531 research outputs found
On group theory for quantum gates and quantum coherence
Finite group extensions offer a natural language to quantum computing. In a
nutshell, one roughly describes the action of a quantum computer as consisting
of two finite groups of gates: error gates from the general Pauli group P and
stabilizing gates within an extension group C. In this paper one explores the
nice adequacy between group theoretical concepts such as commutators, normal
subgroups, group of automorphisms, short exact sequences, wreath products...
and the coherent quantum computational primitives. The structure of the single
qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, one
discovers that M20, the smallest perfect group for which the commutator
subgroup departs from the set of commutators, underlies quantum coherence of
the two-qubit system. One recovers similar results by looking at the
automorphisms of a complete set of mutually unbiased bases.Comment: 10 pages, to appear in J Phys A: Math and Theo (Fast Track
Communication
Leveraging Automorphisms of Quantum Codes for Fault-Tolerant Quantum Computation
Fault-tolerant quantum computation is a technique that is necessary to build
a scalable quantum computer from noisy physical building blocks. Key for the
implementation of fault-tolerant computations is the ability to perform a
universal set of quantum gates that act on the code space of an underlying
quantum code. To implement such a universal gate set fault-tolerantly is an
expensive task in terms of physical operations, and any possible shortcut to
save operations is potentially beneficial and might lead to a reduction in
overhead for fault-tolerant computations. We show how the automorphism group of
a quantum code can be used to implement some operators on the encoded quantum
states in a fault-tolerant way by merely permuting the physical qubits. We
derive conditions that a code has to satisfy in order to have a large group of
operations that can be implemented transversally when combining transversal
CNOT with automorphisms. We give several examples for quantum codes with large
groups, including codes with parameters [[8,3,3]], [[15,7,3]], [[22,8,4]], and
[[31,11,5]]
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
Performance of binary block codes at low signal-to-noise ratios
The performance of general binary block codes on an unquantized additive white Gaussian noise (AWGN) channel at low signal-to-noise ratios is considered. Expressions are derived for both the block error and the bit error probabilities near the point where the bit signal-to-noise ratio is zero. These expressions depend on the global geometric structure of the code, although the minimum distance still seems to play a crucial role. Examples of codes such as orthogonal codes, biorthogonal codes, the (24,12) extended Golay code, and the (15,6) expurgated BCH code are discussed. The asymptotic coding gain at low signal-to-noise ratios is also studied
Isometry and Automorphisms of Constant Dimension Codes
We define linear and semilinear isometry for general subspace codes, used for
random network coding. Furthermore, some results on isometry classes and
automorphism groups of known constant dimension code constructions are derived
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
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