Finding the global minimum of a function with many local extrema is well known to be a challenging computational task. We propose a hybrid quantum-classical algorithm for this optimisation problem, based on a combination of Grover quantum search and the classical Newton-Raphson method. The approach is tailored to situations with a large number of false minima. The main idea is to use quantum search to efficiently locate the vicinity of the true optimum, after which a classical algorithm can solve the problem in only a few iterations. We specifically address the nontrivial question of choosing the discretisation of the function domain, which is critical for quantum approaches to optimisation of continuous functions. While we do not expect that this method will always be superior, we identify a paradigmatic case of a function for which we demonstrate that, remarkably, our hybrid method quadratically outperforms a purely classical approach (measuring performance by the number of function/oracle calls). We evaluate key performance metrics of the algorithm in numerical experiments. We furthermore discuss its possible extensions as well as the potential relevance of hybrid approaches in the context of quantum cognition.
Finding the global minimum of a function with many local extrema is well known to be a challenging computational task. We propose a hybrid quantum-classical algorithm for this optimisation problem, based on a combination of Grover quantum search and the classical Newton-Raphson method. The approach is tailored to situations with a large number of false minima. The main idea is to use quantum search to efficiently locate the vicinity of the true optimum, after which a classical algorithm can solve the problem in only a few iterations. We specifically address the nontrivial question of choosing the discretisation of the function domain, which is critical for quantum approaches to optimisation of continuous functions. While we do not expect that this method will always be superior, we identify a paradigmatic case of a function for which we demonstrate that, remarkably, our hybrid method quadratically outperforms a purely classical approach (measuring performance by the number of function/oracle calls). We evaluate key performance metrics of the algorithm in numerical experiments. We furthermore discuss its possible extensions as well as the potential relevance of hybrid approaches in the context of quantum cognition.
Abstract Solving optimisation problems is a promising near-term application of quantum computers. Quantum variational algorithms (QVAs) leverage quantum superposition and entanglement to optimise over exponentially large solution spaces using an alternating sequence of classically tunable unitaries. However, prior work has primarily addressed discrete optimisation problems. In addition, these algorithms have been designed generally under the assumption of an unstructured solution space, which constrains their speedup to the theoretical limits for the unstructured Grover’s quantum search algorithm. In this paper, we show that QVAs can efficiently optimise continuous multivariable functions by exploiting general structural properties of a discretised continuous solution space with a convergence that exceeds the limits of an unstructured quantum search. We present the quantum multivariable optimisation algorithm and demonstrate its advantage over pre-existing methods, particularly when optimising high-dimensional and oscillatory functions.
Finding the optimal solution to a complex optimisation problem is of great importance in practically all fields of science, technology, technical design and econometrics. We demonstrate that a modified Grover's quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits. Calculations of the global minimum of simple test functions and Lennard-Jones clusters have been carried out on a quantum computer simulator using a modified Grover's algorithm. The number of function evaluations N reduced from O(N) in classical simulation to O(N 1/2) in quantum simulation. We also show how the Grover's quantum algorithm can be combined with the classical Pivot method for global optimisation to treat larger systems.
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.
We propose a quantum unstructured search algorithm to find the extrema or roots of discrete functions, f(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(x∗)≃f∗, where f∗ is the extremum or zero, to an unstructured search problem for x∗. This is achieved by mapping f(x) to a function u(z) to create a quantum oracle, such that u(z∗)=1 and u(z=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(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(x), it can still find approximate solutions even if the conditions are not exactly satisfied.
In this paper a new optimization algorithm named Adaptive Hybrid Quantum-Inspired Optimization Algorithm (AHQSOA) is presented for solving the complex optimization problems effectively. Stochastic traditional algorithms typically fail to come to be stuck in local optimum solutions and they need quite few iterations for converging. Our method integrates two quantum-inspired methodology: Quantum Particle Swarm Optimization (QPSO) for searching a large set solutions based on properties of quantum mechanics - superposition, point are located as randomly as possible, and Quantum Evolutionary Algorithms (QEA) for refining potential solutions by mutation and crossover operation. An adaptive tuning component based on the reinforcement learning dynamically adjusts the algorithm’s parameters, thus balancing the exploration in the wide interval and exploitation in the precise interval. Assessed on the commonly used benchmark functions like Sphere, Rosenbrock, Rastrigin, AHQSOA exhibits faster convergence (only 400 iterations in comparison to 1200 in standard methods), global search superiority (94% whereas 75–87% the traditional methods), and low computational simplicity. These results show all of our algorithm are nicer and faster and more efficient and robust makes for it a good candidate as for high dimensional and complex optimization problems.
ℓ0 Optimization with Robust Non-Oracular Quantum Search
In this article, we introduce an innovative hybrid quantum search algorithm, the Robust Non-oracle Quantum Search (RNQS), which is specifically designed to efficiently identify the minimum value within a large set of random numbers. Distinct from the Grover’s algorithm, the proposed RNQS algorithm circumvents the need for an oracle function that describes the true solution state, a feature often impractical for data science applications. Building on existing non-oracular quantum search algorithms, RNQS enhances robustness while substantially reducing running time. The superior properties of RNQS have been demonstrated through careful analysis and extensive empirical experiments. Our findings underscore the potential of the RNQS algorithm as an effective and efficient solution to combinatorial optimization problems in the realm of quantum computing.