Geometric algebra and information geometry for quantum computational software

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.

“…Specifically, evaluating the entropic speed v E in Eq. (7) and the total entropy production r E in Eq. (4) along the optimum paths in Eq.…”

Section: Constant Fisher Informationmentioning

confidence: 99%

“…Finally, in Ref. [7], methods of information geometry were used to confirm the superfluity of the Walsh-Hadamard operation and, most importantly, to recover the quadratic speedup relation.…”

Section: Introductionmentioning

confidence: 99%

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

Alsing

2020

Phys. Rev. E

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“…For instance, thanks to our geodesic motion analysis together with the observed link between the information geometric complexity and the speed of convergence to the final state, our work appears to be useful for deepening our limited understanding about the existence of a tradeoff between computational speed and availability loss in an information geometric setting of quantum search algorithms with a thermodynamical flavor as presented in Refs. [35,36]. Furthermore, in view of our study of the geometrical and dynamical features that emerge from distinct metrizations of probability spaces, our comparative analysis can help investigate the unresolved problem of whether the complexity of a convex combination of two distributions is related to the complexities of the individual constituents [37].…”

Section: Final Remarksmentioning

confidence: 99%

Information geometric complexity of entropic motion on curved statistical manifolds under different metrizations of probability spaces

Gassner

Int. J. Geom. Methods Mod. Phys.

2019

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We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian geometric properties and entropic dynamical features of a Gaussian probability space where the two distinct dissimilarity measures between probability distributions are the Fisher-Rao information metric and the α-order entropy metric. In the former case, we observe an asymptotic linear temporal growth of the information geometric entropy (IGE) together with a fast convergence to the final state of the system. In the latter case, instead, we note an asymptotic logarithmic temporal growth of the IGE together with a slow convergence to the final state of the system. Finally, motivated by our findings, we provide some insights on a tradeoff between complexity and speed of convergence to the final state in our information geometric approach to problems of entropic inference.

“…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

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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.