The goals of this paper are to show the following. First, Grover's algorithm can be viewed as a digital approximation to the analog quantum algorithm proposed in "An Analog Analogue of a Digital Quantum Computation", by E. Farhi and S. Gutmann, Phys.Rev. A 57, 2403 - 2406 (1998), quant-ph/9612026. We will call the above analog algorithm the Grover-Farhi-Gutmann or GFG algorithm. Second, the propagator of the GFG algorithm can be written as a sum-over-paths formula and given a sum-over-path interpretation, i.e., a Feynman path sum/integral. We will use nonstandard analysis to do this. Third, in the semi-classical limit $\hbar\to 0$, both the Grover and the GFG algorithms (viewed in the setting of the approximation in this paper) must run instantaneously. Finally, we will end the paper with an open question. In "Semiclassical Shor's Algorithm", by P. Giorda, et al, Phys. Rev.A 70, 032303 (2004), quant-ph/0303037, the authors proposed building semi-classical quantum computers to run Shor's algorithm because the success probability of Shor's algorithm does not change much in the semi-classical limit. We ask the open questions: In the semi-classical limit, does Shor's algorithm have to run instantaneously?
This paper is a written version of a one hour lecture given on Lov Grover's database search algorithm. Table Of Contents 1 ... Problem Definition 2 ... The quantum mechanical perspective 3 ... Properties of the inversion I_|psi> 4 ... The method in Lov's "madness" 5 ... Summary of Grover's algorithm 6 ... An example of Grover's algorithm References
Grover discovered a quantum algorithm for identifying a target element in an unstructured search universe of N items in approximately square-root of N queries to a quantum oracle, thus achieving a square-root speed-up over classical algorithms. We present an information-theoretic analysis of Grover's algorithm and show that the square-root speed-up is the best attainable result using Grover's oracle.
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…
The arguments given in this paper suggest that Grover's and Shor's algorithms are more closely related than one might at first expect. Specifically, we show that Grover's algorithm can be viewed as a quantum algorithm which solves a non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard non-abelian QHS algorithm which is equivalent to Grover's algorithm.
Grover's search algorithm is designed to be executed on a quantum mechanical computer. In this paper, the probabilistic wp-calculus is used to model and reason about Grover's algorithm. It is demonstrated that the calculus provides a rigorous programming notation for modelling this and other quantum algorithms and that it also provides a systematic framework of analysing such algorithms.
I investigate whether it would technologically and economically make sense to build database search engines based on Grover's quantum search algorithm. The answer is not fully conclusive but in my judgement rather negative.
We study the characterisation of efficient and non-efficient families of Grover's algorithms according to the majorization principle. We develop a geometrical interpretation based on the parameters that appears on these algorithms. Using this interpretation we observe a step-by-step majorization in all efficient Grover's algorithms, whereas the non-efficient Grover's algorithms fail to abide by this majorization principle. These Majorization results are first obtained from numerical calculations. Motivated by these numerical results, we also obtained an analytical demonstration. Finally, the g…
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…
It is suggested that the individual outcomes of a measurement process can be understood within standard quantum mechanics in terms of the measuring apparatus, treated as a quantum computer, executing Grover's search algorithm.
In this paper we propose an approach to prepare GHZ states of an arbitrary multi-particle system in terms of Grover's fast quantum searching algorithm. This approach can be regarded as an extension of the Grover's algorithm to find one or more items in an unsorted database.
Generalized Grover's searching algorithm for the case in which there are multiple marked states is demonstrated on a nuclear magnetic resonance (NMR) quantum computer. The entangled basis states (EPR states) are synthesized using the algorithm.
A misunderstanding that an arbitrary phase rotation of the marked state together with the inversion about average operation in Grover's search algorithm can be used to construct a (less efficient) quantum search algorithm is cleared. The $π$ rotation of the phase of the marked state is not only the choice for efficiency, but also vital in Grover's quantum search algorithm. The results also show that Grover's quantum search algorithm is robust.
