Programmable photonic unitary circuits for light computing

Unitarity serves as a fundamental concept for characterizing linear and conservative wave phenomena in both classical and quantum systems. Developing platforms that perform unitary operations on light waves in a universal and programmable manner enables the emulation of complex light-matter interactions and the execution of general-purpose functionalities for wave manipulations, photonic computing, and quantum circuits. Recently, numerous approaches to implementing programmable photonic unitary circuits have been proposed and demonstrated, each employing different design strategies that distinctly impact overall device performance. Here, we review foundational design principles and recent achievements in the implementation of programmable photonic unitary circuits, with a particular focus on integrated photonic platforms. We classify the design strategies based on the dimensionality of nontrivial unit operations in their building blocks: lower-dimensional unitary units, such as SU(2) operations, and higher-dimensional ones, such as Fourier transforms. In each category, recent efforts to leverage alternative physical axes, such as the temporal and frequency domains, to address scalability challenges are also reviewed. We discuss the underlying concepts, design procedures, and trade-offs of each design strategy, especially in relation to light-based computing.