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Acta Sinica Quantum Optica

Quantum Partial Search of a Database with Several Target Items——All the target itemsare asymmetrically divided into two target block

Guo Guang-Can

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

By quantum Grover search algorithm,people can find a target item in a database faster than any classical algorithm.In this search algorithm the initial state is the equally weighted superposition of all basis states,which is operated by the unitary operator repeatedly.Consequently,non-target states transfer their amplitude to the target state,so we can find the target item with larger probability.However,sometimes we just want to know partial information,so we only need to find the subdatabase(a block) containing the target item,which is partial search.A partial search algorithm was suggested by Grover and Radhakrishnan,which is denoted as Grover-Radhakrishnan Algorithm of Partial Search(GRK),in which they considered the database contains one target item.In our paper,we study the database which contains several target items,and they are randomly distributed in two target blocks.Our algorithm is based on GRK.We minimize the number of queries to the oracle for blocks of large size,and compare this case with the one that each target block contains the same number of target items.

Related papers

HIERARCHICAL QUANTUM SEARCH

V. E. Korepin, YING XU

Database search has wide applications and is used as a subroutine in many important algorithms. In this paper, we will consider a database with a single target item. Quantum algorithm (Grover) locates the target item faster than any classical algorithm. In addition to a full (Grover) search, it frequently occurs that one is looking for a group of items (a block) containing the target item, rather than the target item itself. This problem is known as partial search. As a generalization of the full search, partial search is of particular importance in practice. Partial search trades accuracy for speed, i.e., it works faster than a full search. There exists different versions of partial search. We will study the optimized version of the algorithm discovered by Grover and Radhakrishnan and call it GRK. GRK can be applied successively (in a sequence). First, the database is partitioned into blocks and GRK is applied to find the target block. This target block is then partitioned into subblocks and GRK is used again to find the target subblock. This procedure can be repeated if the database is large enough. (This sequence of GRK's is called a hierarchy.) Another possibility is to partition the database into subblocks directly and use GRK to find the target subblock once. In this paper, we will prove that the latter is faster (makes less queries to the Oracle).

International Journal of Modern Physics B•Dec 20, 2007•10 citations

QUANTUM SEARCH ALGORITHMS

V. E. Korepin, YING XU

This article reviews recent progress in quantum database search algorithms. The subject is presented in a self-contained and pedagogical way. The problem of searching a large database (a Hilbert space) for a target item is performed by the famous Grover algorithm which locates the target item with high probability and a quadratic speed-up compared with the corresponding classical algorithm. If the database is partitioned into blocks and one is searching for the block containing the target item instead of the target item itself, then the problem is referred to as partial search. Partial search trades accuracy for speed and the most efficient version is the Grover–Radhakrishnan–Korepin (GRK) algorithm. The target block can be further partitioned into sub-blocks so that GRK's can be performed in a sequence called a hierarchy. We study the Grover search and GRK partial search in detail and prove that a GRK hierarchy is less efficient than a direct GRK partial search. Both the Grover search and the GRK partial search can be generalized to the case with several target items (or target blocks for a GRK). The GRK partial search algorithm can also be represented in terms of group theory.

International Journal of Modern Physics B•Dec 20, 2009•7 citations

Optimization of partial search

V. E. Korepin

A quantum Grover search algorithm can find a target item in a database faster than any classical algorithm. One can trade accuracy for speed and find a part of the database (a block) containing the target item even faster; this is partial search. A partial search algorithm was recently suggested by Grover and Radhakrishnan. Here we optimize it. Efficiency of the search algorithm is measured by the number of queries to the oracle. The author suggests a new version of the Grover–Radhakrishnan algorithm which uses a minimal number of such queries. The algorithm can run on the same hardware that is used for the usual Grover algorithm.

arXiv:quant-ph/0503238•Journal of Physics A Mathematical and General•Oct 19, 2005•27 citations

Asymptotic optimality of Grover-Radhakrishnan-Korepin algorithm

Kun Zhang, Kang-Yuan Chen, Xiao-Hui Wang +1 more

Grover's algorithm is a cornerstone of quantum algorithms and is strictly optimal in oracle-query complexity. While the full search problem admits no further improvement, one may trade accuracy for speed in the partial search problem, where the task is to identify only the block containing the target item. The best known quantum algorithm for the partial search problem is the Grover-Radhakrishnan-Korepin (GRK) algorithm, whose optimality has long been conjectured but not proved. In this work, we prove the optimality of GRK in the large-block limit. We formulate partial search as a time-optimal control problem and apply the Pontryagin maximum principle to derive the switching-function dynamics, establish the bang-bang structure of regular extremals, and exclude non-optimal switching patterns. As a result, we show that the optimal regular extremal has the global-local-global form, which yields a control-theoretic proof of the asymptotic optimality of the GRK algorithm in oracle-query complexity.

quant-pharXiv:2604.15886•Journal of Physics A Mathematical and Theoretical•Apr 20, 2026•0 citations

Quantum Partial Searching Algorithm of a Database with Several Target Items

Zhong Pu-Cha, Wan‐Su Bao, Wei-Yun Ma

Choi and Korepin [Quantum Information Processing 6(2007)243] presented a quantum partial search algorithm of a database with several target items which can find a target block quickly when each target block contains the same number of target items. Actually, the number of target items in each target block is arbitrary. Aiming at this case, we give a condition to guarantee performance of the partial search algorithm to be performed and the number of queries to oracle of the algorithm to be minimized. In addition, by further numerical computing we come to the conclusion that the more uniform the distribution of target items, the smaller the number of queries.

