Wasserstein Solution Quality and the Quantum Approximate Optimization Algorithm: A Portfolio Optimization Case Study

Optimizing of a portfolio of financial assets is a critical industrial problem which can be approximately solved using algorithms suitable for quantum processing units (QPUs). We benchmark the success of this approach using the Quantum Approximate Optimization Algorithm (QAOA); an algorithm targeting gate-model QPUs. Our focus is on the quality of solutions achieved as determined by the Normalized and Complementary Wasserstein Distance, $\eta$, which we present in a manner to expose the QAOA as a transporter of probability. Using $\eta$ as an application specific benchmark of performance, we measure it on selection of QPUs as a function of QAOA circuit depth $p$. At $n = 2$ (2 qubits) we find peak solution quality at $p=5$ for most systems and for $n = 3$ this peak is at $p=4$ on a trapped ion QPU. Increasing solution quality with $p$ is also observed using variants of the more general Quantum Alternating Operator Ans\"{a}tz at $p=2$ for $n = 2$ and $3$ which has not been previously reported. In identical measurements, $\eta$ is observed to be variable at a level exceeding the noise produced from the finite number of shots. This suggests that variability itself should be regarded as a QPU performance benchmark for given applications. While studying the ideal execution of QAOA, we find that $p=1$ solution quality degrades when the portfolio budget $B$ approaches $n/2$ and increases when $B \approx 1$ or $n-1$. This trend directly corresponds to the binomial coefficient $nCB$ and is connected with the recently reported phenomenon of reachability deficits. Derivative-requiring and derivative-free classical optimizers are benchmarked on the basis of the achieved $\eta$ beyond $p=1$ to find that derivative-free optimizers are generally more effective for the given computational resources, problem sizes and circuit depths.