Matrix Algebra for Quantum Search Algorithm: Non Unitary Symmetries and Entanglement

Abstract

An algebraic reformulation of the quantum search algorithm associated to a k-valued 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 vector-induced 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.

Parametric quantum search algorithm by CP maps: algebraic, geometric and complexity aspects

“…Here U is a general U (N) unitary matrix, and The oracle algebra. Next, we reconsider the algorithm from an algebraic viewpoint, and to this end, we define the matrix oracle algebra [23]. Let the set = {1, 2, .…”

Entanglement in Quantum Search Database: Periodicity Variations and Counting

“…], N = 2 n } → {0, 1}, c.f. [14][15][16][17]. The work proceeds by treating a set S of cardinality N as a search-able database with no additional structure.…”

Faster Together: Collective Quantum Search

“…Joined quantum searches, all of which have equal Hilbert space dimension N and share the same shape complexity, are displayed as a pattern of bold typed integer partitions from 3 to 7 within the Young lattice. The pattern of equal complexities is independent from N. We define as the matrix oracle algebra A f with respect to the characteristic function f of I ⊂ ∆, the set A f = {A : A = αΣ 0 (f ) + βΣ 1 (f ) + γΣ 2 (f ) + δΣ 3 (f )} where α, β, γ, δ ∈ R are arbitrary real [9,15].…”