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

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 on ternary quantum systems: (i) a model based on magic state distillation that assumes the availability of the ternary Clifford gates, projective measurements, classical control and (ii) a model based on a metaplectic topological quantum computer (MTQC). Arguably, 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 simply emulate the standard binary version of the algorithm by encoding each qubit in a three level system. In this paper we address this emulation approach and analyze the complexity of implementing Shor’s period finding function in both models, (i) and (ii). We compare the costs in terms of magic state counts required in each mode and find that a binary emulation implementation of Shor’s algorithm on a ternary quantum computer requires slightly smaller circuit depth than the corresponding implementation in the binary Clifford+T framework. The reason for this are simplifications for binary arithmetic that can be leveraged over ternary gate sets. We also highlight that magic state preparation on MTQC requires magic state preprocessor of asymptotically smaller size which gives the MTQC solution a significant advantage over the binary framework.