  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.
