88 research outputs found
A note on the minimum distance of quantum LDPC codes
We provide a new lower bound on the minimum distance of a family of quantum
LDPC codes based on Cayley graphs proposed by MacKay, Mitchison and
Shokrollahi. Our bound is exponential, improving on the quadratic bound of
Couvreur, Delfosse and Z\'emor. This result is obtained by examining a family
of subsets of the hypercube which locally satisfy some parity conditions
Tradeoffs for reliable quantum information storage in surface codes and color codes
The family of hyperbolic surface codes is one of the rare families of quantum
LDPC codes with non-zero rate and unbounded minimum distance. First, we
introduce a family of hyperbolic color codes. This produces a new family of
quantum LDPC codes with non-zero rate and with minimum distance logarithmic in
the blocklength. Second, we study the tradeoff between the length n, the number
of encoded qubits k and the distance d of surface codes and color codes. We
prove that kd^2 is upper bounded by C(log k)^2n, where C is a constant that
depends only on the row weight of the parity-check matrix. Our results prove
that the best asymptotic minimum distance of LDPC surface codes and color codes
with non-zero rate is logarithmic in the length.Comment: 10 page
A Construction of Quantum LDPC Codes from Cayley Graphs
We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison
and Shokrollahi. It is based on the Cayley graph of Fn together with a set of
generators regarded as the columns of the parity-check matrix of a classical
code. We give a general lower bound on the minimum distance of the Quantum code
in where d is the minimum distance of the classical code.
When the classical code is the repetition code, we are able to
compute the exact parameters of the associated Quantum code which are .Comment: The material in this paper was presented in part at ISIT 2011. This
article is published in IEEE Transactions on Information Theory. We point out
that the second step of the proof of Proposition VI.2 in the published
version (Proposition 25 in the present version and Proposition 18 in the ISIT
extended abstract) is not strictly correct. This issue is addressed in the
present versio
Constructions and Noise Threshold of Hyperbolic Surface Codes
We show how to obtain concrete constructions of homological quantum codes
based on tilings of 2D surfaces with constant negative curvature (hyperbolic
surfaces). This construction results in two-dimensional quantum codes whose
tradeoff of encoding rate versus protection is more favorable than for the
surface code. These surface codes would require variable length connections
between qubits, as determined by the hyperbolic geometry. We provide numerical
estimates of the value of the noise threshold and logical error probability of
these codes against independent X or Z noise, assuming noise-free error
correction
Ramanujan Complexes and bounded degree topological expanders
Expander graphs have been a focus of attention in computer science in the
last four decades. In recent years a high dimensional theory of expanders is
emerging. There are several possible generalizations of the theory of expansion
to simplicial complexes, among them stand out coboundary expansion and
topological expanders. It is known that for every d there are unbounded degree
simplicial complexes of dimension d with these properties. However, a major
open problem, formulated by Gromov, is whether bounded degree high dimensional
expanders, according to these definitions, exist for d >= 2. We present an
explicit construction of bounded degree complexes of dimension d = 2 which are
high dimensional expanders. More precisely, our main result says that the
2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders.
Assuming a conjecture of Serre on the congruence subgroup property, infinitely
many of them are also coboundary expanders.Comment: To appear in FOCS 201
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