We propose a different methodology towards approaching a Maze problem. We convert the problem into a Quantum Search Problem (QSP), and its solutions are sought for using the iterative Grover's Search Algorithm. Though the category of mazes we are looking at are of the NP complete class, we have redirected such a NP complete problem into a QSP. Our solution deals with two dimensional perfect mazes with no closed loops. We encode all possible individual paths from the starting point of the maze into a quantum register. A quantum fitness operator applied on the register encodes each individual with its fitness value. We propose an oracle design which marks all the individuals above a certain fitness value and use the Grover search algorithm to find one of the marked states. Iterating over this method, we approach towards the optimum solution.
We propose a different methodology towards approaching a Maze problem. We convert the problem into a Quantum Search Problem (QSP), and its solutions are sought for using the iterative Grover's Search Algorithm. Though the category of mazes we are looking at are of the NP complete class, we have redirected such a NP complete problem into a QSP. Our solution deals with two dimensional perfect mazes with no closed loops. We encode all possible individual paths from the starting point of the maze into a quantum register. A quantum fitness operator applied on the register encodes each individual with its fitness value. We propose an oracle design which marks all the individuals above a certain fitness value and use the Grover search algorithm to find one of the marked states. Iterating over this method, we approach towards the optimum solution.
Many large image processing and data processing scenarios can soon develop into maze problems requiring, say, finding the longest possible path, which are unresponsive, intractable, and challenging to analyze! Such maze problems, of which the traveling salesman decision problem is a special case, are of the Non-determinate Polynomial (NPcomplete) class of problems that are often impossible to solve with finite time and storage. We propose a novel methodology to approach this class of NP-complete problems. We convert a suitably formulated maze problem into a Quantum Search Problem (QSP), and the desired solutions are then sought using the iterative Grover’s Search Algorithm. Thus, we reformulate the entire class of such NP-problems into QSPs. Our current solution deals with two-dimensional perfect mazes with no closed loops. We encode all possible individual paths from the starting point of the maze into a quantum register. A quantum fitness operator applied to the register encodes each qubit with its fitness value. We propose the design of an optical oracle that marks all entities above a certain fitness value and uses the Grover search algorithm to find the optimal marked state in an iterative manner.
Abstract A quantum algorithm for solving the traveling salesman problem by the quantum phase estimation and quantum search method is considered. An approach is developed that was previously proposed for solving this problem. A quantum register is used to encode the eigenstates of a unitary operator whose phase determines the length of each possible route. The quantum phase estimation algorithm is used to estimate the length of a route. Then, to find the minimum route length, the measured values of length are encoded into the states of the second quantum register, and the search for the optimal route is carried out using a modified Grover algorithm. Numerical simulation of the proposed quantum algorithm is carried out using the Qiskit library for one and two iterations of the modified Grover algorithm.
A quantum algorithm for solving the traveling salesman problem by the quantum phase estimation and quantum search method is considered. An approach is developed that was previously proposed for solving this problem. A quantum register is used to encode the eigenstates of a unitary operator whose phase determines the length of each possible route. The quantum phase estimation algorithm is used to estimate the length of a route. Then, to find the minimum route length, the measured values of length are encoded into the states of the second quantum register, and the search for the optimal route is carried out using a modified Grover algorithm. Numerical simulation of the proposed quantum algorithm is carried out using the Qiskit library for one and two iterations of the modified Grover algorithm.
Abstract Quantum Genetic Algorithms (QGAs) integrate genetic programming and quantum computing to address search and optimization problems. Most QGA approaches add quantum features to genetic algorithm operators, such as selection, crossover, or mutation. Oppositely, the Reduced Quantum Genetic Algorithm (RQGA) is a fully quantum algorithm that encodes the entire search space (i.e., population) as a superposition of all possible solutions (chromosomes) using the individuals’ quantum register; the fitness function takes the individuals’ quantum register as input to generate corresponding fitness values within the chromosomes’ superposition. RQGA finds the best fitness value and its corresponding chromosome (i.e., the solution or one of the solutions) using Grover’s algorithm. The complexity of Grover’s algorithm is O(2N⁄2), where N is the number of superposed individual chromosomes. However, RQGA operates on an exponentially-big search space (N = 2n, where n is the number of qubits in the individuals’ quantum register), which entails an exponential runtime of O(2n⁄2). This paper introduces an optimization solution for RQGA that controls the algorithm complexity by selecting a limited number of qubits in the individuals’ register and fixing the remaining ones as classical values of ’0’ and ’1’ with a genetic algorithm. We also improve the performance of the RQGA by discarding unfit solutions and bounding the search only in the valid individuals’ area. Putting it all together, we introduce a novel quantum genetic algorithm—Hybrid Quantum Algorithm with Genetic Optimization (HQAGO)—that solves search problems in O(2(n-k)⁄2) oracle queries, where k is the number of fixed classical bits in the individuals’ register. We illustrate instantiations of HQAGO that solve the NP-hard knapsack and graph coloring problems, analyze the complexity of the new algorithm, and study the convergence of its heuristic part.
Grover's algorithm has achieved great success. But quantum search algorithms still are not complete algorithms because of Grover's Oracle. We concerned on this problem and present a new quantum search algorithm in adiabatic model without Oracle. We analyze the general difficulties in quantum search algorithms and show how to solve them in the present algorithm. As well this algorithm could deal with both single-solution and multi-solution searches without modification. We also implement this algorithm on NMR quantum computer. It is the first experiment which perform a real quantum database search rather than a marked-state search.
Grover's quantum algorithm was developed to solve the problem of searching an unstructured database for a certain unique element. In general, this problem can be formulated as follows: an unstructured database consists of elements and contains one unique element that has a certain property that can be tested with polynomial complexity, and which must be found with minimal time and complexity. Classical methods require querying the database to find the desired element, while Grover's algorithm allows you to perform only approximately steps, which are iterations of the procedure, to the database and be sure that the resulting element will be exactly the desired element with a probability close to 1. As in quantum algorithms aimed at cryptanalysis of symmetric transformations, Grover's algorithm repeats the procedure (Grover's iteration) to increase the probability of obtaining the correct result. Similarly to such algorithms in Grover's method, when iterations are continued after reaching the required number of iterations, the probability of obtaining the correct result decreases. This is due to the fact that during the execution of Grover's iteration, a rotation is performed in the complex space. Thus, each iteration, making a turn, brings the register closer and closer to the desired state, but at a certain point a maximum closeness is reached, at which the continued use of iterations will lead to a turn past the desired state, which will move the state of the system away from the desired state. The database for this algorithm can be any search space consisting of a certain number of elements. So, for example, it can be applied to find a secret key for a symmetric cryptograms formation or to find a collision for a hashing function.