Quantum search with modular variables

TL;DR

This paper extends Grover's quantum search algorithm to systems of any dimension using modular variables, demonstrating that the quadratic speedup is preserved beyond qubits, including in continuous variable systems.

Contribution

It provides a dimension-independent formulation of quantum search, applicable to continuous variables via modular variables, maintaining optimal quadratic speedup.

Findings

Quadratic speedup preserved across all system dimensions

Adaptation of the protocol to continuous variable systems

Detailed illustration using modular variable formalism

Abstract

We give a dimension independent formulation of the quantum search algorithm introduced in [L. K. Grover, Phys. Rev. Lett. {\bf 79}, 325 (1997)]. This algorithm provides a quadratic gain when compared to its classical counterpart by manipulating quantum two--level systems, qubits. We show that this gain, already known to be optimal, is preserved, irrespectively of the dimension of the system used to encode quantum information. This is shown by adapting the protocol to Hilbert spaces of any dimension using the same sequence of operations/logical gates as its original qubit formulation. Our results are detailed and illustrated for a system described by continuous variables, where qubits can be encoded in infinitely many distinct states using the modular variable formalism.