The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [PRA 60, 2742 (1999)] is generalized. It is applied to the large class of Grover's type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution is expressed using first order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, P_max. Both T and P_max are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.
In this work I describe a classical analog of Grover's quantum searching algorithm, explaining why a quantum algorithm should be able to perform search in O(sqrtN) steps and also acting as a useful pedagogic demonstration.
It is well-known that when searching one out of four, the original Grover's search algorithm is exact; that is, it succeeds with certainty. It is natural to ask the inverse question: If we are not searching one out of four, is Grover's algorithm definitely not exact? In this article we give a complete answer to this question through some rationality results of trigonometric functions.
We firstly investigate the multipartite entanglement features of the quantum states, including the iteration states achieved by repeated application of Grover iteration and the Oracle ones into which the above iteration states evolve by applying single Oracle operation, employed in Grover's search algorithm by means of the separable degree and the entanglement measure. Then we give the quantitative and qualitative descriptions of the entanglement dynamics in Grover's search algorithm. Our results show that for most instances (i) the separable degrees of these states and ranges of their maximum…
We prove that it is impossible to built a universal quantum machine that produces reflections about an unknown state. We then point out a connection between this result and the optimality of Grover's search algorithm: if such reflection machines were available, it would be possible to accelerate Grover's search algorithm to exponential speedups.
Grover's search algorithm is the optimal quantum algorithm that can search an unstructured database quadratically faster than any known classical algorithm. The role of entanglement and correlations in the search algorithm have been studied in great detail and it is known that entanglement between the qubits is necessary to gain a quadratic speedup, for pure state implementation of the Grover search algorithm. Here, we systematically investigate the behavior of quantum coherence and monogamy of entanglement in the discrete analogue of the $\textit{analog analogue of Grover search algorithm}$. …
We study the entanglement content of the states employed in the Grover algorithm after the first oracle call when a few searched items are concerned. We then construct a link between these initial states and hypergraphs, which provides an illustration of their entanglement properties.
In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We further show that the final state can be expressed in terms of a single oracle operator of the Grover-se…
Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a new depth optimization method for quantum search algorithms. We show that Grover's algorithm is not optimal in depth. We propose a quantum search algorithm, which can be divided into several stages. Each stage has a new initialization, which is a rescaling of the database. This decreases errors. The multistage design is natural for p…
Grover's algorithm is a quantum search algorithm that proceeds by repeated applications of the Grover operator and the Oracle until the state evolves to one of the target states. In the standard version of the algorithm, the Grover operator inverts the sign on only one state. Here we provide an exact solution to the problem of performing Grover's search where the Grover operator inverts the sign on N states. We show the underlying structure in terms of the eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial state to perform the Grover evolution. This allows us to us…
A quantum circuit to construct all maximal cliques using Grover search algorithm is presented. This oracle circuit takes as input an n-qubit state |x> and the adjacency matrix data A of an n-node network, and outputs the state (-1)^f(x) |x> where f(x)=1 if |x> is a maximal clique, and f(x)=0, otherwise. The oracle workspace requires 2n^2 qubits. Each oracle call makes about 10n^2 calls to the Toffoli gates. The overall Grover search algorithm takes as input a uniform superposition n-qubit state and the adjacency matrix data, and outputs a uniform superposition of all maximal clique states.
The present letter proposes a modification in the well known Grover's search algorithm, which searches a database of $N$ unsorted items in $O(\sqrt{N/M})$ steps, where $M$ represents the number of solutions to the search problem. Concurrency control techniques and extra registers for marking and storing the solutions are used in the modified algorithm. This requires additional space but it is shown that the use of extra register and marking techniques can reduce the time complexity to $O(M+\log N)$.
