269 research outputs found
The disjointness of stabilizer codes and limitations on fault-tolerant logical gates
Stabilizer codes are a simple and successful class of quantum
error-correcting codes. Yet this success comes in spite of some harsh
limitations on the ability of these codes to fault-tolerantly compute. Here we
introduce a new metric for these codes, the disjointness, which, roughly
speaking, is the number of mostly non-overlapping representatives of any given
non-trivial logical Pauli operator. We use the disjointness to prove that
transversal gates on error-detecting stabilizer codes are necessarily in a
finite level of the Clifford hierarchy. We also apply our techniques to
topological code families to find similar bounds on the level of the hierarchy
attainable by constant depth circuits, regardless of their geometric locality.
For instance, we can show that symmetric 2D surface codes cannot have non-local
constant depth circuits for non-Clifford gates.Comment: 8+3 pages, 2 figures. Comments welcom
Computing the Weight Distribution of the Binary Reed-Muller Code
We compute the weight distribution of the by combining
the approach described in D. V. Sarwate's Ph.D. thesis from 1973 with knowledge
on the affine equivalence classification of Boolean functions. To solve this
problem posed, e.g., in the MacWilliams and Sloane book [p. 447], we apply a
refined approach based on the classification of Boolean quartic forms in
variables due to Ph. Langevin and G. Leander, and recent results on the
classification of the quotient space
due to V. Gillot and Ph. Langevin
Group homomorphisms as error correcting codes
We investigate the minimum distance of the error correcting code formed by
the homomorphisms between two finite groups and . We prove some general
structural results on how the distance behaves with respect to natural group
operations, such as passing to subgroups and quotients, and taking products.
Our main result is a general formula for the distance when is solvable or
is nilpotent, in terms of the normal subgroup structure of as well as
the prime divisors of and . In particular, we show that in the above
case, the distance is independent of the subgroup structure of . We
complement this by showing that, in general, the distance depends on the
subgroup structure .Comment: 13 page
Suboptimum decoding of block codes
This paper investigates a class of decomposable codes, their distance and structural properties. it is shown that this class includes several classes of well known and efficient codes as subclasses. Several methods for constructing decomposable codes or decomposing codes are presented. A two-stage soft decision decoding scheme for decomposable codes, their translates or unions of translates is devised. This two-stage soft-decision decoding is suboptimum, and provides an excellent trade-off between the error performance and decoding complexity for codes of moderate and long block length
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