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arXiv:1706.04262 · Physical Review A

Optimization by a quantum reinforcement algorithm

Abolfazl Ramezanpour

Published Nov 3, 2017• 11 citations
arXivHTMLPDF

Abstract

A reinforcement algorithm solves a classical optimization problem by introducing a feedback to the system, which slowly changes the energy landscape and converges the algorithm to an optimal solution in the configuration space. Here, we use this strategy to concentrate (localize) the wave function of a quantum particle, which explores the configuration space of the problem, preferentially on an optimal configuration. We examine the method by solving numerically the equations governing the evolution of the system, which are similar to the nonlinear Schr\"odinger equations, for small problem sizes. In particular, we observe that reinforcement increases the minimal energy gap of the system in a quantum annealing algorithm. Our numerical simulations and the latter observation show that such kind of quantum feedback might be helpful in solving a computationally hard optimization problem by a quantum reinforcement algorithm.

DOI
10.1103/physreva.96.052307

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A reinforcement algorithm solves a classical optimization problem by introducing a feedback to the system, which slowly changes the energy landscape and converges the algorithm to an optimal solution in the configuration space. Here, we use this strategy to concentrate (localize) the wave function of a quantum particle, which explores the configuration space of the problem, preferentially on an optimal configuration. We examine the method by solving numerically the equations governing the evolution of the system, which are similar to the nonlinear Schr"odinger equations, for small problem sizes. In particular, we observe that reinforcement increases the minimal energy gap of the system in a quantum annealing algorithm. Our numerical simulations and the latter observation show that such kind of quantum feedback might be helpful in solving a computationally hard optimization problem by a quantum reinforcement algorithm.
Providing an optimal path to a quantum annealing algorithm is key to finding good approximate solutions to computationally hard optimization problems. Reinforcement is one of the strategies that can be used to circumvent the exponentially small energy gaps of the system in the annealing process. Here a time-dependent reinforcement term is added to the Hamiltonian in order to give lower energies to the most probable states of the evolving system. In this study, we take local entropy in the configuration space for the reinforcement and apply the algorithm to a number of easy and hard optimization problems. The reinforced algorithm performs better than the standard quantum annealing algorithm in the quantum search problem, where the optimal parameters behave very differently depending on the number of solutions. Moreover, the reinforcements can change the discontinuous phase transitions of the mean-field ppp-spin model (p>2p>2p>2) to a continuous transition. The algorithm's performance in the binary perceptron problem is also superior to that of the standard quantum annealing algorithm, which already works better than a classical simulated annealing algorithm.
The standard quantum annealing algorithm tries to approach the ground state of a classical system by slowly decreasing the hopping rates of a quantum random walk in the configuration space of the problem, where the on-site energies are provided by the classical energy function. In a quantum reinforcement algorithm, the annealing works instead by increasing gradually the strength of the on-site energies according to the probability of finding the walker on each site of the configuration space. Here, by using the path-integral Monte Carlo simulations of the quantum algorithms, we show that annealing via reinforcement can significantly enhance the success probability of the quantum walker. More precisely, we implement a local version of the quantum reinforcement algorithm, where the system wave function is replaced by an approximate wave function using the local expectation values of the system. We use this algorithm to find solutions to a prototypical constraint satisfaction problem (XORSAT) close to the satisfiability to unsatisfiability phase transition. The study is limited to small problem sizes (a few hundreds of variables), nevertheless, the numerical results suggest that quantum reinforcement may provide a useful strategy to deal with other computationally hard problems and larger problem sizes even as a classical optimization algorithm.
Quantum Machine Learningby Jordi RIU This project applies reinforcement learning techniques to optimize a quantum algorithm.The upcoming quantum computers are capable of optimization tasks in an exponentially large parameter space.Having access to such a large optimization space potentially allows an advantage of Quantum methods over classical algorithms, but at the price of problems in the optimization process -just to name a few, the presence of local minima and the low convergence speeds-.This project explores the application of reinforcement learning techniques to control and optimize the operation of a quantum computer solving a hard combinatorial problem, namely the MaxCut problem.Our methods use a collection of quantum observables to feed the state of the agent, inspired on a similar classical approach to the problem applied to the antiferromagnetic Ising model lattice.Our results show how an agent using these observables can optimally control the operation of a quantum device to obtain quasi-optimal solutions when in combination with the Quantum Approximate Optimization Algorithm (QAOA).v
Abstract The performance of quantum simulations heavily depends on the efficiency of noise mitigation techniques and error correction algorithms. Reinforcement has emerged as a powerful strategy to enhance the efficiency of learning and optimization algorithms. In this study, we demonstrate that a reinforced quantum dynamics can exhibit significant robustness against interactions with a noisy environment. We study a quantum annealing process in which reinforcement encourages the system to maintain its current state or follow a noise-free evolution. A learning algorithm is used to derive a concise approximation of these reinforced dynamics, thereby reducing the total evolution time and, consequently, the system’s exposure to noisy interactions. This also avoids the complexities associated with implementing quantum feedback in such reinforcement algorithms. The efficacy of our method is demonstrated through numerical simulations of reinforced quantum annealing with one-qubit and two-qubit systems under Pauli noise.
Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Annealing strategies, either classical or quantum, explore the configuration space by evolving the system under the influence of thermal or quantum fluctuations. The thermal annealing dynamics can rapidly freeze the system into a low-energy configuration, and it can be simulated well on a classical computer, but it easily gets stuck in local minima. Quantum annealing, on the other hand, can be guaranteed to find the true ground state and can be implemented in modern quantum simulators; however, quantum adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here, we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such a hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasidegenerate ground states.
This paper investigates the application of quantum annealing for solving complex combinatorial optimization problems. The core claim is that quantum annealing leverages its global search capabilities to efficiently address such problems. The proposed mechanism involves formulating the combinatorial optimization problem as a quantum annealing problem and utilizing the simulated annealing process within quantum annealing to conduct a search for the optimal solution. This research explores the potential of quantum annealing in practical applications, acknowledging that its application in combinatorial optimization is still in its exploratory phase. The algorithm is designed to minimize the energy of a system described by a Hamiltonian, which evolves over time according to the time-dependent Schrödinger equation. The core of the algorithm lies in mapping the problem's constraints and objective function into this Hamiltonian. The search process is governed by quantum fluctuations, represented by the operator H^\hat{H}H^, which encourages transitions to lower energy states, effectively navigating the solution space. The effectiveness of the algorithm is evaluated based on the quality of the solution found and the computational resources required. This work provides a foundational exploration of quantum annealing's potential within the field of combinatorial optimization.