Optical Implementation of 2 × 2 Universal Unitary Matrix Transformations

Abstract

Unitary operations are a specific class of linear transformations that have become an essential ingredient for the realization of classical and quantum information processing. The ability of implementing any n-dimensional unitary signal transformation by using a reconfigurable optical hardware has recently led to the pioneering concept of programmable linear optical processor, whose basic building block (BB) must be correctly designed to guarantee that the whole system is able to perform n × n universal (i.e., arbitrary) unitary matrix transformations. Here, it is demonstrated that the present architectures of the BB do not fulfil the universal unitary property (at least) in 2 × 2 optical processors, limiting the number of unitary matrix transformations that may be generated. Aiming to solve this fundamental constraint, the theoretical tools required to analyze and design 2 × 2 universal unitary optical circuits and their corresponding BBs are presented. The consequences of this mathematical framework are explored, obtaining a simple route to implement different BB architectures, all of them guaranteeing a true universal unitary functionality in the resulting 2 × 2 optical processors. These findings may pave the way to revisit the design of high-dimensional unitary optical processors, unleashing the potential of programmable integrated photonics technology.

Analog Programmable‐Photonic Computation

“…APC revolves around the idea of performing operations on a new unit of information, the anbit , which must be easily implementable using PIP technology. Since the building block of PIP is usually an optical circuit carrying out 2×2 matrix transformations, [ 12–14,31 ] the input and output signals of this system are 2D vectors. Thus, it seems reasonable to define an anbit as a 2D vector function $${{\bm \psi }}( t ){\mathrm{\coloneq }}{\psi }_0( t ){{{\bf \hat{e}}}}_0 + {\psi }_1( t ){{{\bf \hat{e}}}}_1$$, where ψ 0, 1 are scalar complex functions referred to as the anbit amplitudes and $${{{\bf \hat{e}}}}_{0,1}$$ are constant orthonormal vectors.…”

“…b) Minimal circuit architecture (MCA) of a U‐gate, implementing the universal unitary matrix of Equation () via the Euler factorization $$U\ = {e}^{i\delta }\ {R}_{{{\bf \hat{n}}}}( \alpha ) \equiv {e}^{i\delta }{R}_{{{\bf \hat{z}}}}( {{\alpha }_3} ){R}_{{{\bf \hat{x}}}}( {{\alpha }_2} ){R}_{{{\bf \hat{z}}}}( {{\alpha }_1} )$$. [ 31 ] The U‐gate generates a rotation around an arbitrary unit vector $${{\bf \hat{n}}}$$ of the GBS, preserving the norm of the input anbit. c) MCA of a G‐ and M‐gate, based on the singular value decomposition.…”

“…The U‐ and G‐gates belong to the U(2) and $${\mathrm{GL}}( {2,\mathbb{C}} )$$ Lie groups, respectively, while an M‐gate belongs to the $$\mathfrak{g}\mathfrak{l}( {2,\mathbb{C}} )$$ Lie algebra. [ 57 ] Using the fundamentals of these algebraic structures, [ 10,31,57,58 ] it is straightforward to find a universal (or arbitrary) matrix in each class of gate, which must be able to describe all the possible 2×2 matrix transformations associated to the class when varying the value of its entries, encoded by parameters. In particular, a universal matrix of a U‐gate reads as follows: [ 10,31 ] $$$$\begin{equation}U\ = {e}^{i\delta }\ \!\!$$…”

Opportunities and Challenges for Large-Scale Phase-Change Material Integrated Electro-Photonics

“…Any arbitrary unitary transformation can be implemented optically using an array of programmable units consisting of a tunable beam splitter and a phase shifter . Using these optical unitary transformations, we can implement matrix-vector multiplication at the heart of optical signal processing, ,, neural networks, and even quantum simulation . Although these systems have been demonstrated using the thermo-optic effect in the past, ,,, the number of programmable units remains limited due to thermal crosstalk, massive energy consumption, and complex control circuit.…”

Programmable wave-based analog computing machine: a metastructure that designs metastructures

“…In terms of their mathematical abilities, the above examples demonstrate wave-based analog computing with functionalities such as integration/differentiation in space (19)(20)(21)(22) and time (23), matrix-vector multiplication (24), emulating equations through physical phenomena (25,26), or acting as platforms for neural network functionalities (3,6,27,28). The intersection with the metamaterial paradigm delivered a series of remarkable analog computing devices with matrix multiplication (4,19,29) and ultimately equation solving (matrix inversion) capabilities (30).…”