Quite often in database search, we only need to extract portion of the information about the satisfying item. Recently Radhakrishnan & Grover [RG] considered this problem in the following form: the database of $N$ items was divided into $K$ equally sized blocks. The algorithm has just to find the block containing the item of interest. The queries are exactly the same as in the standard database search problem. [RG] invented a quantum algorithm for this problem of partial search that took about $0.33\sqrt{N/K}$ fewer iterations than the quantum search algorithm. They also proved that the best any quantum algorithm could do would be to save $0.78 \sqrt(N/K)$ iterations. The main limitation of the algorithm was that it involved complicated analysis as a result of which it has been inaccessible to most of the community. This paper gives a simple analysis of the algorithm. This analysis is based on three elementary observations about quantum search, does not require a single equation and takes less than 2 pages.
Partial search has been proposed recently for finding the target block containing a target element with fewer queries than the full Grover search algorithm which can locate the target precisely. Since such partial searches will likely be used as subroutines for larger algorithms their success rate is important. We propose a partial search algorithm which achieves success with unit probability.
Quantum algorithm can find target item in a database faster than any classical. 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. One can think of partial search in following terms: an exact address of the target item is given by a sequence of many bites, but we want to find only several first bites of the address. We consider different partial search algorithms consisting of a sequence of global and local searches and suggest the optimal one. Global search consists of the standard Grover iterations for th…
It is questionable that Grover algorithm may be more valuable than a classical one, when a partial information is given in a unstructured database. In this letter, to consider quantum search when a partial information is given, we replace the Fourier transform in the Grover algorithm with the Haar wavelet transform. We then, given a partial information $L$ to a unstructured database of size $N$, show that there is the improved speedup, $O(\sqrt{N/L})$.
In this paper, we will use a quantum operator which performs the inversion about the mean operation only on a subspace of the system ({\it Partial Diffusion Operator}) to propose a quantum search algorithm runs in $O(\sqrt N/M})$ for searching unstructured list of size $N$ with $M$ matches such that, $1\le M \le N$. We will show that the performance of the algorithm is more reliable than known quantum search algorithms especially for multiple matches within the search space. A performance comparison with Grover's algorithm will be provided.
We consider a database separated into blocks. Blocks containing target items are called target blocks. Blocks without target items are called non-target blocks. We consider a case, when each target block has the same number of target items. We present a fast quantum algorithm, which finds one of the target blocks. Our algorithm is based on Grover-Radhakrishnan algorithm of partial search. We minimize the number of queries to the oracle.
We investigate the generalisation of quantum search of unstructured and totally ordered sets to search of partially ordered sets (posets). Two models for poset search are considered. In both models, we show that quantum algorithms can achieve at most a quadratic improvement in query complexity over classical algorithms, up to logarithmic factors; we also give quantum algorithms that almost achieve this optimal reduction in complexity. In one model, we give an improved quantum algorithm for searching forest-like posets; in the other, we give an optimal O(sqrt(m))-query quantum algorithm for sea…
Consider the unstructured search of an unknown number l of items in a large unsorted database of size N. The multi-object quantum search algorithm consists of two parts. The first part of the algorithm is to generalize Grover's single-object search algorithm to the multi-object case and the second part is to solve a counting problem to determine l. In this paper, we study the multi-object quantum search algorithm (in continuous time), but in a more structured way by taking into account the availability of partial information. The modeling of available partial information is done simply by the …
The partial adiabatic search algorithm was introduced in [A. Tulsi, Phys. Rev. A 80, 052328 (2009)] as a modification of the usual adiabatic algorithm for quantum search with the idea that most of the interesting computation only happens over a very short range of the adiabatic path. By focussing on that restricted range, one can potentially gain advantage by reducing the control requirements on the system, enabling a uniform rate of evolution. In this comment, we point out an oversight in the original work [1] that invalidates its proof. However, the argument can be corrected, and the calcula…
In this paper, we will define a quantum operator that performs the inversion about the mean only on a subspace of the system (Partial Diffusion Operator). This operator is used in a quantum search algorithm that runs in O(sqrt{N/M}) for searching an unstructured list of size N with M matches such that 1<= M<=N. We will show that the performance of the algorithm is more reliable than known {fixed operators quantum search algorithms} especially for multiple matches where we can get a solution after a single iteration with probability over 90% if the number of matches is approximately more than o…
A quantum search algorithm based on the partial adiabatic evolution\cite{Tulsi2009} is provided. We calculate its time complexity by studying the Hamiltonian in a two-dimensional Hilbert space. It is found that the algorithm improves the time complexity, which is $O(\sqrt{N/M})$, of the local adiabatic search algorithm\cite{Roland2002}, to $O(\sqrt{N}/M)$.
In this paper, we consider the partial database search problem where given a database on N items, we are required to determine the first k bits of an address x such that f(x)=1. We derive an algorithm and a lower bound for this problem in the quantum circuits model. Let q(k,N) be the minimum number of queries needed to find the first k bits of the required address x. We show that there exist constants c_k and d_k such that (pi/4) (1 - d_k/sqrt{K}) sqrt{N} <= q(k,n) <= (pi/4) (1 - c_k/sqrt{K}) sqrt{N}, where K=2^k. Thus, it is always easier to determine a few bits of the target address than to …
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 number of queries to the oracle. The author suggests new version of Grover-Radhakrishnan algorithm which uses minimal number of queries to the oracle. The algorithm can run on the same hardware which is us…
Quantum partial search algorithm is approximate search. It aims to find a target block (which has the target items). It runs a little faster than full Grover search. In this paper, we consider quantum partial search algorithm for multiple target items unevenly distributed in database (target blocks have different number of target items). The algorithm we describe can locate one of the target blocks. Efficiency of the algorithm is measured by number of queries to the oracle. We optimize the algorithm in order to improve efficiency. By perturbation method, we find that the algorithm runs the fas…
Searching and sorting used as a subroutine in many important algorithms. Quantum 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. An example is the following: exact address of the target item is given by a sequence of many bits, but we need to know only some of them. More generally partial search considers the following problem: a database is separated into several blocks. We want to find a block with the target item, not …
Database search has wide applications and is used as a subroutine in many important algorithms. We shall consider a database with one target item. Quantum algorithm finds the target item in a database faster than any classical algorithm. It frequently occurs in practice that only a portion of information about the target item is interesting, or we need to find a group of items sharing some common feature as the target item. This problem is in general formulated as search for a part of the database [a block] containing the target item, instead of the item itself. This is partial search. Partial…
