# High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments

## TL;DR

This paper demonstrates the experimental realization of a complete set of high-dimensional quantum gates on single photons with orbital angular momentum, advancing quantum information processing beyond qubits.

## Contribution

It introduces the first experimental implementation of a four-dimensional generalization of the Pauli X-gate and its powers, completing the set of high-dimensional quantum gates.

## Key findings

- Successfully implemented a four-dimensional X-gate on single photons.
- Demonstrated the complete set of high-dimensional quantum gates experimentally.
- The X-gate concept can be generalized to other quantum systems.

## Abstract

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 multi-level 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 easily generalized to other photonic degrees-of-freedom, as well as to other quantum systems such as ions and superconducting circuits.