It has been shown in recent years that quantum information has a topological nature (\cite{AC}, \cite{Co}, \cite{Co2}). In \cite{V}, Vicary undergoes the study of quantum algorithms using this new topological approach. The advantage of this new formalism is that it allows new proofs of correctness of the algorithms such as Deutsch-Jozsa, hidden subgroup and single-shot Grover algorithms. It also provide more clear insight to the generalizations of these algorithms. In this paper, we consider (multi-step) Grover algorithm from this new perspective.
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.
A modification of Grover's algorithm is proposed, which can be used directly as a fast database search. An explicit two q-bit example is displayed in detail. We discuss the case where the database has multiple entries corresponding to the same target value.
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.
By designing a proper unitary operator U, we synthesize NMR analogues of Einstein-Podolsky-Rosen states (pseudo-EPR states) using generalized Grover's algorithm on a nuclear magnetic resonance (NMR) quantum computer. Experiments also demonstrate generalized Grover's algorithm for the case in which there are multiple marked states.
L. K. Grover's search algorithm in quantum computing gives an optimal, quadratic speedup in the search for a single object in a large unsorted database. In this paper, we generalize Grover's algorithm in a Hilbert-space framework for both continuous and discrete time cases that isolates its geometrical essence to the case where more than one object satisfies the search criterion.
We review Grover's algorithm by means of a detailed geometrical interpretation and a worked out example. Some basic concepts of Quantum Mechanics and quantum circuits are also reviewed. This work is intended for non-specialists which have basic knowledge on undergraduate Linear Algebra.
This paper studies a well-known $π$ machine illustrated by Fig.(1). It is shown that the $π$ machine can compute digits of $π$ if the ratio of block weights, $m_2/m_1$, satisfies certain conditions, and that dynamics of the $π$ machine is identical to that of Grover's algorithm [1] in quantum computing.
We review the basic theoretical underpinnings of the Grover algorithm, providing a rigorous and well motivated derivation. We then present a generalization of Grover's algorithm that searches an arbitrary subspace of the multi-dimensional Hilbert space using a diffusion operation and an amplitude amplification procedure that has been biased by unitary {\em steering operators}. We also outline a generalized Grover's algorithm that takes into account higher level correlations that could exist between database elements. In the traditional Grover algorithm, the Hadamard gate selects a uniform samp…
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.
L. K. Grover's search algorithm in quantum computing gives an optimal, square-root speedup in the search for a single object in a large unsorted database. In this paper, we expound Grover's algorithm in a Hilbert-space framework that isolates its geometrical essence, and we generalize it to the case where more than one object satisfies the search criterion.
In this summary we discuss two new algorithms for Grover's unsorted database search problem that claimed to have reached exponential speedup over Grover's original algorithm. One is in the quantum setting with "power queries" that allow for exponential reduction in the number of queries over Grover's original algorithm with "bit queries". The other is to use "dubit queries" on a duality computer - a new computing model uses a quantum system's wave-particle duality, which is able to achieve even greater computing power and better capability than existent quantum computers we have been discussin…
Grover's quantum search algorithm is considered as one of the milestone in the field of quantum computing. The algorithm can search for a single match in a database with $N$ records in $O(\sqrt{N})$ assuming that the item must exist in the database with quadratic speedup over the best known classical algorithm. This review paper discusses the performance of Grover's algorithm in case of multiple matches where the problem is expected to be easier. Unfortunately, we will find that the algorithm will fail for $M>3N/4$, where $M$ is the number of matches in the list.
We analyze three different quantum search algorithms, the traditional Grover's algorithm, its continuous-time analogue by Hamiltonian evolution, and finally the quantum search by local adiabatic evolution. We show that they are closely related algorithms in the sense that they all perform a rotation, at a constant angular velocity, from a uniform superposition of all states to the solution state. This make it possible to implement the last two algorithms by Hamiltonian evolution on a conventional quantum circuit, while keeping the quadratic speedup of Grover's original algorithm.
