Ternary and Binary Representation of Coordinate and Momentum in Quantum Mechanics

TL;DR

This paper explores representing quantum coordinates and momenta using binary and ternary series expansions, enabling discretization of continuous observables on quantum computers and addressing divergence issues.

Contribution

It introduces a novel method of expanding quantum observables in binary and ternary series, leading to automatic renormalization of divergent integrals.

Findings

Binary and ternary expansions lead to finite values for divergent series

Series coefficients are Hermitian operators

Method facilitates discretization of continuous quantum observables

Abstract

To simulate a quantum system with continuous degrees of freedom on a quantum computer based on quantum digits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to discrete observables. We consider this problem based on expanding quantum observables in series in powers of two and three analogous to the binary and ternary representations of real numbers. The coefficients of the series ("digits") are, therefore, Hermitian operators. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary and ternary expansions of quantum observables automatically leads to renormalization of some divergent integrals and series (giving them finite values).