Low depth measurement-based quantum computation beyond two-level systems

TL;DR

This paper explores low-depth measurement-based quantum computation with qudits, establishing a precise relationship with qudit quantum circuits and demonstrating advantages similar to qubit models, highlighting qudits' potential for efficient universal quantum computing.

Contribution

It introduces a qudit unbounded fan-out circuit model, relates measurement-based computation depth to circuit complexity, and shows a logarithmic depth separation, extending known qubit advantages to qudits.

Findings

Qudit measurement-based model has equivalent depth complexity to unbounded fan-out circuits.

A procedure reduces quantum computational depth in the measurement-based model.

Logarithmic depth separation between measurement-based and bounded-gate circuits.

Abstract

Low depth measurement-based quantum computation with qudits ($d$-level systems) is investigated and a precise relationship between this powerful model and qudit quantum circuits is derived in terms of computational depth and size complexity. To facilitate this investigation a qudit `unbounded fan-out' circuit model, in which a qudit may be quantum-copied into an arbitrary number of ancillas in a single time-step, is introduced and shown to be capable of implementing interesting $n$-qudit unitaries in constant depth. A procedure for reducing the quantum computational depth in the measurement-based model is then proposed and using this it is then shown that there is a logarithmic depth separation between the depth complexity of qudit measurement-based computation and circuits composed of gates act on a bounded number of qudits. The relationship is made precise by showing that the depth complexity of the qudit measurement-based model is exactly equivalent to that of unbounded fan-out circuits. These results illustrate that the well-known advantages inherent in qubit measurement-based quantum computation are also applicable to the higher-dimensional generalisation. As qudits are both naturally available and have been shown to provide fundamental advantages over binary logic encodings, this then suggests that the qudit measurement-based model is a particularly appealing paradigm for universal quantum computation.