Universality of Photonic Interlacing Architectures for Learning Discrete Linear Unitaries

TL;DR

This paper proves that photonic interlacing architectures can universally implement any discrete linear unitary, enabling simple optical devices for complex quantum and classical information processing.

Contribution

It provides a formal proof that a finite sequence of phase and propagator layers can decompose any $U(N)$, demonstrating the universality of photonic interlacing architectures.

Findings

The architecture can exactly reconstruct arbitrary unitary matrices.

A finite number of layers (equal to N) suffices for universality.

Proposed optical devices perform logic operations with a single-layer design.

Abstract

Recent investigations suggest that the discrete linear unitary group $U(N)$ can be represented by interlacing a finite sequence of diagonal phase operations with an intervening unitary operator. However, despite rigorous numerical justifications, no formal proof has been provided. Here, we show that elements of $U(N)$ can be decomposed into a sequence of $N$-parameter phases alternating with $1$-parameter propagators of a lattice Hamiltonian. The proof is based on building a Lie group by alternating these two operators and showing its completeness to represent $U(N)$ for a finite number of layers, which is numerically found to be exactly $N$. This architecture can be implemented using elementary optical components and can successfully reconstruct arbitrary unitary matrices. We propose example devices such as optical logic gates, which perform logic gate operations using a single-layer lossless and passive optical circuit design.