Universal construction for the unsorted quantum search algorithms
Xijia Miao
备注：
45 pages and 1 PS figure. Add Appendix C that how to solve quantum search problem on an NMR quantum computer by starting at the thermal equilibrium state in a spin ensemble of linear molecules
The multiple-quantum operator algebra formalism has been exploited to construct generally an unsorted quantum search algorithm. The exponential propagator and its corresponding effective Hamiltonian are constructed explicitly that describe in quantum mechanics the time evolution of a multi- particle two-state quantum system from the initial state to the output of the unsorted quantum search problem. The exponential propagator usually may not be compatible with the mathematical structure and principle of the search problem and hence is not a real quantum search network, but it can be further decomposed into a product of a series of the oracle unitary operations such as the selective phase-shift operations and the nonselective unitary operations which can be expressed further as a sequence of elementary building blocks such as one-qubit quantum gates and the two-qubit diagonal phase gates, resulting in that the decomposed propagator is compatible with the mathematical structure and principle of the search problem and thus, becomes a real quantum search network. The decomposition for the propagator can be achieved with the help of the operator algebra structure and symmetry of the effective Hamiltonian, and the properties of the multiple-quantum operator algebra spaces, especially the characteristic transformation behavior of the multiple-quantum operators under the z-axis rotations. It has been shown that the computational complexity of the search algorithm is dependent upon that of the numerical multidimensional integration and hence it is believed that the search algorithm could solve efficiently the unsorted search problem. An NMR device is also proposed to solve efficiently the unsorted search problem in polynomial time.
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。
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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 $O(M+\log N)$.
We numerically study the quantum walk search algorithm of Shenvi, Kempe and Whaley [PRA \textbf{67} 052307] and the factors which affect its efficiency in finding an individual state from an unsorted set. Previous work has focused purely on the effects of the dimensionality of the dataset to be searched. Here, we consider the effects of interpolating between dimensions, connectivity of the dataset, and the possibility of disorder in the underlying substrate: all these factors affect the efficiency of the search algorithm. We show that, as well as the strong dependence on the spatial dimension …
Quantum computing has noteworthy speedup over classical computing by taking advantage of quantum parallelism, i.e., the superposition of states. In particular, quantum search is widely used in various computationally hard problems. Grover's search algorithm finds the target element in an unsorted database with quadratic speedup than classical search and has been proved to be optimal in terms of the number of queries to the database. The challenge, however, is that Grover's search algorithm leads to high numbers of quantum gates, which make it infeasible for the Noise-Intermediate-Scale-Quantum…
Detection of symmetry is vital to problem solving. Most of the problems of computer vision and computer graphics and machine intelligence in general, can be reduced to symmetry detection problem. Unstructured search problem can also be looked upon from symmetry detection point of view. Unstructured search can be thought as searching a binary string satisfying some search condition in an unsorted list of binary strings. In this paper unstructured search problem is reduced to function bilateral symmetry detection problem with polynomial overhead in terms of the size of the input. Keywords: Unstr…
One of the significant breakthroughs in quantum computation is Grover's algorithm for unsorted database search. Recently, the applications of Grover's algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed so that the search is accelerated with improved threshold values. By taking advantage of the quantum tunneling effect, bett…
Generic quantum search algorithm searches for target entity in an unsorted database by repeatedly applying canonical Grover's quantum rotation transform to reach near the vicinity of the target entity. Thus, upon measurement, there is a high probability of finding the target entity. However, the number of times quantum rotation transform is to be applied for reaching near the vicinity of the target is a function of the number of target entities present in an unsorted database, which is generally unknown. A wrong estimate of the number of target entities can lead to overshooting or undershootin…
The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can…
Quantum searching for one of $N$ marked items in an unsorted database of $n$ items is solved in $\mathcal{O}(\sqrt{n/N})$ steps using Grover's algorithm. Using nonlinear quantum dynamics with a Gross-Pitaevskii type quadratic nonlinearity, Childs and Young discovered an unstructured quantum search algorithm with a complexity $\mathcal{O}( \min \{ 1/g \, \log (g n), \sqrt{n} \} ) $, which can be used to find a marked item after $o(\log(n))$ repetitions, where $g$ is the nonlinearity strength [PhysRevA.93.022314]. In this work we develop a structured search on a complete graph using a time depen…
Quantum Computing offers an entirely new way of doing computation governed by the rules of quantum mechanics like Superposition and Entanglement. These rules allow us to do computation over all the possible states simultaneously. Hence, offering exponentially higher computation power than the present classical computers. Quantum computing algorithms are entirely different from classical algorithms due to quantum parallel computing derived from quantum state superposition and entanglement, which has natural advantages over classical image processing. Grover algorithm is a quantum-based search a…
We propose a non-Hermitian quantum annealing algorithm which can be useful for solving complex optimization problems. We demonstrate our approach on Grover's problem of finding a marked item inside of unsorted database. We show that the energy gap between the ground and excited states depends on the relaxation parameters, and is not exponentially small. This allows a significant reduction of the searching time. We discuss the relations between the probabilities of finding the ground state and the survival of a quantum computer in a dissipative environment.
