  High-dimensional quantum information processing promises capabilities beyond
the current state of the art, but addressing individual information-carrying
modes presents a significant experimental challenge. Here we demonstrate
effective high-dimensional operations in the time-frequency domain of
non-classical light. We generate heralded photons with tailored temporal-mode
structures through ultrafast pulse shaping of a parametric downconversion pump.
We then implement a quantum pulse gate, enabled by dispersion-engineered
sum-frequency generation, to project onto programmable temporal modes,
reconstructing the quantum state in seven dimensions. We also manipulate the
time-frequency structure by selectively removing temporal modes, explicitly
demonstrating the effectiveness of engineered nonlinear processes for
mode-selective manipulation of quantum states.
