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Physical Review A

Bayesian learning for optimal control of quantum many-body states in optical lattices

Yan-Jun Xie, Han-Ning Dai, Zhen-Sheng Yuan, Youjin Deng, Xiaopeng Li, Yu-Ao Chen, Jian-Wei Pan

Published Jul 22, 2022• 9 citations
Publisher

Abstract

Strongly correlated quantum many-body states provide invaluable resources for state-of-the-art quantum information science, ranging from quantum simulation to quantum metrology. In this work, we propose a supervised machine learning algorithm for optimal control of quantum many-body atomic states in optical lattices by numerical simulations based on the Bayesian method of Gaussian process regression (GPR). We combine this method with the time evolving block decimation (TEBD) algorithm for the preparation of the Heisenberg antiferromagnetic state in a system of bosonic atoms confined with a one-dimensional optical lattice. The quantum many-body ground state of 80 atoms is efficiently optimized within a few hundred machine learning iterations, reaching a state fidelity above $96%$. With a multistep learning strategy, we find the machine-learning-based optimal control method is scalable to large systems owing to its transferability. Its robustness against noise is demonstrated by considering imperfections that are typically present in optical lattice experiments. In the application of the GPR method to the preparation of the two-dimensional (2D) antiferromagnetic state, a state fidelity of $94%$ is reached for a 2D array of $6\ifmmode\times\else\texttimes\fi{}6$ spins through the time dependent variational principle (TDVP) algorithm, confirming the generalizability of our method. We further optimize the Hamiltonian ramping sequence crossing a quantum phase transition of the quantum $XXZ$ spin chain with the exact diagonalization method, from which a control protocol for generating the long-sought atomic Greenberger-Horne-Zeilinger state is obtained. We believe the proposed optimal control method for quantum Hamiltonian ground-state preparation would benefit present ultracold-atom experiments in the study of strongly correlated many-body physics.

DOI
10.1103/physreva.106.013316

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We present a self-consistent algorithm for optimal control simulations of many-body quantum systems. The algorithm features a two-step synergism that combines discrete real-time machine learning (DRTL) with Quantum Optimal Control Theory (QOCT) using the time-dependent Schrödinger equation. Specifically, in step (1), DRTL is employed to identify a compact working space (i.e., the important portion of the Hilbert space) for the time evolution of the many-body quantum system in the presence of a control field (i.e., the initial or previously updated field), and in step (2), QOCT utilizes the DRTL-determined working space to find a newly updated control field for a chosen objective. Steps 1 and 2 are iterated until a self-consistent control objective value is reached such that the resulting optimal control field yields the same targeted objective value when the corresponding working space is systematically enlarged. To demonstrate this two-step self-consistent DRTL-QOCT synergistic algorithm, we perform optimal control simulations of strongly interacting 1D as well as 2D Heisenberg spin systems. In both scenarios, only a single spin (at the left end site for 1D and the upper left corner site for 2D) is driven by the time-dependent control fields to create an excitation at the opposite site as the target. It is found that, starting from all spin-down zero excitation states, the synergistic method is able to identify working spaces and convergence of the desired controlled dynamics with just a few iterations of the overall algorithm. In the cases studied, the dimensionality of the working space scales only quasi-linearly with the number of spins.
Preparation of a target quantum many-body state on quantum simulators is one of the significant steps in quantum science and technology. With a small number of qubits, a few quantum states, such as the Greenberger-Horne-Zeilinger state, have been prepared, but fundamental difficulties in systems with many qubits remain, including the Lieb-Robinson bounds for the number of quantum operations. Here, we provide one algorithm with an implementation of a deep learning process and achieve to prepare the target ground states with many qubits. Our strategy is to train a machine-learning model and predict parameters with many qubits by utilizing a pattern of quantum states from the corresponding quantum states with small numbers of qubits. For example, we demonstrate that our algorithm with the Quantum Approximate Optimization Ansatz can effectively generate the ground state for a 1D XY model with 64 spins. We also demonstrate that the reduced density operator of two qubits can be utilized to capture the pattern of quantum many-body states such as correlation lengths even for quantum critical states.
The preparation of quantum states is crucial for enabling quantum computations and simulations. In this work, we present a general framework for preparing ground states of many-body systems by combining the measurement-feedback control process (MFCP) with machine learning techniques. Specifically, we employ Bayesian optimization (BO) to enhance the efficiency of determining the measurement and feedback operators within the MFCP. As an illustration, we study the ground state preparation of the one-dimensional Bose–Hubbard model. Through BO, we are able to identify optimal parameters that can effectively drive the system towards low-energy states with a high probability across various quantum trajectories. Our results open up new directions for further exploration and development of advanced control strategies for quantum computations and simulations.
The ability to prepare a physical system in a desired quantum state is central to many areas of physics such as nuclear magnetic resonance, cold atoms, and quantum computing. Yet, preparing states quickly and with high fidelity remains a formidable challenge. In this work we implement cutting-edge Reinforcement Learning (RL) techniques and show that their performance is comparable to optimal control methods in the task of finding short, high-fidelity driving protocol from an initial to a target state in non-integrable many-body quantum systems of interacting qubits. RL methods learn about the underlying physical system solely through a single scalar reward (the fidelity of the resulting state) calculated from numerical simulations of the physical system. We further show that quantum state manipulation, viewed as an optimization problem, exhibits a spin-glass-like phase transition in the space of protocols as a function of the protocol duration. Our RL-aided approach helps identify variational protocols with nearly optimal fidelity, even in the glassy phase, where optimal state manipulation is exponentially hard. This study highlights the potential usefulness of RL for applications in out-of-equilibrium quantum physics.