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arXiv:2505.14645

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

Matsakos, Titos, Lomas, Adrian

Published May 20, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2505.14645

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In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation and experimental results obtained on real quantum hardware. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems.
This thesis is devoted to four modern quantum optimisation algorithms and their applications. In the first chapter, we provide a complete Quantum Approximate Optimisation Algorithm (QAOA) pipeline for the discrete global minimum-variance portfolio (DGMVP) problem. We introduce an ansatz with efficient initial-state preparation, new hard mixing operators satisfying the constraints of the DGMVP problem, and scalable cost operators utilising binary encoding of the qubits. Benchmarking the method with the most effective optimisation strategies---dual annealing and unfrozen layerwise optimisation---indicates a scaling advantage over classical methods in finding the global optimum of the DGMVP problem. In the second chapter, we show an end-to-end quantum/quantum-inspired Goemans-Williamson semidefinite programming (SDP) method for relaxing Quadratic Unconstrained Binary Optimisation (QUBO) problems and extracting their rounding energy. Our method identifies a class of QUBO instances that is effectively solvable in polylogarithmic time relative to the problem dimension, under the assumptions of Pauli sparsification of the QUBO cost matrix, reducibility of the diagonal constraints, and efficient evaluation of the corresponding Gibbs-state expectation values. Further, we identify the commuting 1D Hamiltonian and the Kronecker graph as practically important instances. Our numerical experiments show that our method solves the SDP and QUBO with up to 2502^{50}250 variables for 1D Hamiltonian instances and achieves results competitive with the best classical solver for Kronecker graphs, giving strong evidence of the power of our algorithm. In the third chapter, we improve the current QAOA method with an iterative warm-start initial-state preparation method. We implement the ttt-th order statistic state as the warm-start initial state generated from each QAOA optimisation loop. We achieve an improved approximation ratio when solving the 3-regular MaxCut problem, and find that the worst-case approximation ratio converges to that of the best classical algorithm for p=1p=1p=1 QAOA. Additionally, our method achieves more favourable scaling than standalone QAOA, single-string warm-started QAOA, and a classical constrained-sampling approach when solving the DGMVP problem. In the fourth chapter, we provide a hard-constrained Grover adaptive search (GAS) method for fixed-sum quadratic binary optimisation problems, and an extension of the alternating direction method of multipliers (ADMM) framework for fixed-sum quartic binary optimisation. Our method guarantees at least a quadratic speed-up over the classical method, and exponentially reduces the gate count for solving fixed-sum binary optimisation when compared with the soft-constrained GAS in the fault-tolerant quantum computing regime. In summary, our contributions include efficient algorithms and novel frameworks that substantially improve performance and broaden the applicability of these quantum optimisation methods. The proposed methods exhibit favourable scaling behaviour and indicate potential quantum advantages in solving challenging optimisation problems, thereby motivating further research at the interface of quantum computing and optimisation.
Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.
Grover's algorithm stands as a cornerstone of quantum computing, offering a theoretical quadratic speedup for unstructured search problems. This allows for finding a target item in an unsorted database of N entries in approximately the square root of N steps, a significant improvement over classical algorithms which require a number of operations on the order of N in the worst case. This speedup becomes particularly compelling for very large datasets, where classical exhaustive search becomes computationally intractable. The algorithm achieves this by leveraging fundamental quantum mechanical principles, primarily superposition and interference. It operates by preparing qubits in a uniform superposition of all possible states, then iteratively applying a quantum oracle to mark the desired state by flipping its phase, followed by a diffusion operator that amplifies the probability of the marked state. Despite its theoretical promise, real-world implementation faces substantial challenges stemming from Noisy Intermediate-Scale Quantum (NISQ) hardware, which is characterized by high error rates and short coherence times. Consequently, achieving a true, scalable quantum advantage remains a future endeavor, contingent on significant hardware advancements. Nevertheless, Grover's algorithm holds significant potential across diverse fields, including enhancing brute-force attacks on cryptographic systems, accelerating solutions for optimization problems, and speeding up search processes in machine learning and unstructured database queries.
Abstract This guide is based on the IBM Quantum course Phase Estimation and Factoring and focuses on Grover’s algorithm, a quantum algorithm for unstructured search problems that provides a quadratic speedup over classical approaches. In an unstructured search problem, we are given a function (often represented as an oracle) that marks one or more “correct” solutions among NNN possible candidates. A classical algorithm requires on the order of O(N)O(N)O(N) evaluations in the worst case. In contrast, Grover’s algorithm finds a solution using only O(N)O(\sqrt{N})O(N) oracle queries. This quadratic improvement implies: The quantum algorithm requires roughly the square root of the number of operations needed classically. Any classical algorithm for unstructured search must have cost at least quadratic relative to Grover’s quantum procedure. Core Idea of the Algorithm Grover’s algorithm relies on: Preparation of a uniform superposition (typically using Hadamard gates). An oracle that marks valid solutions via phase inversion. The Grover diffusion operator, which amplifies the probability amplitude of the marked states through interference. By repeatedly applying the Grover operator, the amplitude of the desired solution grows while others diminish, allowing measurement to reveal the correct answer with high probability. Practical Considerations Despite its theoretical elegance and wide applicability, the quadratic speedup of Grover’s algorithm is unlikely to yield near-term practical advantage. Modern classical hardware operates at extremely high clock rates and is vastly more mature than current quantum devices. For problem sizes accessible to today’s quantum computers, classical systems remain dominant. However, history shows that even sub-quadratic classical improvements — such as: Fast Fourier Transform (FFT) Efficient sorting algorithms like quicksort and mergesort have had transformative practical impact. The crucial distinction is that Grover’s algorithm depends on fundamentally different hardware — quantum processors — which are still in early development. Long-Term Perspective As quantum hardware advances in qubit count, coherence time, and error correction, Grover’s quadratic advantage may become practically significant for: Cryptographic key search Optimization heuristics Combinatorial search problems Amplitude amplification subroutines in broader quantum algorithms Through hands-on experimentation on IBM Q, this lesson demonstrates how theoretical amplitude amplification translates into executable quantum circuits, reinforcing the relationship between interference, probability amplitudes, and computational complexity.
Portfolio optimization, a core combinatorial optimization problem in quantitative finance, requires balancing return maximization and risk minimization within discrete asset sets. Classical algorithms often struggle with scalability and local optima traps when handling large-scale asset pools, limiting their practicality in dynamic markets. This study proposes a complete Quantum Approximate Optimization Algorithm (QAOA) workflow for portfolio optimization, from problem formalization to experimental validation. We first transform the portfolio optimization problem into a Quadratic Unconstrained Binary Optimization (QUBO) framework, then design a parameterized QAOA circuit with adjustable layers to balance expressiveness and noise resilience. We propose a methodology using Qiskit for quantum simulation and classical optimizers for parameter tuning, we propose implementing the algorithm on real-world S&P 500 stock data. We could also analyze the impact of quantum hardware noise and propose mitigation strategies, providing actionable insights for Noisy intermediate-scale quantum computing (NISQ)-era implementations. This work bridges quantum algorithm design with real-world financial applications, offering a theoretical framework for quantum-driven portfolio optimization.