Generalised Quantum Gates for Qudits and their Application in Quantum Fourier Transform

TL;DR

This paper introduces a universal formulation of quantum gates for qudits of any dimension, enabling the implementation of the Quantum Fourier Transform and expanding the potential for advanced quantum algorithms and error correction.

Contribution

It presents a novel, dimension-independent framework for qudit gates, including explicit universal gate sets and their application to the Quantum Fourier Transform.

Findings

Validated the generalized gates through QFT implementation for arbitrary d

Demonstrated the correctness and utility of the approach

Expanded the design space for qudit-based quantum computing

Abstract

Quantum computing with qudits, quantum systems with $d > 2$ levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels $d$, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary $d$, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.