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

Paper: arXiv 2505.14645

Authors: Titos Matsakos, Adrian Lomas

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.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical formulation of the Quantum Search for Extrema and Roots Algorithm (QSERA) for discrete optimization, extending beyond standard QUBO limits. However, it lacks empirical validation on real-world market datasets or physical quantum hardware backtests, remaining primarily theoretical.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve Discrete & Portfolio Optimisation via Quantum Unstructured Search"] --> B["Inputs: Discrete Function f(x) & A Priori Target f*"]
    B --> C["Map f(x) to Oracle Function u(z) <br> (u=1 for z*, u=0 for z≠z*)"]
    C --> D["Execute Grover's Search Algorithm <br> (Quadratic Speed-Up)"]
    D --> E["Assemble Quantum Circuit for <br> Multi-Asset Portfolio Optimisation"]
    E --> F["Outcomes: Finds Extrema/Roots Beyond QUBO Limits & Handles Approximate Solutions"]

Cited References (53)

1. Challenges and opportunities in quantum optimization (2024) — link
2. A Structured Survey of Quantum Computing for the Financial Industry (2022) — link
3. Quantum algorithm for finding minimum values in a Quantum Random Access Memory (2023) — link
4. Quantum algorithm for credit valuation adjustments (2022) — link
5. Wasserstein Solution Quality and the Quantum Approximate Optimization Algorithm: A Portfolio Optimization Case Study (2022) — link
6. Grover’s Quantum Algorithm Applied to Global Optimization (2005) — link
7. A review on Quantum Approximate Optimization Algorithm and its variants (2024) — link
8. Quantum Algorithms for Mixed Binary Optimization Applied to Transaction Settlement (2021) — link

0 of 53 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.