  Even a minor boost in solving combinatorial optimization problems can greatly
benefit multiple industries. Quantum computers, with their unique information
processing capabilities, hold promise for delivering such enhancements. The
Filtering Variational Quantum Eigensolver (F-VQE) is a variational hybrid
quantum algorithm designed to solve combinatorial optimization problems on
existing quantum computers with limited qubit number, connectivity, and
fidelity. In this work we employ Instantaneous Quantum Polynomial circuits as
our parameterized quantum circuits. We propose a hardware-efficient
implementation that respects limited qubit connectivity and show that they
halve the number of circuits necessary to evaluate the gradient with the
parameter-shift rule. To assess the potential of this protocol in the context
of combinatorial optimization, we conduct extensive numerical analysis. We
compare the performance against three classical baseline algorithms on weighted
MaxCut and the Asymmetric Traveling Salesperson Problem (ATSP). We employ
noiseless simulators for problems encoded on 13 to 29 qubits, and up to 37
qubits on the IBMQ real quantum devices. The ATSP encoding employed reduces the
number of qubits and avoids the need of constraints compared to the standard
QUBO / Ising model. Despite some observed positive signs, we conclude that
significant development is necessary for a practical advantage with F-VQE.
