Factoring with Qutrits: Shor's Algorithm on Ternary and Metaplectic Quantum Architectures

TL;DR

This paper evaluates the resource costs of implementing Shor's factorization algorithm on ternary quantum computers using two models: a fault-tolerant magic state approach and a topological quantum computer, comparing their efficiencies.

Contribution

It provides a detailed complexity analysis of Shor's algorithm on ternary architectures and compares two natural models for fault-tolerant quantum computation.

Findings

Ternary implementation can be more efficient than binary encoding.

Metaplectic topological quantum computers have lower asymptotic costs for universality.

Complexity of Shor's algorithm varies significantly between models.

Abstract

We determine the cost of performing Shor's algorithm for integer factorization on a ternary quantum computer, using two natural models of universal fault-tolerant computing: (i) a model based on magic state distillation that assumes the availability of the ternary Clifford gates, projective measurements, classical control as its natural instrumentation set; (ii) a model based on a metaplectic topological quantum computer (MTQC). A natural choice to implement Shor's algorithm on a ternary quantum computer is to translate the entire arithmetic into a ternary form. However, it is also possible to emulate the standard binary version of the algorithm by encoding each qubit in a three-level system. We compare the two approaches and analyze the complexity of implementing Shor's period finding function in the two models. We also highlight the fact that the cost of achieving universality through magic states in MTQC architecture is asymptotically lower than in generic ternary case.