Linear combination of unitaries (LCU for short) is one of the most important techniques in designing quantum algorithms. In this paper, we propose a new quantum algorithm in three different forms to achieve LCU. Different from previous algorithms [Childs-linear-system,Clader,Long11], the complexity now only depends on the number of the unitaries and the precision. So it will play more important role in the design of quantum algorithms when the number of unitaries is small, such as quantum iteration algorithms. %Since in iteration algorithms, $m$ often refers to the iteration steps, which is small if the iteration algorithms are efficient. Moreover, as an application of the new LCU, three new quantum algorithms to the searching problem will be proposed, which will provide us new insights into Grover's searching algorithm. We also show that the problem of LCU is closely related to the problem of if we can efficiently implement $U^t$ for $0<t<1$ when $U$ is an efficiently implemented unitary operator? This problem is not hard to solve. However, it becomes inefficient when it contains a strict requirement on precision, such as in Grover's algorithm. Finally, as an application of the new LCU technique, we will show that the quantum state of any real classical vector can be prepared efficiently in quantum computer. So this solves the "input problem" in quantum computer efficiently.
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.
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.
I improve the tight bound on quantum searching by Boyer et al. (quant-ph/9605034) to a matching bound, thus showing that for any probability of success Grovers quantum searching algorithm is optimal. E.g. for near certain success we have to query the oracle pi/4 sqrt{N} times, where N is the size of the search space. I also show that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers. Earlier results left open the possibility of a more efficient parallelization.
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.
We introduce the concepts of Grover operators and Grover kernels to systematically analyse Grover's searching algorithms. Then, we investigate a one-parameter family of quantum searching algorithms of Grover's type and we show that the standard Grover's algorithm is a distinguished member of this family. We show that all the algorithms of this class solve the searching problem with an efficiency of order $O(\sqrt{N})$, with a coefficient which is class-dependent. The analysis of this dependence is a test of the stability and robustness of the algorithms. We show the stability of this construct…
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 studying the attribute of the inversion about average operation in quantum searching algorithm, we find the similarity between the quantum searching and the course of two rigid bodies'collision. Some related questions are discussed from this similarity.
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.
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.
Grover search algorithm drives a quantum system from an initial state to a desired final state by using selective phase inversions of these two states. In (1), we studied a generalization of Grover algorithm which relaxes the assumption of the efficient implementation of the selective phase inversion of the initial state, also known as diffusion operator. This assumption is known to become a serious handicap in cases of physical interest (2,3,4,5). Our general search algorithm works with almost arbitrary diffusion operator with only restriction of having the initial state as one of its eigenst…
We provide a tight analysis of Grover's recent algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine the number of iterations necessary to achieve almost certainty of finding the answer. Furthermore, we analyse the behaviour of the algorithm when the element to be found appears more than once in the table and we provide a new algorithm to find such an element even when the number of solutions is not known ahead of time. Using techniques from Shor's quantum …
Exact quantum algorithm for searching of multiple sets intersection
School of Mathematical Sciences, Hangzhou Dianzi University（杭州电子科技大学数学科学学院）；School of Computer and Computing Science, Hangzhou City University（杭州城市大学计算机与计算科学学院）
We investigate the issue of speed-up and the necessity of entanglement in Grover's quantum search algorithm. We find that in a pure state implementation of Grover's algorithm entanglement is present even though the initial and target states are product states. In pseudo-pure state implementations, the separability of the states involved defines an entanglement boundary in terms of a bound on the purity parameter. Using this bound we investigate the necessity of entanglement in quantum searching for these pseudo-pure state implementations. If every active molecule involved in the ensemble is `c…
We study some extensions of Grover's quantum searching algorithm. First, we generalize the Grover iteration in the light of a concept called amplitude amplification. Then, we show that the quadratic speedup obtained by the quantum searching algorithm over classical brute force can still be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform approximate counting, which can be seen as an amplitude estimation process.
Given two sets A and B and two oracles O(A) and O(B) that can identify the elements of these sets respectively, the goal is to find an element common to both sets using minimum number of oracle queries. Each application of either O(A) or O(B) is taken as a single oracle query. This is basically a search problem and a straightforward application of Grover algorithm can solve this problem but its performance is slow compared to the optimal one by a constant factor of 1.57. Here we present a variant of Grover algorithm which achieves the optimal performance in not too restrictive cases.
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…
One of the most important quantum algorithms ever discovered is Grover's algorithm for searching an unordered set. We give a new lower bound in the query model which proves that Grover's algorithm is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound.
We discuss the quantum search algorithm using complex queries that has recently been published by Grover (quant-ph/9706005). We recall the algorithm adding some details showing which complex query has to be evaluated. Based on this version of the algorithm we discuss its complexity.
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 present a scheme to prepare a quantum state in a ion trap with probability approaching to one by means of ion trap quantum computing and Grover's quantum search algorithm acting on trapped ions.
