21 research outputs found
On Self-Dual Quantum Codes, Graphs, and Boolean Functions
A short introduction to quantum error correction is given, and it is shown
that zero-dimensional quantum codes can be represented as self-dual additive
codes over GF(4) and also as graphs. We show that graphs representing several
such codes with high minimum distance can be described as nested regular graphs
having minimum regular vertex degree and containing long cycles. Two graphs
correspond to equivalent quantum codes if they are related by a sequence of
local complementations. We use this operation to generate orbits of graphs, and
thus classify all inequivalent self-dual additive codes over GF(4) of length up
to 12, where previously only all codes of length up to 9 were known. We show
that these codes can be interpreted as quadratic Boolean functions, and we
define non-quadratic quantum codes, corresponding to Boolean functions of
higher degree. We look at various cryptographic properties of Boolean
functions, in particular the propagation criteria. The new aperiodic
propagation criterion (APC) and the APC distance are then defined. We show that
the distance of a zero-dimensional quantum code is equal to the APC distance of
the corresponding Boolean function. Orbits of Boolean functions with respect to
the {I,H,N}^n transform set are generated. We also study the peak-to-average
power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove
that PAR_IHN of a quadratic Boolean function is related to the size of the
maximum independent set over the corresponding orbit of graphs. A construction
technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It
is finally shown that both PAR_IHN and APC distance can be interpreted as
partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure
The LU-LC conjecture is false
The LU-LC conjecture is an important open problem concerning the structure of
entanglement of states described in the stabilizer formalism. It states that
two local unitary equivalent stabilizer states are also local Clifford
equivalent. If this conjecture were true, the local equivalence of stabilizer
states would be extremely easy to characterize. Unfortunately, however, based
on the recent progress made by Gross and Van den Nest, we find that the
conjecture is false.Comment: Added a new part explaining how the counterexamples are foun
Boolean Functions, Projection Operators and Quantum Error Correcting Codes
This paper describes a fundamental correspondence between Boolean functions
and projection operators in Hilbert space. The correspondence is widely
applicable, and it is used in this paper to provide a common mathematical
framework for the design of both additive and non-additive quantum error
correcting codes. The new framework leads to the construction of a variety of
codes including an infinite class of codes that extend the original ((5,6,2))
code found by Rains [21]. It also extends to operator quantum error correcting
codes.Comment: Submitted to IEEE Transactions on Information Theory, October 2006,
to appear in IEEE Transactions on Information Theory, 200
Codeword stabilized quantum codes: algorithm and structure
The codeword stabilized ("CWS") quantum codes formalism presents a unifying
approach to both additive and nonadditive quantum error-correcting codes
(arXiv:0708.1021). This formalism reduces the problem of constructing such
quantum codes to finding a binary classical code correcting an error pattern
induced by a graph state. Finding such a classical code can be very difficult.
Here, we consider an algorithm which maps the search for CWS codes to a problem
of identifying maximum cliques in a graph. While solving this problem is in
general very hard, we prove three structure theorems which reduce the search
space, specifying certain admissible and optimal ((n,K,d)) additive codes. In
particular, we find there does not exist any ((7,3,3)) CWS code though the
linear programming bound does not rule it out. The complexity of the CWS search
algorithm is compared with the contrasting method introduced by Aggarwal and
Calderbank (arXiv:cs/0610159).Comment: 11 pages, 1 figur
Graph-Based Classification of Self-Dual Additive Codes over Finite Fields
Quantum stabilizer states over GF(m) can be represented as self-dual additive
codes over GF(m^2). These codes can be represented as weighted graphs, and
orbits of graphs under the generalized local complementation operation
correspond to equivalence classes of codes. We have previously used this fact
to classify self-dual additive codes over GF(4). In this paper we classify
self-dual additive codes over GF(9), GF(16), and GF(25). Assuming that the
classical MDS conjecture holds, we are able to classify all self-dual additive
MDS codes over GF(9) by using an extension technique. We prove that the minimum
distance of a self-dual additive code is related to the minimum vertex degree
in the associated graph orbit. Circulant graph codes are introduced, and a
computer search reveals that this set contains many strong codes. We show that
some of these codes have highly regular graph representations.Comment: 20 pages, 13 figure