Photonic interlacing architectures for learning discrete linear unitaries

Recent investigations suggest that the discrete linear unitary group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>U</a:mi><a:mo>(</a:mo><a:mi>N</a:mi><a:mo>)</a:mo></a:mrow></a:math> 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>U</b:mi><b:mo>(</b:mo><b:mi>N</b:mi><b:mo>)</b:mo></b:mrow></b:math> can be decomposed into a sequence of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>N</c:mi></c:math>-parameter phases alternating with one-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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>U</d:mi><d:mo>(</d:mo><d:mi>N</d:mi><d:mo>)</d:mo></d:mrow></d:math> for a finite number of layers. This is numerically verified by using Haar random matrices as targets, showing a convergence for exactly <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>N</e:mi></e:math> layers. As a specific application, we propose an integrated all-optical logic gate device that performs OR, NAND, XOR, and XAND tasks within a lossless and passive optical circuit design.