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Recursive multiport schemes for implementing quantum algorithms with photonic integrated circuits

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.