Grover's search algorithm was originally proposed for circuit-based quantum computers. A crucial part of it is to query an oracle -- a black-box unitary operation. Generation of this oracle is formally beyond the original algorithm design. Here, we propose a realization of Grover's oracle for a large class of searching problems using a quantum annealing step. The time of this step grows only logarithmically with the size of the database. This suggests an efficient path to application of Grover's algorithm to practically important problems, such as finding the ground state of a Hamiltonian with…
Grover's algorithm is usually described in terms of the iteration of a compound operator of the form $Q = - H I_{0} H I_{x_0}$. Although it is quite straightforward to verify the algebra of the iteration, this gives little insight into why the algorithm works. What is the significance of the compound structure of $Q$? Why is there a minus sign? Later it was discovered that $H$ could be replaced by essentially any unitary $U$. What is the freedom involved here? We give a description of Grover's algorithm which provides some clarification of these questions.
In this paper, we give a quantum circuit for calculating the mean value of a function $A(x^n)\in \mathbb{C}$, where $x^n\in \{0,1\}^n$. Known classical algorithms for calculating the mean value of a structureless function $A(x^n)$ take ${\cal O}(2^n)$ steps. Our quantum algorithm is based on a Grover-like algorithm and it takes ${\cal O}(\sqrt{2^n})$ steps. Our algorithm differs significantly from previously proposed quantum algorithms for calculating the mean value of a function via Grover's algorithm.
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…
This paper concerns the Grover algorithm that permits to make amplification of quantum states previously tagged by an Oracle. Grover's algorithm allows searches in an unstructure database of n entries finding a marked element with a quadratic speedup. The algorithm requires a predefined number of runs to succeed with probability close to one.This article provides a description of the amplitude amplification quantum algorithm mechanism in a very short computational way, based on tensor products and provides a geometric presentation of the successive system states. All the basis changes are full…
External environment influences on Grover's search algorithm modeled by quantum noise are investigated. The algorithm is shown to be robust under that external dissipation. Explicitly we prove that the resulting search positive maps acting on unsorted N-dimensional database made of projective density matrices depend on x the strength of the environment, and that there are infinitely many x values for which search is successful after O(\sqrt{N}) queries. These algorithms are quantum entropy increasing.
Dense coding has been implemented using the generalized Grover's algorithm and its inverse operation. Exploiting the superpositions of two Einstein-Podolsky-Rosen (EPR) states, messages that are possible to be transmitted increase. Our scheme is demonstrated using nuclear magnetic resonance (NMR). Experimental results show a good agreement between theory and experiment.
There are major advantages in a newer version of Grover's quantum algorithm utilizing a general unitary transformation in the search of a single object in a large unsorted database. In this paper, we generalize this algorithm to multiobject search. We show the techniques to achieve the reduction of the problem to one on an invariant subspace of dimension just equal to two.
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…
Quantum algorithm is a key tool for cryptanalysis. At present, people are committed to building powerful quantum algorithms and tapping the potential of quantum algorithms, so as to further analyze the security of cryptographic algorithms under quantum computing. Recombining classical quantum algorithms is one of the current ideas to construct quantum algorithms. However, they cannot be easily combined, the feasibility of quantum algorithms needs further analysis in quantum environment. This paper reanalyzes the existing combined algorithm Grover-meets-Simon algorithm in terms of the principle…
Evolution of entanglement with the proceeding of quantum algorithms affects the outcome of the algorithm. Particularly, the performance of Grover's search algorithm gets worsened if the initial state of the algorithm is an entangled one. The success probability of search can be seen as an operational measure of entanglement. This paper demonstrates an entanglement measure based on the performance of Grover's search algorithm for three and five qubit systems. We also show that although the overall pattern shows growth of entanglement, its rise to a maximum and then consequent decay, the presenc…
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 $\textit{analog analogue of Grover search algorithm}$. …
Distributed computing seems to be a natural approach to overcome size limitations of quantum computers in terms of number of qubits. But one lacks an efficient distribution approach to deal systematically with potential algorithms. This work proposes suitable quantum probes to reduce measurements in distributed sub-systems. Probes are introduced in the context of distributed Grover search. These are shown to be indeed efficient. The latter can be generalized to be applicable to other distributed algorithms of interest.
