Geometry of Quantum Computation with Qutrits

TL;DR

This paper explores the geometric structure of quantum circuits with qutrits, revealing that optimal circuits correspond to shortest paths in a curved geometry of the special unitary group, with detailed analysis for three-qutrit systems.

Contribution

It introduces a geometric framework for understanding quantum circuit complexity with qutrits, extending previous qubit-based approaches to higher-dimensional systems.

Findings

Optimal quantum circuits are shortest paths in a curved geometry of SU(3^n)

Detailed analysis of three-qutrit systems

Provides a geometric perspective on quantum circuit complexity

Abstract

Determining the quantum circuit complexity of a unitary operation is an important problem in quantum computation. By using the mathematical techniques of Riemannian geometry, we investigate the efficient quantum circuits in quantum computation with $n$ qutrits. We show that the optimal quantum circuits are essentially equivalent to the shortest path between two points in a certain curved geometry of $SU(3^n)$. As an example, three-qutrit systems are investigated in detail.