Constructing all qutrit controlled Clifford+T gates in Clifford+T

TL;DR

This paper presents a method to construct all qutrit controlled Clifford+T gates exactly and efficiently without ancillae, enabling resource-effective implementations of complex quantum operations.

Contribution

It introduces a polynomial T-count construction for multi-controlled qutrit gates using Clifford+T, surpassing qubit limitations and enabling new quantum circuit designs.

Findings

Exact, unitary construction of multi-controlled qutrit gates

Polynomial T-count scaling with number of controls

Implementation of ancilla-free multi-controlled T and Toffoli gates

Abstract

For a number of useful quantum circuits, qudit constructions have been found which reduce resource requirements compared to the best known or best possible qubit construction. However, many of the necessary qutrit gates in these constructions have never been explicitly and efficiently constructed in a fault-tolerant manner. We show how to exactly and unitarily construct any qutrit multiple-controlled Clifford+T unitary using just Clifford+T gates and without using ancillae. The T-count to do so is polynomial in the number of controls $k$, scaling as $O(k^{3.585})$. With our results we can construct ancilla-free Clifford+T implementations of multiple-controlled T gates as well as all versions of the qutrit multiple-controlled Toffoli, while the analogous results for qubits are impossible. As an application of our results, we provide a procedure to implement any ternary classical reversible function on $n$ trits as an ancilla-free qutrit unitary using $O(3^n n^{3.585})$ T gates.