Recursive multiport schemes for implementing quantum algorithms with photonic integrated circuits

Abstract: We present recursive multiport schemes for implementing quantum Fourier transforms and the inversion step in Grover's algorithm on an integrated linear optics device. In particular, each scheme shows how to execute a quantum operation on $2d$ modes using a pair of circuits for the same operation on $d$ modes. The circuits operate on path-encoded qudits and realize $d$-dimensional unitary transformations on these states using linear optical networks with $O\left(d^2\right)$ optical elements. To evaluate the schemes against realistic errors, we ran simulations of proof-of-principle experiments using a simple fabrication model of silicon-based photonic integrated devices that employ directional couplers and thermo-optic modulators for beam splitters and phase shifters, respectively. We find that high-fidelity performance is achievable with our multiport circuits for $2$-qubit and $3$-qubit quantum Fourier transforms, and for quantum search on four-item and eight-item databases.