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.
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.
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.
Quantum computing is quickly becoming a field that can change the game. It can completely change how businesses solve optimisation problems. We will be looking at three different quantum algorithms in great detail: the Quantum Approximate Optimisation Algorithm (QAOA), the Variational Quantum Eigensolver (VQE), and Grover's Algorithm. We look into how these algorithms work on the inside, how they compare to more traditional methods, and how they might be used in areas like energy, banking, and logistics. The piece then talks about current research projects that are trying to fix the technical issues and hardware limits of quantum technology. In the end, we look ahead to possible future developments that might help solve optimisation problems, such as better quantum gear and more complex quantum algorithms. By combining what has already been written with what is new, this study aims to shed light on how quantum computing could help solve tough optimisation problems and spark new ideas.
Abstract Quantum-assisted algorithms are expected to improve the computing performance of classical computers. A quantum genetic algorithm utilizes the advantages of quantum computation by combining the truncation selection in a classical genetic algorithm with the quantum Grover’s algorithm. The parallelism of evaluation can create global search and reduce the need of crossover and mutation in a conventional genetic algorithm. In this work, we aim to demonstrate and simulate the performance of an implementable quantum-assisted genetic algorithm. The algorithm was tested by using quadratic unconstrained binary optimization (QUBO) for 100 iterations; and the results were compared with those from a classical counterpart for 2000 iterations, where both simulations were performed over 100 repetitions. The results showed that the quantum algorithm converges to the optimal solution faster. While the variance is higher at early stage, it quickly and greatly reduces as the algorithm converges. The histograms of possible solutions consistently exhibits this behavior.
The study examines the use of Grover's quantum algorithm to optimize global search problems, demonstrating its quadratic speedup over classical methods for improved efficiency. The research tackles practical challenges, including quantum noise, scalability, and resource constraints, by proposing innovative solutions. The techniques developed integrate quantum search principles into large-scale optimization tasks across diverse fields. This work lays a foundational framework for applying quantum algorithms to solve complex scientific and industrial problems.
Quantum computing offers an accelerated mechanism for solving combinatorial optimization problems that could be hardly solvable with classical tools, in this context, this paper highlights the utility of quantum computing in solving NP-complete problems efficiently and shows up the importance of employing quantum algorithms achieving parallel calculation in the purpose of reducing the algorithmic complexity and simplifying the development of optimization applications. Within this publication, an optimized version of quantum circuit conception for developing the oracle of the Grover's search algorithm is introduced for solving the vertex cover problem, and it was also tested using the Qiskit library and applied to a set of instances to evaluate its performance.
Classical computer simulations of quantum systems are not efficient-well-known examples range from the Hubbard model to lattice QCD-and Feynman argued that quantum simulations would do far better [1].The essence of the argument is that quantum simulations sum multiple evolutionary paths (in superposition) contributing to a quantum process at one go, while classical simulations evaluate these paths one by one.Formalisation of this advantage for simulating physical Hamiltonians, in terms of computational complexity, has improved over the years step by step [2,3,4,5,6].Here, treating Grover's quantum search algorithm as a Hamiltonian evolution problem, we expose the physical reasons behind the improvement in computational complexity.Computational complexity of a problem is a measure of the resources needed to solve it.Conventionally, the computational complexity of a decision problem is specified in terms of the size of its input, noting that the size of its output is only one bit.Problems with different output requirements are reduced to a sequence of decision problems, with gradually narrowing bounds on the output adding one bit of precision for every decision made.In such a scenario, the number of decision problems solved equals the number of output bits, and it is appropriate to specify the complexity of the original problem in terms of the size of its input as well as its output.Generalising the conventional classification, the computational algorithm then can be labeled efficient if the required resources are polynomial in terms of the size of both its input and its output.Popular importance sampling methods are not efficient according to our criterion, because the number of iterations needed in the computational effort has a negative power-law dependence on the precision ε (i.e.N iter ∝ ε -2 as per the central limit theorem).On the other hand, finding zeroes of a function by bisection is efficient (i.e.N iter ∝ log ε), and finding them by Newton's method is super-efficient (i.e.N iter ∝ log log ε). PoS(LATTICE2014)324 Optimisation of Quantum Evolution AlgorithmsApoorva Patel