  We study quantum interior point methods (QIPMs) for second-order cone
programming (SOCP), guided by the example use case of portfolio optimization
(PO). We provide a complete quantum circuit-level description of the algorithm
from problem input to problem output, making several improvements to the
implementation of the QIPM. We report the number of logical qubits and the
quantity/depth of non-Clifford T-gates needed to run the algorithm, including
constant factors. The resource counts we find depend on instance-specific
parameters, such as the condition number of certain linear systems within the
problem. To determine the size of these parameters, we perform numerical
simulations of small PO instances, which lead to concrete resource estimates
for the PO use case. Our numerical results do not probe large enough instance
sizes to make conclusive statements about the asymptotic scaling of the
algorithm. However, already at small instance sizes, our analysis suggests
that, due primarily to large constant pre-factors, poorly conditioned linear
systems, and a fundamental reliance on costly quantum state tomography,
fundamental improvements to the QIPM are required for it to lead to practical
quantum advantage.