Entanglement plays a crucial role in quantum processes particularly those pertaining to quantum information and computation. An analytical expression for an entanglement measure defined in terms of success rate of Grover's search algorithm has been obtained for qutrit systems with real coefficients and the calculated results agree well with the conventional entropy based measure. The entanglement measure is further generalised for qudit (d-level) systems with real coefficients.
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 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.
In this paper, we give a quantum circuit that calculates symmetrized functions. Our algorithm applies the original Grover's algorithm or a variant thereof such as AFGA (adaptive fixed point Grover's algorithm). Our algorithm uses AFGA in conjunction with two new techniques we call "targeting two hypotheses" and "blind targeting". Suppose AFGA drives the starting state $|s\rangle$ to the target state $|t\rangle$. When targeting two hypotheses, $|t\rangle$ is a superposition $a_0|0\rangle + a_1|1\rangle$ of two orthonormal states or hypotheses $|0\rangle$ and $|1\rangle$. When targeting blindly,…
In a recent paper (quant-ph/0506105), A S Gupta, M. Gupta and A. Pathak proposed a modified Grover algorithm that would exponentially accelerate the unsorted database search problem if the number of marked items is known. If this were true, it would represent a major fundamental breakthrough in computer science, mathematics, quantum information and other related branches of sciences. However the algorithm is not valid. We will explain it in this brief comment.
In this article we consider an experimental study showing the influence of emotion regulation strategies on human memory performance: part of such experimental results are difficult to explain within a classic cognitive allocation model. We provide a first attempt to build a model of human memory processes based on a quantum algorithm, the Grover's algorithm, which allows to search a particular item within an unsorted set of items more efficiently than any classic search algorithm. Based on Grover's algorithm paradigm, this new memory model results to have interesting features: it is an iterat…
We simulate Grover's algorithm in a classical computer by means of a stochastic method using the Hubbard-Stratonovich decomposition of n-qubit gates into one-qubit gates integrated over auxiliary fields. The problem reduces to finding the fixed points of the associated system of Langevin differential equations. The equations are obtained automatically for any number of qubits by employing a computer algebra program. We present the numerical results of the simulation for a small search space.
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.
Grover's search algorithm searches a database of $N$ unsorted items in $O(\sqrt{N/M})$ steps where $M$ represents the number of solutions to the search problem. This paper proposes a scheme for searching a database of $N$ unsorted items in $O(logN)$ steps, provided the value of $M$ is known. It is also shown that when $M$ is unknown but if we can estimate an upper bound of possible values of $M$, then an improvement in the time complexity of conventional Grover's algorithm is possible. In that case, the present scheme reduces the time complexity to $O(MlogN)$.
We analyze the role played by entanglement in the dynamical evolution of Grover's search algorithm in the space of qubits. We show that the algorithm can be equivalently described as an iterative change of the entanglement between the qubits, which governs the evolution of the initial state towards the target state, and where the entanglement can be quantified in terms a single entanglement monotone. We also provide a necessary and sufficient condition for the quadratic speedup, which illustrates how the change in the bipartite entanglement of the state after each iteration determines the corr…
We investigate the necessary and sufficient conditions in order that a unitary operator can amplify a pre-assigned component relative to a particular basis of a generic vector at the expense of the other components. This leads to a general method which allows, given a vector and one of its components we want to amplify, to choose the optimal unitary operator which realizes that goal. Grover's quantum algorithm is shown to be a particular case of our general method. However the general structure of the unitary we find is remarkably similar to that of Grover's one: a sign flip of one component c…
