Phase context decomposition of diagonal unitaries for higher-dimensional systems

TL;DR

This paper extends an efficient diagonal operator decomposition method to qudit systems, optimizing circuit complexity by using phase-context awareness and signed base-$d$ representations, significantly improving scalability for higher-dimensional quantum systems.

Contribution

It generalizes Welch et al.'s decomposition method to qudits, optimizing multi-controlled INC-gate circuits with phase-context awareness and improved complexity bounds.

Findings

Circuit complexity scales as O(d n^2 k) for qudits.

Decomposition method is optimized for phases with k distinct values.

Compared to qubit case, the method offers better scalability for higher dimensions.

Abstract

We generalize an efficient decomposition method for diagonal operators by Welch et al. to qudit systems. The phase-context aware method focusses on cascaded entanglers whose decomposition into multi-controlled INC-gates can be optimized by the choice of a proper signed base-$d$ representation for the natural numbers. While the gate count of the best known decomposition method for general diagonal operators on qubit systems scales with $\mathcal{O}(2^n)$, the circuits synthesized by the Welch algorithm for diagonal operators with $k$ distinct phases are upper-bounded by $\mathcal{O}(n^2k)$, which is generalized to $\mathcal{O}(dn^2k)$ for the qudit case in this paper.