An attempt has been made in this paper to modify Grover's Algorithm to find the binary string solutions approximating a target cost value. In that direction, new Controlled Oracle and the Local Diffusion Operator are suggested, apart from incorporating suitable ancilla qubits. A possible strategy to estimate eigenvalues and eigenstates of a given cost Hamiltonian, extending the reasoning of the methodology, is also pointed out. Typical results and relevant discussions are captured to support the propositions.
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.
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 $O(M+\log N)$.
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 $O(MlogN)$.
In this paper, we propose a quantum amplitude estimation method that uses a modified Grover operator and quadratically improves the estimation accuracy in the ideal case, as in the conventional one using the standard Grover operator. Under the depolarizing noise, the proposed method can outperform the conventional one in the sense that it can in principle achieve the ultimate estimation accuracy characterized by the quantum Fisher information in the limit of a large number of qubits, while the conventional one cannot achieve the same value of ultimate accuracy. In general this superiority requ…
The translation of Grover's search algorithm from its standard version, designed for implementation on a single quantum system amenable to projective measurements, into one suitable for an ensemble of quantum computers, whose outputs are expectation values of observables, is described in detail. A filtering scheme, which effectively determines expectation values on a limited portion of the quantum state, is presented and used to locate a single item for searches involving more than one marked item. A truncated version of Grover's algorithm, requiring fewer steps than the translated standard ve…
By allowing measurements of observables other than the state of the qubits in a quantum computer, one can find eigenvectors very quickly. If a unitary operation U is implemented as a time-independent Hamiltonian, for instance, one can collapse the state of the computer to a nearby eigenvector of U with a measurement of the energy. We examine some recent proposals for quantum computation using time-independent Hamiltonians and show how to convert them into ``artificial orbitals'' whose energy eigenstates match those of U. This system can be used to find eigenvectors and eigenvalues with a singl…
A new notion of controllability, eigenstate controllability, is defined for finite-dimensional bilinear quantum mechanical systems which are neither strongly completely controllably nor completely controllable. And a quantum control algorithm based on Grover iteration is designed to perform a quantum control task of steering a system, which is eigenstate controllable but may not be (strongly) completely controllable, from an arbitrary state to a target state.
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 …
The desired interference required for quantum computing may be modified by the wave function oscillations for the implementation of quantum algorithms[Phys.Rev.Lett.84(2000)1615]. To diminish such detrimental effect, we propose a scheme with trapped ion-pairs being qubits and apply the scheme to the Grover search. It can be found that our scheme can not only carry out a full Grover search, but also meet the requirement for the scalable hot-ion quantum computing. Moreover, the ion-pair qubits in our scheme are more robust against the decoherence and the dissipation caused by the environment tha…
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…
Grover's quantum search algorithm, involving a large number of qubits, is highly sensitive to errors in the physical implementation of the unitary operators. This poses an intrinsic limitation to the size of the database that can be practically searched. The lack of robustness of Grover's algorithm for a large number of qubits is due to quite stringent "phase-matching" condition. To overcome this limitation, Tulsi suggested a modified search algorithm [PRA 78, 022332] which succeeds as long as the errors are reproducible and reversible while Grover's algorithm fails. Such systematic errors ari…
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…
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…
Finding the optimal solution to a complex optimization problem is of great importance in practically all fields of science, technology, technical design and econometrics. We demonstrate that a modified Grover's quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits. Calculations of the global minimum of simple test functions and Lennard-Jones clusters have been carried out on a quantum computer simulator using a modified Grover's algorithm. The number of function evaluations $N$ reduced from O(N) in classical simulation to $O(\sqrt{N}…
We propose a circuit-model quantum algorithm for eigenpath traversal that is based on a combination of concepts from Grover's search and adiabatic quantum computation. Our algorithm deploys a sequence of reflections determined from eigenspaces of instantaneous Hamiltonians defined along an adiabatic schedule in order to prepare a ground state of a target problem Hamiltonian. We provide numerical evidence suggesting that, for combinatorial search problems, our algorithm can find a solution faster, on average, than Grover's search. We demonstrate our findings by applying both algorithms to solvi…
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 propose a strategy to achieve the Grover search algorithm by adiabatic passage in a very efficient way. An adiabatic process can be characterized by the instantaneous eigenvalues of the pertaining Hamiltonian, some of which form a gap. The key to the efficiency is based on the use of parallel eigenvalues. This allows us to obtain non-adiabatic losses which are exponentially small, independently of the number of items in the database in which the search is performed.
