Quantum Algorithms for Portfolio Optimization

TL;DR

This paper introduces the first quantum algorithm for constrained portfolio optimization, offering potential polynomial and near-linear speedups over classical methods depending on problem accuracy and instance characteristics.

Contribution

The paper presents a novel quantum algorithm for portfolio optimization with complexity bounds and experimental insights into its practical speedup potential.

Findings

Quantum algorithm achieves polynomial speedup for moderate accuracy solutions.

Experiments suggest potential $O(n)$ speedup for most instances.

Complexity depends on problem-specific parameters and well-conditioning.

Abstract

We develop the first quantum algorithm for the constrained portfolio optimization problem. The algorithm has running time $\widetilde{O} \left( n\sqrt{r} \frac{\zeta \kappa}{\delta^2} \log \left(1/\epsilon\right) \right)$, where $r$ is the number of positivity and budget constraints, $n$ is the number of assets in the portfolio, $\epsilon$ the desired precision, and $\delta, \kappa, \zeta$ are problem-dependent parameters related to the well-conditioning of the intermediate solutions. If only a moderately accurate solution is required, our quantum algorithm can achieve a polynomial speedup over the best classical algorithms with complexity $\widetilde{O} \left( \sqrt{r}n^\omega\log(1/\epsilon) \right)$, where $\omega$ is the matrix multiplication exponent that has a theoretical value of around $2.373$, but is closer to $3$ in practice. We also provide some experiments to bound the problem-dependent factors arising in the running time of the quantum algorithm, and these experiments suggest that for most instances the quantum algorithm can potentially achieve an $O(n)$ speedup over its classical counterpart.