Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes

TL;DR

This paper explores how Fisher information decreases in physical systems and uses information geometry to analyze quantum evolution equations, providing insights into the speed and thermodynamic efficiency of quantum algorithms.

Contribution

It introduces an information geometric framework to characterize the behavior of Fisher information in quantum and statistical systems, linking it to computational speed and thermodynamic efficiency.

Findings

Fisher information decreases along solutions of the Boltzmann equation.

Quantum search algorithms follow geodesic paths with constant Fisher information.

The framework relates Fisher information decay to speed and thermodynamic trade-offs.

Abstract

The relevance of the concept of Fisher information is increasing in both statistical physics and quantum computing. From a statistical mechanical standpoint, the application of Fisher information in the kinetic theory of gases is characterized by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case. From a quantum mechanical standpoint, the output state in Grover's quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. Additionally, Grover's algorithm is specified by constant Fisher information. In this paper, we present an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and power-law decay. Furthermore, for each case, we compute both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts. Finally, we briefly comment on the possibility of using the proposed methods of information geometry to help identify a suitable trade-off between speed and thermodynamic efficiency in quantum search algorithms.