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.
Quantum computation and quantum information are known well to be very hot interdisciplinary fields [1], where a number of approaches have been taken. In a series of papers [2, 3, 4, 5], the author takes geometric and dynamical-system approaches to differential equations on the quantum information space (QIS) raised from algorithms in engineering. One of the keys to the papers is the fact that the SLD-Fisher metric of the QIS is obtained from the natural Riemannian metric of the space of ordered tuples of multiqubits through a geometric reduction [2]. On the QIS, three types of geodesics are of much interest; the m-geodesics, the e-geodesics and the Levi-Civita geodesics. As a continuation of the paper [5] showing that the Grover search geodesic for a given ordered tuple of multiqubits is reduced to an m-geodesic on the QIS, this talk presents presents the classes curves which are reduced to the m-, the e-, and the Levi-Civita geodesics on the QIS. In particular, the curves related with the m-geodesics are discussed in detail. Since the QIS consists of positive-(semi)definite matrices with unit trace, computer algebra would be worth applying to matrix computations on the QIS. Applicability of computer algebra will be discussed also in this talk.
geodesic in (2 n+1 -1)-dimensional sphere.Further, the reduced search sequence is given rise to the complex projective space CP 2 n -1 through a geometric reduction, which is also shown to be on a geodesic in CP 2 n -1 .Roughly speaking, the reduction in [15] is made through the phase-factor elimination from quantum states, so that CP 2 n -1 is thought of as the space of rays.Note that the geodesics above are associated with the standard metric on (2 n+1 -1)-dimensional sphere and with the Fubini-Study metric on CP 2 n -1 , respectively.The Fubini-Study metric on CP 2 n -1 is utilized also in [15] to measure the minimum distance from each state involved in the search sequence to the submanifold consisting of non-entangled states, which characterizes the entanglement of the states along the search.As expected benefits of geometric and dynamical views on quantum algorithmic studies like [15], the following would be worth listed;1 By revealing underlying geometry of quantum algorithms (not necessarily universal), numbers of results in geometry are expected to be applied to make advances in quantum computation and information.2 On looked upon the iterations made in algorithms as (discrete) time-evolutions of states, numbers of results in dynamical systems are expected to be applied to make advances in quantum computation and information.3 In view of a close connection between geometry and dynamical systems, geometric and dynamical-systems studies on quantum algorithms may provide interesting examples of dynamical systems.It would be worth noting here that there exists another approach to quantum searches using adiabatic evolution [16][17][18].That approach, however, is outside the scope of this chapter since the search dealt with in this chapter is organized on the so-called amplitude magnification technique [8] which differs from the adiabatic evolution.
We show that quantum information geometry can be used to characterize Grover's searching algorithm. Specifically, quantifying the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric, we uncover that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuos approximation of the parametric quantum out-put state in Grover's algorithm.
The theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory: CPTP, PPT and Schwarz maps. First, we develop the general structure that equivariant maps
Φ:A→B between
C∗-algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality
Φ(X∗X)≥Φ(X)∗Φ(X) and form an intermediate class between positive and completely positive maps. We completely classify
U(n)-equivariant on
Mn(C) and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker
DU(n)-equivariance (diagonal unitary symmetry) and for tensor-product symmetries
U(n1)⊗U(n2). In each case, the parameter regions where
Φ is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the
U(n)-equivariant family satisfies
PPT⟺EB, while the
DU(2), symmetric
DU(3),
U(2)⊗U(2) and
U(2)⊗U(3), families obey the
PPT2 conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the
PPT2 property holds.