Programmable nonlinear function synthesis on a photonic processor with quantum signal processing

Programmable photonic integrated circuits are emerging as increasingly large and versatile interferometric processors operating at room temperature. However, their native operations are linear, while many computational tasks require nonlinear input-output transformations that typically rely on nonlinear optical materials or resonant devices. Here, we establish a direct mapping between quantum signal processing (QSP) and the native two-mode operations of programmable interferometric photonic integrated circuits, showing that the $SU(2)$ structure required by QSP can be realized through programmable phase control and mode mixing. We experimentally demonstrate this correspondence using dual-rail single-photon encoding on a 24-mode programmable photonic integrated circuit, realizing QSP sequences up to depth $L=11$ and synthesizing STEP, ReLU and SELU functions through programmable phase control. The optical transformation remains linear, while repeated encoding of the input variable and coherent interference produce a nonlinear dependence of the output probabilities on the encoded variable. Across all accessible circuit depths, the measured responses follow the programmed QSP transformations, with hardware-induced mean squared errors between $10^{-3}$ and $10^{-2}$, all remaining below the intrinsic finite-depth approximation error. These results establish an algorithm-to-hardware mapping of QSP on programmable photonic processors. They also demonstrate an algorithmic route to programmable nonlinear functions of encoded variables on linear photonic hardware.