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 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.
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 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…
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 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}…
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）
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.
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）
The model of quantum associative memories (QAM) we propose here consists in simplifying and generalizing that of Rigui Zhou \etal \cite{zhou2012} who uses the quantum matrix with binary decision diagram and nonlinear search algorithm in his model. It is worth noting that David Rosenbaum put forth the quantum matrix with binary decision diagram \cite{Rosenbaum2010} and Abrams and Llyod did the nonlinear algorithm. \cite{Abrams1998} Our model gives the possibility to retrieve one of the sought states in multi-values retrieving scheme when a measure on the first register is done. It is better tha…
This MS thesis explores the effects and origins of a 'noise with memory' in the dynamics of an open quantum system. The system considered here is a multi-qubit register performing the Grover's quantum search algorithm. We show that a Markovian-correlated noise can enhance the efficiency of the algorithm over a time correlation-less noise. We also analytically find the set of necessary and sufficient conditions for the algorithm's success probability to remain invariant with respect to the number of noisy sites in the register and point out that these conditions hold irrespective of the presenc…
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…
