A quantum unstructured search algorithm for discrete optimisation: the use case of portfolio optimisation

TL;DR

This paper introduces QSERA, a quantum algorithm leveraging Grover's search to efficiently find extrema in discrete functions, demonstrated through a portfolio optimisation case, surpassing traditional quadratic models.

Contribution

The paper presents QSERA, a novel quantum unstructured search algorithm capable of handling higher-order objective functions in discrete optimisation.

Findings

QSERA achieves quadratic speed-up over classical methods.

It can handle higher-order terms than QUBO frameworks.

It provides approximate solutions even with imperfect prior knowledge.

Abstract

We propose a quantum unstructured search algorithm to find the extrema or roots of discrete functions, $f(\mathbf{x})$, such as the objective functions in combinatorial and other discrete optimisation problems. The first step of the Quantum Search for Extrema and Roots Algorithm (QSERA) is to translate conditions of the form $f(\mathbf{x}_*) \simeq f_*$, where $f_*$ is the extremum or zero, to an unstructured search problem for $\mathbf{x}_*$. This is achieved by mapping $f(\mathbf{x})$ to a function $u(z)$ to create a quantum oracle, such that $u(z_*) = 1$ and $u(z \neq z_*) = 0$. The next step is to employ Grover's algorithm to find $z_*$, which offers a quadratic speed-up over classical algorithms. The number of operations needed to map $f(\mathbf{x})$ to $u(z)$ determines the accuracy of the result and the circuit depth. We describe the implementation of QSERA by assembling a quantum circuit for portfolio optimisation, which can be formulated as a combinatorial problem. QSERA can handle objective functions with higher order terms than the commonly-used Quadratic Unconstrained Binary Optimisation (QUBO) framework. Moreover, while QSERA requires some a priori knowledge of the extrema of $f(\mathbf{x})$, it can still find approximate solutions even if the conditions are not exactly satisfied.