Transformations on quantum states form a basic building block of every quantum information system. From photonic polarization to two-level atoms, complete sets of quantum gates for a variety of qubit systems are well known. For multilevel quantum systems beyond qubits, the situation is more challenging. The orbital angular momentum modes of photons comprise one such high-dimensional system for which generation and measurement techniques are well studied. However, arbitrary transformations for such quantum states are not known. Here we experimentally demonstrate a four-dimensional generalization of the Pauli X gate and all of its integer powers on single photons carrying orbital angular momentum. Together with the well-known Z gate, this forms the first complete set of high-dimensional quantum gates implemented experimentally. The concept of the X gate is based on independent access to quantum states with different parities and can thus be generalized to other photonic degrees of freedom and potentially also to other quantum systems.
Babazadeh, A., Erhard, M., Wang, F., Malik, M., Nouroozi, R., Krenn, M., & Zeilinger, A. (2017). High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments. Physical Review Letters, 119(18), . https://doi.org/10.1103/PhysRevLett.119.180510