Quantum simulation of the Riemann-Hurwitz zeta function

TL;DR

This paper introduces a quantum simulation method for the Riemann-Hurwitz zeta function using a tailored Hamiltonian, enabling scalable and accurate emulation of the function's behavior along specific complex plane lines.

Contribution

The authors present a novel quantum simulation approach for the RH zeta function based on a truncated Dirichlet series and engineered Hamiltonians suitable for various physical platforms.

Findings

Hamiltonian reproduces RH function along  line in complex plane

Simulation can be scaled for increased accuracy

Practical limitations include finite lattice sites and decoherence

Abstract

We propose a simple realization of a quantum simulator of the Riemann-Hurwitz (RH) \zeta\ function based on a truncation of its Dirichlet representation. We synthesize a nearest-neighbour-interaction Hamiltonian, satisfying the property that the temporal evolution of the autocorrelation function of an initial bare state of the Hamiltonian reproduces the RH function along the line \sigma+i \omega t of the complex plane, with \sigma>1. The tight-binding Hamiltonian with engineered hopping rates and site energies can be implemented in a variety of physical systems, including trapped ion systems and optical waveguide arrays. The proposed method is scalable, which means that the simulation can be in principle arbitrarily accurate. Practical limitations of the suggested scheme, arising from a finite number of lattice sites N and from decoherence, are briefly discussed.