Matrix Algebra for Quantum Search Algorithm: Non Unitary Symmetries and Entanglement
Abstract
An algebraic reformulation of the quantum search algorithm associated to a kvalued oracle function, is introduced in terms of the so called oracle matrix algebra, by means of which a Bloch sphere like description of search is obtained. A parametric family of symmetric completely positive trace preserving (CPTP) maps, that formalize the presence of quantum noise but preserves the complexity of the algorithm, is determined. Dimensional reduction of representations of oracle Lie algebra is introduced in order to determine the reduced density matrix of subsets of qubits in database. The L1 vectorinduced norm of reduced density matrix is employed to define an index function for the quantum entanglement between database qubits, in the presence of non invariant noise CPTP maps. Analytic investigations provide a causal relation between entanglement and fidelity of the algorithm, which is controlled by quantum noise parameter.
 Publication:

Quantum Communication, Measurement and Computing (QCMC): The Tenth International Conference
 Pub Date:
 October 2011
 DOI:
 10.1063/1.3630150
 Bibcode:
 2011AIPC.1363...73E
 Keywords:

 algorithm theory;
 matrix algebra;
 quantum noise;
 Lie algebras;
 quantum mechanics;
 03.67.Ac;
 02.10.Yn;
 42.50.Lc;
 02.20.Sv;
 03.65.Ud;
 Quantum algorithms protocols and simulations;
 Matrix theory;
 Quantum fluctuations quantum noise and quantum jumps;
 Lie algebras of Lie groups;
 Entanglement and quantum nonlocality