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arXiv:1908.00269

Exact quantum search based on analytical multiphase matching for known number of target items and the experimental demonstration on IBM Q

Tan Li, Xiang-Qun Fu, Yan Wang, Shuo Zhang, Xiang Wang, Du, Yu-Tao, Wan‐Su Bao

Published Aug 1, 2019• 0 citations
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Abstract

In [Phys. Rev. Lett. 113, 210501 (2014)], to achieve the optimal fixed-point quantum search in the case of unknown fraction (denoted by $λ$) of target items, the analytical multiphase matching (AMPM) condition has been proposed. In this paper, we find out that the AMPM condition can also be used to design the exact quantum search algorithm in the case of known $λ$, and the minimum number of iterations reaches the optimal level of existing exact algorithms. Experiments are performed to demonstrate the proposed algorithm on IBM's quantum computer. In addition, we theoretically find two coincidental phases with equal absolute value in our algorithm based on the AMPM condition and that algorithm based on single-phase matching. Our work confirms the practicability of the AMPM condition in the case of known $λ$, and is helpful to understand the mechanism of this condition.

DOI
10.48550/arxiv.1908.00269

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In [Phys. Rev. Lett. 113, 210501 (2014)], to achieve the optimal fixed-point quantum search in the case of unknown fraction (denoted by λλλ) of target items, the analytical multiphase matching (AMPM) condition has been proposed. In this paper, we find out that the AMPM condition can also be used to design the exact quantum search algorithm in the case of known λλλ, and the minimum number of iterations reaches the optimal level of existing exact algorithms. Experiments are performed to demonstrate the proposed algorithm on IBM's quantum computer. In addition, we theoretically find two coincidental phases with equal absolute value in our algorithm based on the AMPM condition and that algorithm based on single-phase matching. Our work confirms the practicability of the AMPM condition in the case of known λλλ, and is helpful to understand the mechanism of this condition.
Grover's algorithm achieves a quadratic speedup over classical algorithms, but it is considered necessary to know the value of $\ensuremath{\lambda}$ exactly [Phys. Rev. Lett. 95, 150501 (2005); Phys. Rev. Lett. 113, 210501 (2014)], where $\ensuremath{\lambda}$ is the fraction of target items in the database. In this paper, we find out that the Grover algorithm can actually apply to the case where one can identify the range that $\ensuremath{\lambda}$ belongs to from a given series of disjoint ranges. However, Grover's algorithm still cannot maintain high success probability when there exist multiple target items. For this problem, we proposed a complementary-multiphase quantum search algorithm in which multiple phases complement each other so that the overall high success probability can be maintained. Compared to the existing algorithms, in the case defined above, our algorithm achieves the following three goals simultaneously: (1) the success probability can be no less than any given value between 0 and 1, (2) the algorithm is applicable to the entire range of $\ensuremath{\lambda}$, and (3) the number of iterations is almost the same as that of Grover's algorithm. Especially compared to the optimal fixed-point algorithm [Phys. Rev. Lett. 113, 210501 (2014)], our algorithm uses fewer iterations to achieve a success probability greater than 82.71%, e.g., when the minimum success probability is required to be 99.25%, the number of iterations can be reduced by 50%.
For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in O ( 1 / λ 0 ) rather than O ( 1 / λ ) , where λ 0 is a known lower bound of λ. (ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.
The success probability of the Grover quantum search algorithm decreases quickly when the fraction of target items exceeds 1/4, where the phase plays a significant role. Therefore, we use multiple phases to complement each other. We obtain three useful properties and an important theorem of the success probability and design a systematic solution of the optimal phases for an arbitrary number of phases. Based on these results, we finally propose a multi-phase quantum search algorithm whose success probability rises with the increase of the number of phases with just a single iteration, and it tends to be 100% when the fraction of target items is over a lower limit.
We investigate the phase matching problem in the fixed-point quantum search algorithm proposed by Grover [Phys. Rev. Lett.95 (2005) 150501]. We show that the optimal phase shift is π/3.61, which replaces the π/3 phase shift in fixed-point quantum search algorithm. The π/3.61-phase algorithm can be achieved in [Formula: see text] with the success probability of at least 94.11%, which offsets disadvantage that the success probability of getting correct results usually decreases with the increase of marked items when original Grover quantum search algorithm is applied to search an unordered database. In the meantime, this work also indicates that Grover quantum search algorithm is considerably robust to certain kinds of perturbations and is robust against modest noise in the initialization procedure.
Fixed-point quantum search algorithms succeed at finding one of MMM target items among NNN total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property---the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.
Grover's algorithm achieves a quadratic speedup for unstructured search when a global oracle for the target set is available. However, its success probability is generally below 100%100\%100%. To overcome this issue, Long presented a modified version of Grover's search algorithm by introducing a phase-matching condition, which can search for the target set with zero theoretical failure rate. Nevertheless, in many applications, the target set is determined by the intersection of multiple constrained sets. Most existing Grover-based methods are limited to the intersection of only two sets and are not always successful in searching for the target set. In this work, based on Long's method, we propose an algorithm for exactly searching the intersection of multiple sets. We derive and prove a phase-matching condition for searching the intersection exactly. Furthermore, we present a quantum circuit that implements exact intersection search. Finally, we support our theoretical results with numerical simulations.