Chinese Physics Letters•Feb 1, 2009•1 citation

Asymptotic bounds on quantum partial search algorithm and its applications to parallel search

Yan-Bo Jiang, Xiao-Hui Wang, Kun Zhang +1 more

Grover's algorithm provides a quadratic speedup over classical algorithms for searching an unstructured database and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is the standard and most extensively studied protocol for this task. In this work, we provide systematic numerical evidence that the GRK operator sequence gives the highest success probability in all examined cases, supporting it as the optimal ansatz among admissible compositions of global and local Grover operators. Guided by this numerically supported GRK ansatz, we derive an asymptotically tight upper bound on the maximal success probability within the GRK family and establish the corresponding lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols yields a strict, though subleading, improvement over the outer parallel Grover scheme. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms.

quant-pharXiv:2603.01462•Physical Review A•Mar 3, 2026•1 citation
By quantum Grover search algorithm,people can find a target item in a database faster than any classical algorithm.In this search algorithm the initial state is the equally weighted superposition of all basis states,which is operated by the unitary operator repeatedly.Consequently,non-target states transfer their amplitude to the target state,so we can find the target item with larger probability.However,sometimes we just want to know partial information,so we only need to find the subdatabase(a block) containing the target item,which is partial search.A partial search algorithm was suggested by Grover and Radhakrishnan,which is denoted as Grover-Radhakrishnan Algorithm of Partial Search(GRK),in which they considered the database contains one target item.In our paper,we study the database which contains several target items,and they are randomly distributed in two target blocks.Our algorithm is based on GRK.We minimize the number of queries to the oracle for blocks of large size,and compare this case with the one that each target block contains the same number of target items.
Database search has wide applications and is used as a subroutine in many important algorithms. In this paper, we will consider a database with a single target item. Quantum algorithm (Grover) locates the target item faster than any classical algorithm. In addition to a full (Grover) search, it frequently occurs that one is looking for a group of items (a block) containing the target item, rather than the target item itself. This problem is known as partial search. As a generalization of the full search, partial search is of particular importance in practice. Partial search trades accuracy for speed, i.e., it works faster than a full search. There exists different versions of partial search. We will study the optimized version of the algorithm discovered by Grover and Radhakrishnan and call it GRK. GRK can be applied successively (in a sequence). First, the database is partitioned into blocks and GRK is applied to find the target block. This target block is then partitioned into subblocks and GRK is used again to find the target subblock. This procedure can be repeated if the database is large enough. (This sequence of GRK's is called a hierarchy.) Another possibility is to partition the database into subblocks directly and use GRK to find the target subblock once. In this paper, we will prove that the latter is faster (makes less queries to the Oracle).
This article reviews recent progress in quantum database search algorithms. The subject is presented in a self-contained and pedagogical way. The problem of searching a large database (a Hilbert space) for a target item is performed by the famous Grover algorithm which locates the target item with high probability and a quadratic speed-up compared with the corresponding classical algorithm. If the database is partitioned into blocks and one is searching for the block containing the target item instead of the target item itself, then the problem is referred to as partial search. Partial search trades accuracy for speed and the most efficient version is the Grover–Radhakrishnan–Korepin (GRK) algorithm. The target block can be further partitioned into sub-blocks so that GRK's can be performed in a sequence called a hierarchy. We study the Grover search and GRK partial search in detail and prove that a GRK hierarchy is less efficient than a direct GRK partial search. Both the Grover search and the GRK partial search can be generalized to the case with several target items (or target blocks for a GRK). The GRK partial search algorithm can also be represented in terms of group theory.
A quantum Grover search algorithm can find a target item in a database faster than any classical algorithm. One can trade accuracy for speed and find a part of the database (a block) containing the target item even faster; this is partial search. A partial search algorithm was recently suggested by Grover and Radhakrishnan. Here we optimize it. Efficiency of the search algorithm is measured by the number of queries to the oracle. The author suggests a new version of the Grover–Radhakrishnan algorithm which uses a minimal number of such queries. The algorithm can run on the same hardware that is used for the usual Grover algorithm.
Grover's algorithm is a cornerstone of quantum algorithms and is strictly optimal in oracle-query complexity. While the full search problem admits no further improvement, one may trade accuracy for speed in the partial search problem, where the task is to identify only the block containing the target item. The best known quantum algorithm for the partial search problem is the Grover-Radhakrishnan-Korepin (GRK) algorithm, whose optimality has long been conjectured but not proved. In this work, we prove the optimality of GRK in the large-block limit. We formulate partial search as a time-optimal control problem and apply the Pontryagin maximum principle to derive the switching-function dynamics, establish the bang-bang structure of regular extremals, and exclude non-optimal switching patterns. As a result, we show that the optimal regular extremal has the global-local-global form, which yields a control-theoretic proof of the asymptotic optimality of the GRK algorithm in oracle-query complexity.
Choi and Korepin [Quantum Information Processing 6(2007)243] presented a quantum partial search algorithm of a database with several target items which can find a target block quickly when each target block contains the same number of target items. Actually, the number of target items in each target block is arbitrary. Aiming at this case, we give a condition to guarantee performance of the partial search algorithm to be performed and the number of queries to oracle of the algorithm to be minimized. In addition, by further numerical computing we come to the conclusion that the more uniform the distribution of target items, the smaller the number of queries.
Grover's algorithm provides a quadratic speedup over classical algorithms for searching an unstructured database and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is the standard and most extensively studied protocol for this task. In this work, we provide systematic numerical evidence that the GRK operator sequence gives the highest success probability in all examined cases, supporting it as the optimal ansatz among admissible compositions of global and local Grover operators. Guided by this numerically supported GRK ansatz, we derive an asymptotically tight upper bound on the maximal success probability within the GRK family and establish the corresponding lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols yields a strict, though subleading, improvement over the outer parallel Grover scheme. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms.