  We propose novel methods for the exact synthesis of single-qubit unitaries
with high success probability and gate fidelity, considering both time-bin and
frequency-bin encodings. The proposed schemes are experimentally implementable
with a spectral linear-optical quantum computation (S- LOQC) platform, composed
of electro-optic phase modulators and phase-only programmable filters (pulse
shapers). We assess the performances in terms of fidelity and probability of
the two simplest 3-components configurations for arbitrary gate generation in
both encodings and give an exact analytical solution for the synthesis of an
arbitrary single-qubit unitary in the time-bin encoding, using a single-tone
Radio Frequency (RF) driving of the EOMs. We further investigate the
parallelization of arbitrary single-qubit gates over multiple qubits with a
compact experimental setup, both for spectral and temporal encodings. We
systematically evaluate and discuss the impact of the RF bandwidth - that
conditions the number of tones driving the modulators - and of the choice of
encoding for different targeted gates. We moreover quantify the number of high
fidelity Hadamard gates that can be synthesized in parallel, with minimal and
increasing resources in terms of driving RF tones in a realistic system. Our
analysis positions spectral S-LOQC as a promising platform to conduct massively
parallel single qubit operations, with potential applications to quantum
metrology and quantum tomography.
