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International Journal of Quantum Information

A NEW QUANTUM ALGORITHM FOR SOLVING THE MINIMUM SEARCHING PROBLEM

Luis Antônio Brasil Kowada, Carlile Lavor, Renato Portugal, Celina M.H. de Figueiredo

Published Jun 1, 2008• 11 citations
Publisher

Abstract

In this paper, we propose a new quantum algorithm for solving the minimum searching problem. This algorithm has the same order of time and space complexities as the algorithm proposed by Dürr and Høyer, but it provides a quadratic reduction in the number of measurements. In addition to the correctness and complexity analysis of the algorithm, we present simulation results considering an NP-hard problem.

DOI
10.1142/s021974990800361x

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In this paper, we propose a new quantum algorithm for solving the minimum searching problem. This algorithm has the same order of time and space complexities as the algorithm proposed by Dürr and Høyer, but it provides a quadratic reduction in the number of measurements. In addition to the correctness and complexity analysis of the algorithm, we present simulation results considering an NP-hard problem.
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. Furthermore, we provide the corresponding quantum circuits, together with three equivalent simplifications. In circumstances when we can exactly estimate the ratio of the number of solutions M and the searched space N, our method can improve the successful probability close to 100%. Furthermore, compared with DHA, our algorithm shows an advantage in complexity with large databases and in the gate complexity of constructing oracles. Experiments have been executed on an IBM superconducting processor with two qubits, and a practical problem of finding the minimum from Titanic passengers' age was numerically simulated. Both showed that our optimized maximum or minimum performs more efficiently compared with DHA. Our algorithm can serve as an important subroutine in various quantum algorithms which involves searching maximum or minimum.
Abstract We present a quantum algorithm for finding the minimum of a function based on multistep quantum computation, and apply the algorithm for solving optimization problems with continuous variables. We construct the state space of the problem by discretizing the variables of the problem, and divide the state space according to the function values of the vectors of the state space. By comparing the function values of the vectors with a series of threshold values in decreasing order, we construct a sequence of Hamiltonians where the search space of a Hamiltonian is nested in that of the previous one. By applying a multistep quantum computation process for finding the ground state of the last Hamiltonian, the optimal vector of the state space of the problem is located in a small search space and can be determined efficiently. One of the most difficult problems in optimization algorithms is that a trial vector is trapped in a deep local minimum while the global minimum is missed, this problem can be alleviated in our algorithm and the run time is proportional to the number of the steps of the algorithm, provided that the reduction rate of the search spaces is polynomial large. We discuss the implementation of the algorithm and test the algorithm for some test functions.
We develop three new quantum algorithms for searching the desired target state in the unstructured database of size N. The first algorithm requires Log N iterative steps. It constructs two quantum bags of equal size in terms of two quantum states, out of which exactly one quantum state will have nonzero overlap with the target state. This determination of overlap is done by taking the inner product, in Log N time [2], of the implicitly known target state with any one of these two quantum states. The second algorithm requires just one single step which uses a new suitable operator and the choice of this operator is problem dependent, i.e. it depends upon the number of qubits required to be used to represent an element in the index set. The third algorithm again requires only a single step and this algorithm makes use of a fixed (same) operator. It is known that algorithm for unstructured database search can be easily adaptable for solving NP-Complete problems. However, the computational complexity of NP-Complete problems after the adaptations of both the classical as well as quantum [1] search algorithms remains of the exponential order as the exponent for quantum [1] algorithm changes only to one-half times the exponent for classical algorithm. But for our quantum algorithms the exponent falls substantially so that our new quantum algorithms for unstructured search are capable if reducing the computational complexity of NP-Complete problems to polynomial order!
Abstract The knapsack problem is an important example of a combinatorial optimization problem with a wide range of applications in direct practical applications. In this article, the author considers the classical version of the problem (0-1 knapsack) and proposes a quantum algorithm for solving it based on a hybrid approach that combines the Grover quantum search algorithm with the procedure for finding the Durr–Hoyer minimum. The paper presents an analysis of the computational complexity of the proposed algorithm, and in addition to theoretical analysis, the article contains an explicit description of the quantum circuit implementing the proposed algorithm.