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Annalen der Physik

On the Success Probability of Quantum Partial Adiabatic Search

Jie Sun

Published Oct 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1002/andp.70316

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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.
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 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 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.
A quantum search algorithm based on the partial adiabatic evolution [Phys. Rev. A 80, 052328 (2009)] is provided. We calculate its time complexity and find that the algorithm improves the time complexity, which is O(N/M)O(\sqrt{N/M})O(N/M​), of the local adiabatic search algorithm [Phys. Rev. A 65, 042308 (2002)] to O(N/M)O(\sqrt{N}/M)O(N​/M).
Abstract In this paper, we study a kind of nonlinear model of adiabatic evolution in quantum search problem. As will be seen here, for this problem, there always exists a possibility that this nonlinear model can successfully solve the problem, while the linear model can not. Also in the same setting, when the overlap between the initial state and the final stare is sufficiently large, a simple linear adiabatic evolution can achieve O(1) time efficiency, but infinite time complexity for the nonlinear model of adiabatic evolution is needed. This tells us, it is not always a wise choice to use nonlinear interpolations in adiabatic algorithms. Sometimes, simple linear adiabatic evolutions may be sufficient for using.