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arXiv:2006.09044

Quantum Ground States from Reinforcement Learning

Ariel Barr, Willem Gispen, Austen Lamacraft

Published Jun 16, 2020• 9 citations
arXivHTMLPDF

Abstract

Finding the ground state of a quantum mechanical system can be formulated as an optimal control problem. In this formulation, the drift of the optimally controlled process is chosen to match the distribution of paths in the Feynman--Kac (FK) representation of the solution of the imaginary time Schr\"odinger equation. This provides a variational principle that can be used for reinforcement learning of a neural representation of the drift. Our approach is a drop-in replacement for path integral Monte Carlo, learning an optimal importance sampler for the FK trajectories. We demonstrate the applicability of our approach to several problems of one-, two-, and many-particle physics.

DOI
10.17863/cam.54883

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Finding the ground state of a quantum mechanical system can be formulated as an optimal control problem. In this formulation, the drift of the optimally controlled process is chosen to match the distribution of paths in the Feynman--Kac (FK) representation of the solution of the imaginary time Schr"odinger equation. This provides a variational principle that can be used for reinforcement learning of a neural representation of the drift. Our approach is a drop-in replacement for path integral Monte Carlo, learning an optimal importance sampler for the FK trajectories. We demonstrate the applicability of our approach to several problems of one-, two-, and many-particle physics.
We introduce reinforcement learning (RL) formulations of the problem of finding the ground state of a many-body quantum mechanical model defined on a lattice. We show that stoquastic Hamiltonians - those without a sign problem - have a natural decomposition into stochastic dynamics and a potential representing a reward function. The mapping to RL is developed for both continuous and discrete time, based on a generalized Feynman-Kac formula in the former case and a stochastic representation of the Schrödinger equation in the latter. We discuss the application of this mapping to the neural representation of quantum states, spelling out the advantages over approaches based on direct representation of the wavefunction of the system.
The progress in machine learning during the last decade has had a considerable impact on many areas of science and technology, including quantum technology. This work explores the application of reinforcement learning (RL) methods to the quantum control problem of state transfer in a single qubit. The goal is to create an RL agent that learns an optimal policy and thus discovers optimal pulses to control the qubit. The most crucial step is to mathematically formulate the problem of interest as a Markov decision process (MDP). This enables the use of RL algorithms to solve the quantum control problem. Deep learning and the use of deep neural networks provide the freedom to employ continuous action and state spaces, offering the expressivity and generalization of the process. This flexibility helps to formulate the quantum state transfer problem as an MDP in several different ways. All the developed methodologies are applied to the fundamental problem of population inversion in a qubit. In most cases, the derived optimal pulses achieve fidelity equal to or higher than 0.9999, as required by quantum computing applications. The present methods can be easily extended to quantum systems with more energy levels and may be used for the efficient control of collections of qubits and to counteract the effect of noise, which are important topics for quantum sensing applications.
Machine learning is rapidly finding its way into the field of computational quantum physics. One of the most popular and widely studied approaches in this direction is to use neural networks to model quantum states (NQS) in the Hamiltonian formulation of quantum mechanics. However, an alternative angle of attack to leverage machine learning in physics is through the path integral formulation, which has so far received far more limited attention. In this paper, we explore how reinforcement learning can be used to compute a class of Euclidean path integrals that yield the thermal density matrix of a quantum system, thereby enabling the computation of the free energy or other thermal expectation values. In particular, we propose a two-step approach with the unique feature that after a variational approximation for a quantity is obtained in a first step, it can then be used to efficiently compute the exact result in a second step. We benchmark this method on several simple systems and then apply it to the quantum rotor chain.
Abstract Emerging reinforcement learning techniques using deep neural networks have shown great promise in control optimization. They harness non-local regularities of noisy control trajectories and facilitate transfer learning between tasks. To leverage these powerful capabilities for quantum control optimization, we propose a new control framework to simultaneously optimize the speed and fidelity of quantum computation against both leakage and stochastic control errors. For a broad family of two-qubit unitary gates that are important for quantum simulation of many-electron systems, we improve the control robustness by adding control noise into training environments for reinforcement learning agents trained with trusted-region-policy-optimization. The agent control solutions demonstrate a two-order-of-magnitude reduction in average-gate-error over baseline stochastic-gradient-descent solutions and up to a one-order-of-magnitude reduction in gate time from optimal gate synthesis counterparts. These significant improvements in both fidelity and runtime are achieved by combining new physical understandings and state-of-the-art machine learning techniques. Our results open a venue for wider applications in quantum simulation, quantum chemistry and quantum supremacy tests using near-term quantum devices.
Solving the quantum many-body Schrödinger equation is a fundamental and challenging problem in the fields of quantum physics, quantum chemistry, and material sciences. One of the common computational approaches to this problem is Quantum Variational Monte Carlo (QVMC), in which ground-state solutions are obtained by minimizing the energy of the system within a restricted family of parameterized wave functions. Deep learning methods partially address the limitations of traditional QVMC by representing a rich family of wave functions in terms of neural networks. However, the optimization objective in QVMC remains notoriously hard to minimize and requires second-order optimization methods such as natural gradient. In this paper, we first reformulate energy functional minimization in the space of Born distributions corresponding to particle-permutation (anti-)symmetric wave functions, rather than the space of wave functions. We then interpret QVMC as the Fisher-Rao gradient flow in this distributional space, followed by a projection step onto the variational manifold. This perspective provides us with a principled framework to derive new QMC algorithms, by endowing the distributional space with better metrics, and following the projected gradient flow induced by those metrics. More specifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which uses the gradient flow induced by the Wasserstein metric, rather than Fisher-Rao metric, and corresponds to transporting the probability mass, rather than teleporting it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.
This paper introduces a quantum framework for addressing reinforcement learning (RL) tasks, grounded in the quantum principles and leveraging a fully quantum model of the classical Markov decision process. By employing quantum concepts and a quantum search algorithm, this work presents the implementation and optimization of the agent-environment interactions entirely within the quantum domain, eliminating reliance on classical computations. Key contributions include the quantum-based state transitions, return calculation, and trajectory search mechanism that utilize quantum principles to demonstrate the realization of RL processes through quantum phenomena. The implementation emphasizes the fundamental role of quantum superposition in enhancing computational efficiency for RL tasks. Results demonstrate the capacity of a quantum model to achieve quantum enhancement in RL, highlighting the potential of fully quantum implementations in decision-making tasks. This work not only underscores the applicability of quantum computing in machine learning but also contributes to the field of quantum reinforcement learning by offering a robust framework for understanding and exploiting quantum computing in RL systems.