In this work we address two questions concerning Grover's algorithm. In the first we give an answer to the question how to employ Grover's algorithm for actual search over database. We introduce a quantum model of an unordered phone book (quantum database) with programmable queries to search in the phone book either for a number, or for a name. In the second part we investigate how successful the algorithm can be if the number of elements of the database is not known precisely. This question reduces to analysis of the distinguishability of states occurring during Grover's algorithm. We found t…
Comment on quant-ph/0506105: The modified Grover algorithm cannot speedup unsorted database search
在最近的一篇论文（quant-ph/0506105）中，A S Gupta，M. Gupta和A. Pathak提出了一种改进的Grover算法，如果已知标记项的数量，该算法将指数加速未排序数据库搜索问题。如果这是真的，这将是计算机科学、数学、量子信息和其他相关科学分支的重大突破。然而，该算法是无效的。我们将在这个简短的评论中解释它。
Quantum Information Theory promises to speed up computation so is observed in real quantum computers as proved to its classical counterpart. This revolutionizes every field linked directly or indirectly with computation. Grover algorithm in quantum information gives quadratic speed up in unstructured database search. With the availability of public online resources for quantum computers like IBM, quantum image processing came into the picture for making use of quantum computers in the image processing field. Our research interest is to find all darker pixels in a 2x2 grayscale image using Grov…
This is a note accompanying "CS 410/510: INTRO TO QUANTUM COMPUTING" I taught at Portland State University in Spring 2017. It is a review and summary of some early results related to Grover's quantum search algorithm in a consistent way. I had to go back and forth among several books, notes and original papers to sort out various details when preparing the lectures, which was a pain. This is the motivation behind this note. I would like to thank Peter Høyer for valuable feedback on this note.
Grover算法通常用$Q = - H I_{0} H I_{x_0}$形式的复合算子的迭代来描述。虽然验证迭代的代数是非常简单的，但这对算法的工作原理几乎没有帮助。$Q$的化合物结构有什么意义？为什么有个负号？后来发现$H$基本上可以被任何酉$U$所取代。这里涉及的自由是什么？我们给出了一个描述格罗弗的算法，提供了一些澄清这些问题。
Grover's quantum search algorithm provides a quadratic quantum advantage over classical algorithms across a broad class of unstructured search problems. The original protocol is probabilistic, returning the desired result with significant probability on each query, but in general, requiring several iterations of the algorithm. We present a modified version to return the correct result with certainty without having user control over the quantum search oracle. Our deterministic, two-parameter "D2p" protocol utilizes generalized phase rotations replacing the phase inversions after a standard orac…
We point out that the main results-the analytic expressions for the Groverian Measure of Entanglement, in the above mentioned paper are erroneous. The technical mistake of the paper is discussed. It is shown by an explicit example that the formula for calculating the Groverian measure yields G(|ψ>) = 0 for some entangled states.
We present efficient methods to implement the quantum computing Grover search algorithm using the Rydberg blockade interaction. We show that simple pi-pulse excitation sequences between ground and Rydberg excited states readily produce the key conditional phase shift and inversion-about-the mean unitary operations for the Grover search. Multi-qubit implementation schemes suitable for different properties of the atomic interactions are identifed and the error scaling of the protocols with system size is found to be promising for immediate experimental investigation.
The success probability of a search of $M$ targets from a database of size $N$, using Grover's search algorithm depends critically on the number of iterations of the composite operation of the oracle followed by Grover's diffusion operation. Although the required number of iterations scales as $\mathcal{O}(\sqrt{N})$ for large $N$, the asymptote is not a good indicator of the optimal number of iterations when $\sqrt{M/N}$ is not small. A scheme for the determination of the exact number of iterations, subject to a threshold set for the success probability of the search (probability of detecting…
We calculate the amount of entanglement of the multiqubit quantum states employed in the Grover algorithm, by following its dynamics at each step of the computation. We show that genuine multipartite entanglement is always present. Remarkably, the dynamics of any type of entanglement as well as of genuine multipartite entanglement is independent of the number n of qubits for large n, thus exhibiting a scale invariance property. We compare this result with the entanglement dynamics induced by a fixed-point quantum search algorithm. We also investigate criteria for efficient simulatability in th…
In order to understand the bounds of utilization of the Grover's search algorithm for the large unstructured data in presence of the quantum computer noise, we undertake a series of simulations by inflicting various types of noise, modelled by the IBM QISKit. We apply three forms of Grover's algorithms: (1) the standard one, with 4-10 qubits, (2) recently published modified Grover's algorithm, set to reduce the circuit depth, and (3) the algorithms in (1) and (2) with multi-control Toffoli's modified by addition of an ancilla qubit. Based on these simulations, we find the upper bound of noise …
Using Grover algorithm, this work investigates a technique for encoding search phrases used in wildcard searches. The technique involves creating a phase Oracle that loads data into a quantum circuit together with the search terms that have been encoded. The method entails constructing a phase Oracle programmatically using encoded input information and encoded search terms. By combining Grover diffusion operator with the phase Oracle, Hadamard gates, and zero-initialized three-qubit states, a complete quantum circuit is created. Trapped ion and superconducting qubit quantum computers, which we…