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…
This paper discusses an improvement to Grover's algorithm for searches where target states are Hamming weight eigenstates and search space is not ordered. It is shown that under these conditions search efficiency depends on the smaller number of 0's and 1's, not the total length, of binary string of target state, and that Grover's algorithm can be improved whenever number of 0's and number 1's are not equal. In particular, improvement can be exponential when number of 0's or number of 1's is very small relative to binary string length. One interesting application is that Dicke state preparatio…
Grover's quantum search algorithm evolves a quantum system from a known source state $|s\rangle$ to an unknown target state $|t\rangle$ using the selective phase inversions, $I_{s}$ and $I_{t}$, of these two states. In one of the generalizations of Grover's algorithm, $I_{s}$ is replaced by a general diffusion operator $D_{s}$ having $|s\rangle$ as an eigenstate and $I_{t}$ is replaced by a general selective phase rotation $I_{t}^ϕ$. A fast quantum search is possible as long as the operator $D_{s}$ and the angle $ϕ$ satisfies certain conditions. These conditions are very restrictive in nature.…
Sun Yat-sen University（中山大学）；Tsinghua University（清华大学）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））；QUDOOR Co, Ltd.（QUDOOR有限公司）
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 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…
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.
The radial part of the Klein-Gordon equation for the generalized Woods-Saxon potential is solved by using the Nikiforov-Uvarov method in the case of spatially dependent mass within the new approximation scheme to the centrifugal potential term. The energy eigenvalues and corresponding normalized eigenfunctions are computed. The solutions in the case of constant mass are also studied to check out the consistency of our new approximation scheme.
We consider discrete-time evolution equations in which the stochastic operator of a classical random walk is replaced by a unitary operator. Such a problem has gained much attention as a framework for coined quantum walks that are essential for attaining the Grover limit for quantum search algorithms in physically realizable, low-dimensional geometries. In particular, we analyze the exact real-space renormalization group (RG) procedure recently introduced to study the scaling of quantum walks on fractal networks. While this procedure, when implemented numerically, was able to provide some deep…
The iterative algorithm recently proposed by Waxman for solving eigenvalue problems, which relies on the method of moments, has been modified to improve its convergence considerably without sacrificing its benefits or elegance. The suggested modification is based on methods to calculate low-lying eigenpairs of large bounded hermitian operators or matrices.
Motivated by the challenge of seeking a rigorous foundation for the bulk-boundary correspondence for free fermions, we introduce an algorithm for determining exactly the spectrum and a generalized-eigenvector basis of a class of banded block quasi-Toeplitz matrices that we call corner-modified. Corner modifications of otherwise arbitrary banded block-Toeplitz matrices capture the effect of boundary conditions and the associated breakdown of translational invariance. Our algorithm leverages the interplay between a non-standard, projector-based method of kernel determination (physically, a bulk-…
Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search
Beijing International Center for Mathematical Research, Peking University（北京大学数学中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum spe…
Considering Grover's Search Algorithm (GSA) with the standard diffuser stage applied, we revisit the $3$-qubit unique String Detection Problem (SDP) and extend the algorithm to $4$-qubit SDP with multiple winners. We then investigate unstructured search problems with non-uniform distributions and define the Orthogonal Vector Problem (OVP) under quantum settings. Although no numerically stable results is reached under the original GSA framework, we provide intuition behind our implementation and further observations on OVP. We further perform a special case analysis under the modified GSA frame…