We describe an implementation of Grover's fixed-point quantum search algorithm on a nuclear magnetic resonance (NMR) quantum computer, searching for either one or two matching items in an unsorted database of four items. In this new algorithm the target state (an equally weighted superposition of the matching states) is a fixed point of the recursive search operator, and so the algorithm always moves towards the desired state. The effects of systematic errors in the implementation are briefly explored.
In standard Grover's algorithm for quantum searching, the probability of finding the marked item is not exactly 1. In this Letter we present a modified version of Grover's algorithm that searches a marked state with full successful rate. The modification is done by replacing the phase inversion by two phase rotation through angle $ϕ$. The rotation angle is given analytically to be $ϕ=2 \arcsin(\sin{π\over (4J+6)}\over \sinβ)$, where $\sinβ={1\over \sqrt{N}}$, $N$ the number of items in the database, and $J$ an integer equal to or greater than the integer part of $({π\over 2}-β)/(2β)$. Upon mea…
The essential operations of a quantum computer can be accomplished using solely optical elements, with different polarization or spatial modes representing the individual qubits. We present a simple all-optical implementation of Grover's algorithm for efficient searching, in which a database of four elements is searched with a single query. By `compiling' the actual setup, we have reduced the required number of optical elements from 24 to only 12. We discuss the extension to larger databases, and the limitations of these techniques.
The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of N items requires Ω(N) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in O{\sqrt{N}} queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alte…
Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schrödinger's equation are represented as vector fields. In view of optim…
This article highlights some of the key operating principles of Grover algorithm. These principles were used to develop a new oracle function, that illustrates the possibility of using Grover algorithm for solving more realistic and specific search problems, like searching for a solution to a simple mathematical equation.
Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. A practicable necessary condition well-suited for many-body systems has been proven recently in [Phys. Rev. Lett. 119, 200401 (2017)]. Here we tailor this condition for estimating run times of quantum adiabatic algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that thus obtained lower bound …
Grover's algorithm for quantum searching is generalized to deal with arbitrary initial complex amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the marked and unmarked states. These equations are solved exactly. New expressions are derived for the optimal time of measurement and the maximal probability of success. They 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. Our results imply that Grover's algorithm is robust agains…
As the engineering endeavour to realise quantum computers progresses, we consider that such machines need not rely on binary as their de facto unit of information. We investigate Grover's algorithm under a generalised quantum circuit model, in which the information and transformations can be expressed in any arity, and analyse the structural and behavioural properties while preserving the semantics; namely, searching for the unique preimage to an output a function. We conclude by demonstrating that the generalised procedure retains $O(\sqrt{N})$ time complexity.
Withdrawn by the author due to irreparable errors. We present a quantum algorithm that in the black-box model performs a search in an ordered list of N elements. Using 3/4 log N + O(1) queries, it achieves a success probability of at least 1/2, whereas classically, log N - O(1) queries are needed to obtain constant success probability. Moreover, our algorithm employs the Haar transform and thus differs substantially from Grover's search algorithm and from algorithms relying on the quantum Fourier transform.
In this paper we give a quantum mechanical algorithm that can search a database by a single query, when the number of solutions is more than a quarter. It utilizes modified Grover operator of arbitrary phase.
Grover's algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N-r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T \sim O(\sqrt{N/r}) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial a…
An SO(3) picture of the generalized Grover's quantum searching algorithm,with arbitrary unitary transformation and with arbitrary phase rotations, is constructed. In this picture, any quantum search operation is a rotation in a 3 dimensional space. Exact formulas for the rotation angle and rotational axis are given. The probability of finding the marked state is just $(z+1)/2$, where z is the z-component of the state vector. Exact formulas for this probability is easily obtained. The phase matching requirement and the failure of algorithm when phase mismatches are clearly explained.
A fast quantum search algorithm for continuous variables is presented. The result is the quantum continuous variable analog of Grover's algorithm originally proposed for qubits. A continuous variable analog of the Hadamard (i.e., Fourier transform) operation is used in conjunction with inversion about the average of quantum states to allow the approximate identification of an unknown quantum state in a way that gives a square-root speed-up over search algorithms using classical continuous variables. Also, we show that this quantum search algorithm is robust for a generalised Fourier transforma…
We demonstrated the detailed construction of the hybrid quantum-classical computer. Based on this architecture, the useful concept of amplitude interception is illustrated. It is then embedded into a combination of Depth-First Search and Grover's algorithm to generate a novel approach, the Depth-First Grover Search(DFGS), to handle multi-solution searching problems on unstructured databases with an unknown number of solutions. Our new algorithm attains an average complexity of $\mathcal{O}(m\sqrt{N})$ which performs as efficient as a normal Grover Search, and a $\mathcal{O}(\sqrt{p}N)$ complex…
Each iteration in Grover's original quantum search algorithm contains 4 steps: two Hadamard-Walsh transformations and two amplitudes inversions. When the inversion of the marked state is replaced by arbitrary phase rotation θand the inversion for the prepared state |γ> is replaced by rotation through ϕ, we found that these phase rotations must satisfy a matching condition θ=ϕ. Approximate formula for the amplitude of the marked state after an arbitrary number of iterations are also derived. We give also a simple explanation of the phase matching requirement.
