This paper introduces a new approach to improve Grover's search algorithm by utilizing variational learning, with a specific focus on incorporating a partial diffusion operator. Grover's search algorithm is a well-known quantum algorithm that provides a quadratic speedup for searching an unsorted database. However, its performance can be further enhanced by modifying its diffusion operator. In this work, we aim to identify a parameterized quantum circuit that can effectively learn and optimize Grover's search algorithm, incorporating a partial diffusion operator. The key idea behind this approach is to use variational learning, a technique that employs parameterized quantum circuits to optimize the algorithm's performance. By adjusting the parameters of the quantum circuit, the algorithm can be tailored to better solve the search problem. Variational learning is employed to determine the optimal parameters for the partial diffusion operator, enabling the quantum circuit to adapt to different problem instances. Our experimental results demonstrate that the optimized quantum circuit outperforms the traditional Grover’s algorithm that uses the standard partial diffusion operator. The results suggest that the proposed approach offers significant improvements in terms of algorithmic efficiency and performance. This advancement could have important implications for the development of quantum algorithms, particularly in applications related to database search and optimization problems. In conclusion, this paper presents a promising new method to enhance Grover's search algorithm, utilizing variational learning and a partial diffusion operator, with results showing improved performance over the original algorithm.
Grover's algorithm is optimal in terms of oracle queries. We can trade accuracy for speed, which gives rise to the quantum partial-search algorithm. The partial-search algorithm is implemented using two kinds of Grover operators, global and local, where the former is the standard Grover operator for full search and the latter has a diffusion operator acting on the search subspace. The global-local-global sequence, also known as the Grover-Radhakrishnan-Korepin (GRK) algorithm, has been proved optimal in the oracle-query metric. In this work, we show that the alternating sequence of global and local Grover operators, formed by repeatedly applying a fixed product of global and local Grover operators, can achieve a lower expected circuit depth than the GRK algorithm. Through systematic analysis, we obtain its exact success probability, expected depth, and asymptotically optimal parameters. We derive the boundary, characterized by the ratio between the depths of the oracle and the global diffusion operator, separating the depth-optimal and oracle-optimal partial-search algorithms. When the depths of the oracle and the global diffusion operator are comparable, our proposed alternating partial-search sequence can reduce the minimum expected circuit depth by more than 20\% compared with the GRK algorithm.
Grover's algorithm is optimal in query complexity, but not necessarily in circuit depth. We formulate unstructured quantum search as a circuit-depth optimization problem and identify a critical depth ratio separating query optimality from depth optimality. The resulting depth-efficient search operators exhibit a palindromic structure, in which shallow diffusion-like operators symmetrically replace selected Grover diffusion layers while preserving efficient amplitude amplification. This structure yields a simple depth-efficiency criterion and an analytic expression for the minimal expected depth. Applying the framework to X-type mixers, local diffusion operators, and nested local diffusion operators, we obtain substantial depth reductions over standard Grover search. In particular, nested local constructions reduce the total circuit depth by about 40% when the oracle and Grover diffusion operators have comparable depth. These results reveal the resource-dependent nature of quantum-search optimality and establish palindromic constructions as a systematic route to depth-efficient quantum search algorithms.
Grover's search algorithm has attracted great attention due to its quadratic speedup over classical algorithms in unsorted database search problems. However, Grover's algorithm is inefficient in multi-target search problems, except in the case of 1/4 of the data in the database satisfying the search conditions. Long presented a modified version of Grover's search algorithm by introducing a phase-matching condition that can search for the target state with a zero theoretical failure rate. In this work, we present an optimized exact multi-target search algorithm based on the modified Grover's algorithm by transforming the canonical diffusion operator to a more efficient diffusion operator, which can solve the multi-target search problem with a 100% success rate while requiring fewer gate counts and shallower circuit depth. After that, the optimized multi-target algorithm for four different items, including 2-qubit with 2 targets, 5-qubit with 2 targets, 5-qubit with 4 targets, and 6-qubit with 3 targets, are implemented on MindSpore framework. The experimental results show that, compared with Grover's algorithm and the modified Grover's algorithm, the proposed algorithm can reduce the quantum gate count by at least 21.1% and the depth of the quantum circuit by at least 11.4% and maintain a 100% success probability. Our code are available at https://github.com/mindsporelab/models/tree/master/research/arxiv_papers/GROVER-OP. Thanks for the support provided by MindSpore Community.