Improved entanglement-based high-dimensional optical quantum computation with linear optics

Quantum gates are the essential block for quantum computer. High-dimensional quantum gates exhibit remarkable advantages over their two-dimensional counterparts for some quantum information processing tasks. Here we present a family of entanglement-based optical controlled-SWAP gates on mathbb{C}²otimes mathbb{C}^dotimes mathbb{C}^d. With the hybrid encoding, we encode the control qubits and target qudits in photonic polarization and spatial degrees of freedom, respectively. The circuit is constructed using only (2+3d) $dgeq 2$ linear optics, beating an earlier result of 14 linear optics with d=2. The circuit depth 5 is much lower than an earlier result of 11 with d=2. Besides, the fidelity of the presented circuit can reach 99.4%, and it is higher than the previous counterpart with d=2. Our scheme are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Moreover, our approach allows d>2.