On Grover’s search algorithm from a quantum information geometry viewpoint

Alsing

2020

Phys. Rev. E

Self Cite

We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su (2; C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external timedependent magnetic field that specifies the su (2; C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states, and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus, we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.

Section: Introductionmentioning

confidence: 99%

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions

Alsing

2020

Phys. Rev. E

Self Cite

“…The N-dimensional probability distribution vectorp (p 0 (θ ) , p 1 (θ ) ,..., p N 1 (θ )) with p k (θ ) defined in (14) can be regarded as a path characterizing Grover's algorithm on a suitable probability space. In what follows, we show that such a path is indeed a geodesic path for which the quantum Fisher information action functional achieves an extremal value.…”

Section: Grover's Search Algorithmmentioning

confidence: 99%

“…Our analysis explicitly recognizes that the Fubini-Study metric is a quantum version of the Fisher metric [11]. Substituting (18) into (17) and using (13) together with the normalization condition on the probabilities in (14), after some straightforward algebra it turns out that the infinitesimal Wigner-Yanase line element reads,…”

Section: Grover's Algorithm and Quantum Information Geometrymentioning

confidence: 99%

“…We then presented an information geometric characterization of Grover's quantum search algorithm as a geodesic in the parameter space characterizing the pure state model, 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. For a very recent and more detailed analysis concerning the quantum information geometric characterization of Grover's algorithm, we refer to [14]. As pointed out in [1], quantum information geometry is only in his infancy and much more research awaits to be performed.…”

Section: Final Remarksmentioning

confidence: 99%

See 1 more Smart Citation

An information geometric viewpoint of algorithms in quantum computing

AIP Conference Proceedings

Mancini

2012

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

“…For recent discussions on the transition from the digital to analog quantum computational setting for Grover's algorithm, we refer to Ref. [4][5][6]. Ideally, one seeks to achieve unit success probability (that is, unit fidelity) in the shortest possible time in a quantum search problem.…”

Section: Introductionmentioning

confidence: 99%

Transition probabilities in generalized quantum search Hamiltonian evolutions

Gassner

Int. J. Geom. Methods Mod. Phys.

Capozziello

2019

Self Cite

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.