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International Journal of Quantum Information

QUANTUM ALGORITHMS FOR OPTIMIZATION USING ASYMPTOTIC QUANTUM SEARCH

Rubens Viana Ramos, Paulo Benicio Melo de Sousa

Published Aug 1, 2008• 0 citations
Publisher

Abstract

Problems of optimization are very common in several areas like engineering, physics, economics and mathematics and usually, they are very difficult to solve. Basically, one has to find the minimum or maximum of an objective function. In this work, we chose two optimization problems and we present quantum algorithms to solve them. The problems are: (1) to find the best decomposition of a unitary operation achievable by a programmable quantum circuit; (2) to find the minimal distance of a linear code. The quantum algorithms used are asymptotic quantum search and their main property is the fact that only one measurement is required.

DOI
10.1142/s021974990800416x

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Problems of optimization are very common in several areas like engineering, physics, economics and mathematics and usually, they are very difficult to solve. Basically, one has to find the minimum or maximum of an objective function. In this work, we chose two optimization problems and we present quantum algorithms to solve them. The problems are: (1) to find the best decomposition of a unitary operation achievable by a programmable quantum circuit; (2) to find the minimal distance of a linear code. The quantum algorithms used are asymptotic quantum search and their main property is the fact that only one measurement is required.
Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.
Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.
Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.
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.
Optimization is the process of selecting the best option from all possibilities, and problems related to finding the best option among many options are called optimization problems. A few examples of such problems are route picking, partial loading, crew scheduling, and portfolio optimization, and those problems naturally appear everywhere in society and in our daily lives. Most optimization problems could be solved by listing all possibilities and then searching for the best one, but we aim to find a systematic strategy (known as an algorithm) that finds the optimal solution in the least amount of time, as brute-force searching is often too time-consuming. That is, we would like to find optimal algorithms for finding optimal solutions. Quantum physics provides a more accurate explanation and prediction of nature in many cases than classical physics. Therefore, a quantum computer has the potential to be more powerful and useful than a classical one for solving optimization problems, though the extent of this advantage remains an active area of research. Moreover, designing algorithms for a quantum computer requires fundamentally different ideas and concepts than for a classical computer. In this thesis, we explore the uses of quantum computers in solving several fundamental optimization problems. We specifically consider solving linear regression, finding the top eigenvectors of a matrix, and finding the shortest nonzero vector of a lattice. More precisely, we provide near-optimal quantum algorithms that are asymptotically better than the best-possible classical algorithms for linear regression with ℓ1\ell_1ℓ1​-norm constraint (known as Lasso) and for approximating the top eigenvectors of a matrix. Our algorithms for finding the shortest nonzero vector of a lattice are asymptotically better than the best known classical algorithms. On the other hand, there are some problems for which no useful computational advantage is possible, even when we use quantum computers. This situation, however, can sometimes be beneficial if we do not want quantum computers to solve specific problems fast - say, problems relevant to post-quantum cryptography. In such a situation, we would like evidence demonstrating the difficulty of solving these problems on a quantum computer. To this end, we propose variants of Boolean satisfiability problems that are believed to not be more efficiently solvable on a quantum computer, and then via reductions use those (conjectured) quantum lower bounds to study the quantum limitations of several popular optimization problems like lattice problems, quantum strong simulation, and hitting set problems.
Several mathematical problems can be modeled as a search in a database. An example is the problem of finding the minimum of a function. Quantum algorithms for solving this problem have been proposed and all of them use the quantum search algorithm as a subroutine and several intermediate measurements are realized. In this work, it is proposed a new quantum algorithm for finding the minimum of a function in which quantum search is not used as a subroutine and only one measurement is needed. This is also named asymptotic quantum search. As an example, we propose a quantum algorithm based on asymptotic quantum search and quantum counting able to calculate a lower bound of the probability of finding a Diophantine equation with integer solution.