We present in this work, if a set of well organized suboracles is available, an algorithm for multiobject search with certainty in an unsorted database of $N$ items. Depending on the number of the objects, the technique of phase tunning is included in the algorithm. If one single object is to be searched, this algorithm performs a factor of two improvement over the best algorithm for a classical sorted database. While if the number of the objects is larger than one, the algorithm requires slightly less than $\log_{4}N$ queries, but no classical counterpart exists since the resulting state is a superposition of the marked states.
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.
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 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…
An algorithm for structured database searching is presented and used to solve the set partition problem. O(n) oracle calls are required in order to obtain a solution, but the probability that this solution is optimal decreases exponentially with problem size. Each oracle call is followed by a measurement, implying that it is necessary to maintain quantum coherence for only one oracle call at a time.
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…
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.
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）
Finding a maximum or minimum is a fundamental building block in many mathematical models. Compared with classical algorithms, Durr, Hoyer's quantum algorithm (DHA) achieves quadratic speed. However, its key step, the quantum exponential searching algorithm (QESA), which is based on Grover algorithm, is not a sure-success algorithm. Meanwhile, quantum circuits encounter the gate decomposition problem due to variation of the scale of data. In this paper, we propose an optimized quantum algorithm for searching maximum and minimum, based on DHA and the optimal quantum exact search algorithm. Furth…
We investigate the behavior of coherence in scattering quantum walk search on complete graph under the condition that the total number of vertices of the graph is greatly larger than the marked number of vertices we are searching, $N \gg v$. We find that the consumption of coherence represents the increase of the success probability for the searching,also the consumption of coherence is related to the efficiency of the algorithm represented by oracle queries.If no coherence is consumed, the efficiency of the algorithm will be the same as the classical blind search, implying that coherence is r…
In this paper, we improve Bruschweiler's algorithm such that only one query is needed for searching the single object z from N=2^n unsorted elements. Our algorithm construct the new oracle query function g(.) satisfying g(x)=0 for all input x, except for one, say x=z, where g(z)=z. To store z, our algorithm extends from one ancillary qubit to n ancillary qubits. We then measure these ancillary qubits to discover z. We further use our ensemble search algorithm to attack classical public key cryptosystems. Given the ciphertext C=Ek(m, r) which is generated by the encryption function Ek(), a publ…
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 search algorithm gives a quantum attack against block ciphers by searching for a key that matches a small number of plaintext-ciphertext pairs. This attack uses $O(\sqrt{N})$ calls to the cipher to search a key space of size $N$. Previous work in the specific case of AES derived the full gate cost by analyzing quantum circuits for the cipher, but focused on minimizing the number of qubits. In contrast, we study the cost of quantum key search attacks under a depth restriction and introduce techniques that reduce the oracle depth, even if it requires more qubits. As cases in point, we d…
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 propose a classical to quantum information encoding system using non--orthogonal states and apply it to the problem of searching an element in a quantum list. We show that the proposed encoding scheme leads to an exponential gain in terms of quantum resources and, in some cases, to an exponential gain in the number of runs of the protocol. In the case where the output of the search algorithm is a quantum state with some particular physical property, the searched state is found with a single query to the introduced oracle. If the obtained quantum state must be converted back to classical inf…
Searching large databases is an important problem with broad applications. The Grover search algorithm provides a powerful method for quantum computers to perform searches with a quadratic speedup in the number of required database queries over classical computers. It is an optimal search algorithm for a quantum computer, and has further applications as a subroutine for other quantum algorithms. Searches with two qubits have been demonstrated on a variety of platforms and proposed for others, but larger search spaces have only been demonstrated on a non-scalable NMR system. Here, we report res…
The quantum search algorithm of Chen and Diao, which finds with certainty a single target item in an unsorted database, is modified so as to be capable of searching for an arbitrary specified number of target items. If the number of targets, nu_0, is a power of four, the new algorithm will with certainty find one of the targets in a database of n items using (1/2)(3(N/nu_0)^{log_base_4(3)}-1) \approx (1/2)(3(N/nu_0)^{0.7925}-1) oracle calls, where N is the smallest power of four greater than or equal to n. If nu_0 is not a power of four, the algorithm will, with a probability of at least one-h…
Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical algorithm. It is possible to use Grover algorithm, taking profit of its ability to find a specific value in a unordered database, to find, for example, the zero of a logical function; the minimal or maximal value in a database or to recognize if an odd number is prime or not. Here we show quantum algorithms to solve those cited mathematical problems. The soluti…
Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and the vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a novel data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, a useful subset of operator matrices and state vectors can be represented in a form that grows polynomially with the number of qubits. This subset contains, but is not limited to, any equal superposition of n qubits, any computational…
