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International Journal of Quantum Information

Invalid quantum partial adiabatic search can be remedied

Jie Sun, Hui Zheng, Zhiming Zhang, Songfeng Lu

Published Jan 14, 2026• 1 citation
Publisher

Abstract

This paper first demonstrates that by appropriately specifying the evolution interval in quantum partial adiabatic evolution, the time complexity of the resulting quantum search algorithm can achieve the well-known quadratic speedup bound of quantum computation. Then this result is used as a reference to remedy invalid quantum partial adiabatic search schemes, which means its time complexity doesn’t align with the optimal quantum speedup. And, it is found that the two versions of quantum partial adiabatic evolutions, i.e. “global” and “local” ones must be treated separately, despite previous studies indicating only minor differences in algorithmic performance between the two cases. Additionally, for the “local” variant of the quantum partial adiabatic search algorithm, it can be seen that there are other possible choices for the evolution interval besides the referred one. Our findings further show that significant variations in the resulting single-round success probabilities occur between valid and invalid quantum partial adiabatic evolutions.

DOI
10.1142/s0219749926500012

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This paper first demonstrates that by appropriately specifying the evolution interval in quantum partial adiabatic evolution, the time complexity of the resulting quantum search algorithm can achieve the well-known quadratic speedup bound of quantum computation. Then this result is used as a reference to remedy invalid quantum partial adiabatic search schemes, which means its time complexity doesn’t align with the optimal quantum speedup. And, it is found that the two versions of quantum partial adiabatic evolutions, i.e. “global” and “local” ones must be treated separately, despite previous studies indicating only minor differences in algorithmic performance between the two cases. Additionally, for the “local” variant of the quantum partial adiabatic search algorithm, it can be seen that there are other possible choices for the evolution interval besides the referred one. Our findings further show that significant variations in the resulting single-round success probabilities occur between valid and invalid quantum partial adiabatic evolutions.
This paper presents a framework for quantum partial adiabatic evolution and applies it to re-examine the well-known quantum search problem. We particularly focus on a detailed analysis of the algorithm’s success probability, which serves as a clear criterion for differentiating valid implementations from invalid ones. Specifically, when the time complexity aligns with the optimal quantum computation, the algorithm achieves a substantially high success probability. Conversely, so-called “improved” versions that exceed the quadratic speedup characteristic of quantum computing exhibit a negligibly low success probability with the increase of target elements. These findings underscore the critical importance of selecting the appropriate evolution interval and the correct method for calculating the success probability in studies of quantum partial adiabatic evolution.
ABSTRACT In this paper, we revisit the problem of success probability of quantum partial adiabatic search algorithms, which has been studied in a series of our prior works. The conclusions presented here can be divided into positive and negative aspects. On the one hand, for the quantum partial adiabatic evolution with time complexity of , where and denote the sizes of the total elements and the target elements in the search problem, respectively, our choice of narrow evolution paths in the related papers is in fact acceptable. This is reflected in the finite success probability of the quantum algorithm. On the other hand, for the quantum partial adiabatic computation (QPAC) with time complexity of , the choice of narrow evolution paths in the previous paper is indeed invalid, as evidenced by the low success probability. These findings demonstrate that previous criticisms of our work are not fully justified, and they highlight the necessity of a careful treatment of success probability calculations in quantum partial adiabatic evolutions.
Recently, Zhang and Lu provided a quantum search algorithm based on partial adiabatic evolution, which beats the time bound of local adiabatic search when the number of marked items in the unsorted database is larger than one. Later, they found that the above two adiabatic search algorithms had the same time complexity when there is only one marked item in the database. In the present paper, following the idea of Roland and Cerf [Roland J and Cerf N J 2002 Phys. Rev. A 65 042308 ], if within the small symmetric evolution interval defined by Zhang et al., a local adiabatic evolution is performed instead of the original "global" one, this "new" algorithm exhibits slightly better performance, although they are progressively equivalent with M increasing. In addition, the proof of the optimality for this partial evolution based local adiabatic search when M = 1 is also presented. Two other special cases of the adiabatic algorithm obtained by appropriately tuning the evolution interval of partial adiabatic evolution based quantum search, which are found to have the same phenomenon above, are also discussed.
Abstract In this paper, we further verify the validity of the quantum partial adiabatic search algorithm which was initialized in the previous related works by revisiting its quantum circuit model. The main results got here are as follows. When considering implementing quantum partial adiabatic evolution on a quantum circuit, a correction is given for the time slice estimation for the first stage during this approximation in the previous related works, new evidence is provided for a time complexity cost O(N/M)O(\sqrt{N}/M)O(N​/M) O ( N / M ) of quantum partial adiabatic algorithm is impossible, and the correct time complexity O(N/M)O(\sqrt{N/M})O(N/M​) O ( N / M ) of it is emphasized once more according to its circuit correspondence, in which N is the total number of elements in the search problem of which M of them are the marked ones. The findings exposed are hopeful for revisiting quantum partial adiabatic evolution and its connection with the quantum circuit model.
This paper presents and implements a specified partial adiabatic search algorithm on a quantum circuit. It studies the minimum energy gap between the first excited state and the ground state of the system Hamiltonian and it finds that, in the case of M = 1, the algorithm has the same performance as the local adiabatic algorithm. However, the algorithm evolves globally only within a small interval, which implies that it keeps the advantages of global adiabatic algorithms without losing the speedup of the local adiabatic search algorithm.
In this paper, we study two aspects of quantum adiabatic evolution for a prototypical search problem: the optimality of the corresponding algorithm and its relation to the quantum circuit model. Firstly, we propose a general framework for proving the square-root speedup of the quantum adiabatic algorithm to be optimal over classical computation, which is readily applicable to the case of multiple targets. Through this framework, we also find that it is possible to further reduce the time complexity by increasing the physical energy of the system, encompassing results from previous works. Secondly, we find that, on the one hand, when the quantum adiabatic algorithm that achieves quadratic speedup is implemented on a quantum circuit, the time slice needed is always consistent with its time complexity, which also encompasses previous results; on the other hand, however, if a further algorithmic improvement is considered, the time slice always remains invariant. This phenomenon represents a significant observation with potential applications. We anticipate that the main results of this paper will interest the quantum adiabatic computation community and may help us to design efficient quantum algorithms for practical problems in the future.