev.
Magic state distillation in all prime dimensions using quantum Reed-Muller codes
We propose families of protocols for magic state distillation -- important
components of fault tolerance schemes --- for systems of odd prime dimension.
Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford
gates. We find that, in higher dimensions, small and effective codes can be
used that have no direct analogue in qubit (two-dimensional) systems. We
present several concrete protocols, including schemes for three-dimensional
(qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol
is, by many measures, the best magic state distillation scheme yet discovered.
It excels both in terms of error threshold with respect to depolarising noise
(36.3%) and the efficiency measure know as "yield", where, for a large region
of parameters, it outperforms its qubit counterpart by many orders of
magnitude.Comment: Updated from V1 to include results on the remarkable d=5 cas
Sparse Quantum Codes from Quantum Circuits
Sparse quantum codes are analogous to LDPC codes in that their check operators require examining only a constant number of qubits. In contrast to LDPC codes, good sparse quantum codes are not known, and even to encode a single qubit, the best known distance is O(√n log(n)), due to Freedman, Meyer and Luo.
We construct a new family of sparse quantum subsystem codes with minimum distance n[superscript 1 - ε] for ε = O(1/√log n). A variant of these codes exists in D spatial dimensions and has d = n[superscript 1 - ε - 1/D], nearly saturating a bound due to Bravyi and Terhal.
Our construction is based on a new general method for turning quantum circuits into sparse quantum subsystem codes. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local.National Science Foundation (U.S.) (Grant CCF-1111382)United States. Army Research Office (Contract W911NF-12-1-0486
From Skew-Cyclic Codes to Asymmetric Quantum Codes
We introduce an additive but not -linear map from
to and exhibit some of its interesting
structural properties. If is a linear -code, then is an
additive -code. If is an additive cyclic code then
is an additive quasi-cyclic code of index . Moreover, if is a module
-cyclic code, a recently introduced type of code which will be
explained below, then is equivalent to an additive cyclic code if is
odd and to an additive quasi-cyclic code of index if is even. Given any
-code , the code is self-orthogonal under the trace
Hermitian inner product. Since the mapping preserves nestedness, it can be
used as a tool in constructing additive asymmetric quantum codes.Comment: 16 pages, 3 tables, submitted to Advances in Mathematics of
Communication
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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