Transition probabilities in generalized quantum search Hamiltonian evolutions

Abstract

A relevant problem in quantum computing concerns how fast a source state can be driven into a target state according to Schrödinger's quantum mechanical evolution specified by a suitable driving Hamiltonian. In this paper, we study in detail the computational aspects necessary to calculate the transition probability from a source state to a target state in a continuous time quantum search problem defined by a multi-parameter generalized time-independent Hamiltonian. In particular, quantifying the performance of a quantum search in terms of speed (minimum search time) and fidelity (maximum success probability), we consider a variety of special cases that emerge from the generalized Hamiltonian. In the context of optimal quantum search, we find it is possible to outperform, in terms of minimum search time, the well-known Farhi-Gutmann analog quantum search algorithm. In the context of nearly optimal quantum search, instead, we show it is possible to identify sub-optimal search algorithms capable of outperforming optimal search algorithms if only a sufficiently high success probability is sought. Finally, we briefly discuss the relevance of a tradeoff between speed and fidelity with emphasis on issues of both theoretical and practical importance to quantum information processing.

Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

“…Finally, building upon our works presented in Refs. [7,17,18,20], we presented in Ref. [21] an information geometric analysis of geodesic speeds and entropy production rates in geodesic motion on manifolds of parametrized quantum states.…”

“…In Refs. [17,18], instead, we analyzed the possibility of modifying the original Farhi-Gutmann Hamiltonian quantum search algorithm [19] in order to speed up the procedure for producing a suitably distributed unknown normalized quantum mechanical state provided only a nearly optimal fidelity is sought. In Ref.…”

Quantum Groverian Geodesic Paths with Gravitational and Thermal Analogies

“…Fortunately, the comprehension of the connection between the Fisher information and the schedule of a quantum algorithm has been highly enhanced in Refs. [9][10][11]. More specifically, a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems was presented in Ref.…”

“…After a serious reconsideration of the underlying geometrical structure of our previously mentioned works in Refs. [4,[8][9][10][11][12][13], we have arrived at the conclusion that it is the effective and serious exploitation of the link between quantum search algorithms and geodesics in the complex projective Hilbert space that serves as the essential ingredient giving rise to an increasing number of intriguing interdisciplinary investigations in the literature connecting concepts from information geometry, quantum computing, and thermodynamics. For that reason, we propose to reconsider in this article the mathematical derivation of such an important mathematical link that provides an ever increasing number of penetrating physical insights.…”

Information Geometric Aspects of Probability Paths with Minimum Entropy Production for Quantum State Evolution

“…In this paper, driven by the absence of any thermodynamical analysis of quantum searching protocols and building upon our previous research presented in Refs. [42,[44][45][46][47][48], we incorporate the Riemannian geometric concepts of speed and efficiency within both thermodynamical and quantum mechanical settings, in an effort to provide some theoretical perspective on the compromise between efficiency and speed in terms of minimal entropy production channels arising from quantum evolutions. In particular, we present an information geometric characterization of entropy production rates and entropic speeds in geodesic evolution occurring on statistical manifolds of parametrized quantum states, which emerge as output states of su(2; C) Hamiltonian models that approximate various types of continuous-time quantum search protocols.…”