A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To serach a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a phase angle in its amplitude of unity modulus. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel is also studied.
We present an information geometric characterization of Grover's quantum search algorithm. First, we quantify the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric. We then show that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover's algorithm. We also discuss …
We mainly study exceptional configuration for coined quantum walk search. For searching on a two-dimensional grid by AKR algorithm, we find some new classes of exceptional configurations that cannot be found by the AKR algorithm effectively and the known diagonal configuration can be regarded as its special case. Meanwhile, we give two modified quantum walk models that can improve the success probability in the exceptional configurations by numerical simulation. Furthermore, we introduce the concept of generalized exceptional configuration and consider search by quantum walk on a cycle with Gr…
Decoherence in quantum searches, and in the Grover search in particular, has already been extensively studied, leading very quickly to the loss of the quadratic speedup over the classical case, when searching for some target (marked) element within a set of size $N$. The noise models used were, however, global. In this paper we study Grover search under the influence of localized partially dephasing noise of rate $p$. We find, that in the case when the size $k$ of the affected subspace is much smaller than $N$, and the target is unaffected by the noise, namely when $kp\ll\sqrt{N}$, the quadrat…
An important and usual problem is to search all states we want from a database with a large number of states. In such, recall is vital. Grover's original quantum search algorithm has been generalized to the case of multiple solutions, but no one has calculated the query complexity in this case. We will use a generalized algorithm with higher precision to solve such a search problem that we should find all marked states and show that the practical query complexity increases with the number of marked states. In the end we will introduce an algorithm for the problem on a ``duality computer'' and …
This paper presented two general quantum search algorithms. We derived the iterated formulas and the simpler approximate formulas and the precise formula for the amplitude in the desired state. A mathematical proof of Grover's algorithm being optimal among the algorithms with arbitrary phase rotations was given in this paper. This first reported the non-symmetric effects of different rotating angles, and gave the first-order approximate phase condition when rotating angles are different.
Grover's algorithm is normally presented as a method of searching a database, however it would be more accurately described as a method of identifying elements of an interval of the integers which satisfy some logical clause - an example might be identifying binary strings which correspond to the solutions of a Sudoku problem. In this paper we present the first method of performing a true database search using Grover's search algorithm, by first creating a mapping from a set of indices in the range 0:2^n-1 to a set of database elements, then applying the clause to these elements. We then demon…
We present new results on an optical implementation of Grover's quantum search algorithm. This extends previous work in which the transverse spatial mode of a light beam oscillates between a broad initial input shape and a highly localized spike, which reveals the position of the tagged item. The spike reaches its maximum intensity after $\sim\sqrt N$ round trips in a cavity equipped with two phase plates, where $N$ is the ratio of the surface area of the original beam and the area of the phase spot or tagged item. In our redesigned experiment the search space is now two-dimensional. In the ti…
The search operation for a marked state by means of Grover's quantum searching algorithm is shown to be an element of group SU(2) which acts on a 2-dimensional space spanned by the marked state and the unmarked collective state. Based on this underlying structure, those exact bounds of the steps in various quantum search algorithms are obtained in a quite concise way. This reformulation of the quantum searching algorithm also enables a detailed analysis of the decoherence effects caused by its coupling with an environment. It turns out that the environment will not only make the quantum search…
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…
Quantum algorithms use the principles of quantum mechanics, as for example quantum superposition, in order to solve particular problems outperforming standard computation. They are developed for cryptography, searching, optimisation, simulation and solving large systems of linear equations. Here, we implement Grover's quantum algorithm, proposed to find an element in an unsorted list, using a single nuclear 3/2-spin carried by a Single Molecular Magnet (SMM) transistor. The coherent manipulation of this multi-level qudit is achieved by means of electric fields only. Grover's search algorithm w…
The operations of data set, such as intersection, union and complement, are the fundamental calculation in mathematics. It's very significant that designing fast algorithm for set operation. In this paper, the quantum algorithm for intersection is presented. And its running time is sqrt(|A|*|B|*|C|) for set operation C = A intersection B, while classical computation needs O (|A| *|B|) steps of computation in general, where |.| denotes the size of set. The presented algorithm is the combination of Grover's algorithm, classical memory and classical iterative computation, and the combination meth…
Search-base algorithms have widespread applications in different scenarios. Grover's quantum search algorithms and its generalization, amplitude amplification, provide a quadratic speedup over classical search algorithms for unstructured search. We consider the problem of searching with prior knowledge. More preciously, search for the solution among N items with a prior probability distribution. This letter proposes a new generalization of Grover's search algorithm which performs better than the standard Grover algorithm in average under this setting. We prove that our new algorithm achieves t…