The use of superposition of states in quantum computation, known as quantum parallelism, has significant advantage in terms of speed over the classical computation. It can be understood from the early invented quantum algorithms such as Deutsch's algorithm, Deutsch-Jozsa algorithm and its variation as Bernstein-Vazirani algorithm, Simon algorithm, Shor's algorithms etc. Quantum parallelism also significantly speeds up the database search algorithm, which is important in computer science because it comes as a subroutine in many important algorithms. Quantum database search of Grover achieves th…
We discuss the quantum annealing of the fully-connected ferromagnetic $ p $-spin model in a dissipative environment at low temperature. This model, in the large $ p $ limit, encodes in its ground state the solution to the Grover's problem of searching in unsorted databases. In the framework of the quantum circuit model, a quantum algorithm is known for this task, providing a quadratic speed-up with respect to its best classical counterpart. This improvement is not recovered in adiabatic quantum computation for an isolated quantum processor. We analyze the same problem in the presence of a low-…
For the unsorted database quantum search with the unknown fraction $λ$ of target items, there are mainly two kinds of methods, i.e., fixed-point or trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113, 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder's algorithm is actually in $O(1/\sqrt{λ_0})$ rather than $O(1/\sqrtλ)$, where $λ_0$ is a known lower bound of $λ$. (ii) In terms of the trail-and-error method, cu…
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 spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) The walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by…
SRM University AP（SRM大学AP校区）；Center for Advanced Systems Understanding (CASUS)（高级系统理解中心（CASUS））；Helmholtz-Zentrum Dresden-Rossendorf (HZDR)（亥姆霍兹德累斯顿罗森多夫研究中心（HZDR））；Firstsource Solutions Ltd（Firstsource解决方案有限公司）；Indian Institute of Technology Guwahati（印度理工学院古瓦哈蒂分校）
Recent studies have been spurred on by the promise of advanced quantum computing technology, which has led to the development of quantum computer simulations on classical hardware. Grover's quantum search algorithm is one of the well-known applications of quantum computing, enabling quantum computers to perform a database search (unsorted array) and quadratically outperform their classical counterparts in terms of time. Given the restricted access to database search for an oracle model (black-box), researchers have demonstrated various implementations of Grover's circuit for two to four qubits…
We study the case where quantum computing could improve jet clustering by considering two new quantum algorithms that might speed up classical jet clustering algorithms. The first one is a quantum subroutine to compute a Minkowski-based distance between two data points, while the second one consists of a quantum circuit to track the rough maximum into a list of unsorted data. When one or both algorithms are implemented in classical versions of well-known clustering algorithms (K-means, Affinity Propagation and $k_T$-jet) we obtain efficiencies comparable to those of their classical counterpart…
We present a systematic construction of quantum circuits implementing Grover's database search algorithm for arbitrary number of targets. We introduce a new operator which flips the sign of the targets and evaluate its circuit complexity. We find the condition under which the circuit complexity of the database search algorithm based on this operator is less than that of the conventional one.
