7,894 research outputs found
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
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
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