The block-ZXZ synthesis of an arbitrary quantum circuit

TL;DR

This paper introduces a novel quantum circuit synthesis method based on a matrix decomposition theorem, enabling the construction of circuits for any unitary matrix using controlled single-qubit gates, generalizing classical logic synthesis.

Contribution

It extends a recent theorem to synthesize arbitrary quantum circuits with controlled gates, unifying quantum and classical logic circuit synthesis approaches.

Findings

The method applies to any $2^w imes 2^w$ unitary matrix.

Synthesized circuits use controlled 1-qubit gates like NEGATOR and PHASOR.

Reduces to classical controlled NOT gate synthesis for permutation matrices.

Abstract

Given an arbitrary $2^w \times 2^w$ unitary matrix $U$, a powerful matrix decomposition can be applied, leading to four different syntheses of a $w$-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by F\"uhr and Rzeszotnik, generalizing the scaling of single-bit unitary gates ($w=1$) to gates with arbitrary value of~$w$. The synthesized circuit consists of controlled 1-qubit gates, such as NEGATOR gates and PHASOR gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled NOT gates, in the case that $U$ is a permutation matrix.