Quantum Groverian geodesic paths with gravitational and thermal analogies
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Quantum Groverian geodesic paths with gravitational and thermal analogies Carlo Cafaro1,a , Domenico Felice1,2 , Paul M. Alsing3 1 SUNY Polytechnic Institute, Albany, New York 12203, USA 2 I. S. I. S. “ A. Volta”, 81031 Aversa, Italy 3 Air Force Research Laboratory, Information Directorate, Rome, NY 13441, USA
Received: 25 September 2020 / Accepted: 4 November 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We present a unifying variational calculus derivation of Groverian geodesics for both quantum state vectors and quantum probability amplitudes. In the first case, we show that horizontal affinely parametrized geodesic paths on the Hilbert space of normalized vectors emerge from the minimization of the length specified by the Fubini–Study metric on the manifold of Hilbert space rays. In the second case, we demonstrate that geodesic paths for probability amplitudes arise by minimizing the length expressed in terms of the Fisher information. In both derivations, we find that geodesic equations are described by simple harmonic oscillators (SHOs). However, while in the first derivation the frequency of oscillations is proportional to the (constant) energy dispersion E of the Hamiltonian system; in the second √ derivation the frequency of oscillations is proportional to the square root F of the (constant) Fisher information. Interestingly, by setting these two frequencies equal to each other, we recover the well-known Anandan–Aharonov relation linking the squared speed of evolution of an Hamiltonian system with its energy dispersion. Finally, upon transitioning away from the quantum setting, we discuss the universality of the emergence of geodesic motion of SHO type in the presence of conserved quantities by analyzing two specific phenomena of gravitational and thermodynamical origin, respectively.
1 Introduction The concept of Fisher information plays a key role in both physics [1] and information theory [2]. Methods of Fisher information have been widely employed for both classical and quantum physical systems [3]. The increasing importance of the concept of Fisher information in both statistical physics and quantum computing was recently pointed out in [4]. In statistical physics for instance, the application of Fisher information in the kinetic theory of gases is specified by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case [5]. In quantum physics, for example, the output state in Grover’s quantum search algorithm follows a geodesic path emerging from the Fubini– Study metric on the manifold of Hilbert-space rays [6–8]. In Ref. [4], the authors presented an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes
a e-mail: [email protected] (corresponding author)
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originating from special functional forms of the Fisher
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