Contributions to the Theory of Clifford-Cyclotomic Circuits

TL;DR

This paper advances the theory of Clifford-cyclotomic circuits by improving synthesis algorithms, reducing ancilla requirements for certain cases, and extending the methods to new classes of unitaries.

Contribution

It improves the synthesis algorithm for $n=2^k$ by reducing ancilla count and extends the algorithm to handle $n=3 imes 2^k$ cases.

Findings

Reduced ancilla count from $k-2$ to $k-3$ for $n=2^k$, $k extgreater=4$.

Proved minimal ancilla requirement for $k=4$.

Extended synthesis algorithm to $n=3 imes 2^k$ cases.

Abstract

Let $n$ be a positive integer divisible by 8. The Clifford-cyclotomic gate set $\mathcal{G}_n$ consists of the Clifford gates, together with a $z$-rotation of order $n$. It is easy to show that, if a circuit over $\mathcal{G}_n$ represents a unitary matrix $U$, then the entries of $U$ must lie in $\mathcal{R}_n$, the smallest subring of $\mathbb{C}$ containing $1/2$ and $\mathrm{exp}(2\pi i/n)$. The converse implication, that every unitary $U$ with entries in $\mathcal{R}_n$ can be represented by a circuit over $\mathcal{G}_n$, is harder to show, but it was recently proved to be true when $n=2^k$. In that case, $k-2$ ancillas suffice to synthesize a circuit for $U$, which is known to be minimal for $k=3$, but not for larger values of $k$. In the present paper, we make two contributions to the theory of Clifford-cyclotomic circuits. Firstly, we improve the existing synthesis algorithm by showing that, when $n=2^k$ and $k\geq 4$, only $k-3$ ancillas are needed to synthesize a circuit for $U$, which is minimal for $k=4$. Secondly, we extend the existing synthesis algorithm to the case of $n=3\cdot 2^k$ with $k\geq 3$.