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arXiv:2103.08645 · Physical Review A

Tomography of time-dependent quantum Hamiltonians with machine learning

Chen-Di Han, Bryan J. Glaz, Mulugeta Haile, Ying‐Cheng Lai

Published Dec 1, 2021• 14 citations
arXivHTMLPDF

Abstract

Interacting quantum Hamiltonians are fundamental to quantum computing. Data-based tomography of time-independent quantum Hamiltonians has been achieved, but an open challenge is to ascertain the structures of time-dependent quantum Hamiltonians using time series measurements taken locally from a small subset of the spins. Physically, the dynamical evolution of a spin system under time-dependent driving or perturbation is described by the Heisenberg equation of motion. Motivated by this basic fact, we articulate a physics-enhanced machine-learning framework whose core is Heisenberg neural networks. In particular, we develop a deep learning algorithm according to some physics-motivated loss function based on the Heisenberg equation, which ``forces'' the neural network to follow the quantum evolution of the spin variables. We demonstrate that, from local measurements, not only can the local Hamiltonian be recovered, but the Hamiltonian reflecting the interacting structure of the whole system can also be faithfully reconstructed. We test our Heisenberg neural machine on spin systems of a variety of structures. In the extreme case in which measurements are taken from only one spin, the achieved tomography fidelity values can reach about $90%$. The developed machine-learning framework is applicable to any time-dependent systems whose quantum dynamical evolution is governed by the Heisenberg equation of motion.

DOI
10.1103/physreva.104.062404

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Interacting quantum Hamiltonians are fundamental to quantum computing. Data-based tomography of time-independent quantum Hamiltonians has been achieved, but an open challenge is to ascertain the structures of time-dependent quantum Hamiltonians using time series measurements taken locally from a small subset of the spins. Physically, the dynamical evolution of a spin system under time-dependent driving or perturbation is described by the Heisenberg equation of motion. Motivated by this basic fact, we articulate a physics-enhanced machine-learning framework whose core is Heisenberg neural networks. In particular, we develop a deep learning algorithm according to some physics-motivated loss function based on the Heisenberg equation, which ``forces'' the neural network to follow the quantum evolution of the spin variables. We demonstrate that, from local measurements, not only can the local Hamiltonian be recovered, but the Hamiltonian reflecting the interacting structure of the whole system can also be faithfully reconstructed. We test our Heisenberg neural machine on spin systems of a variety of structures. In the extreme case in which measurements are taken from only one spin, the achieved tomography fidelity values can reach about 9090%90. The developed machine-learning framework is applicable to any time-dependent systems whose quantum dynamical evolution is governed by the Heisenberg equation of motion.
Abstract The problem of reconstructing an unknown time-dependent Hamiltonian of a two-level quantum system from data on the system’s dynamics is considered. A method based on a neural network within the Neural ODE framework is proposed, in which the continuous evolution of the quantum state is modeled by integrating the Schrödinger equation with a parametric Hamiltonian model implemented by the network. As input data, measured time series of the expectation values of Pauli operators for a qubit are used. The network is trained by minimizing the difference between the experimentally observed and modeled trajectories of quantum observables. Using the example of simulating spin dynamics in a time-varying magnetic field, it is shown that the proposed approach enables high-accuracy reconstruction of the time-dependent Hamiltonian parameters without interrupting the quantum evolution. The results demonstrate that the Neural ODE model provides continuous tracking of the time parameter and reliable reconstruction of the system’s dynamics under conditions of limited data and noise.
Quantum systems governed by time-dependent Hamiltonians pose significant challenges for the accurate computation of unitary time-evolution operators, which are essential for predicting quantum state dynamics. In this work, we introduce a physics-informed deep learning approach based on Physics-Informed Neural Networks to estimate these operators over the full time domain. By incorporating physical constraints such as unitarity and leveraging the second-order Magnus expansion on the evolution operator, the proposed framework enables the estimation of unitary matrices at different time intervals. The model is trained using simulated unitary operators and evaluated on quantum systems ranging from 2 to 6 qubits. For larger many-body systems, specifically those with 7 and 8 qubits, the same methodology is employed to reconstruct an effective time-dependent Hamiltonian, from which the corresponding time-evolution operator is computed over the entire temporal domain. The proposed framework achieves fidelities exceeding 0.92 using a limited number of unitary samples, indicating a potential reduction in measurement and data acquisition costs. These results highlight the effectiveness of the approach for data-driven simulation and identification of quantum dynamical systems, with direct relevance to quantum computing and quantum simulation applications.
The problem of reconstructing an unknown time-dependent Hamiltonian of a two-level quantum system from its observed dynamics is considered. A method based on a neural network in the Neural ODE framework is proposed, where the continuous evolution of the quantum state is modeled by integrating the Schrodinger equation with a parametric Hamiltonian model implemented by the network. The input data consist of measured time series of the expectation values of Pauli operators for a qubit. The network is trained by minimizing the difference between the experimentally observed and simulated trajectories of quantum observables. Using the example of spin dynamics in a time-varying magnetic field, it is shown that the proposed approach allows accurate reconstruction of the parameters of the time-dependent Hamiltonian without interrupting the quantum evolution. The results demonstrate that the Neural ODE model provides continuous handling of the time parameter and robust reconstruction of system dynamics under conditions of limited data and noise.
Abstract Quantum state tomography is a daunting challenge of experimental quantum computing, even in moderate system size. One way to boost the efficiency of state tomography is via local measurements on reduced density matrices, but the reconstruction of the full state thereafter is hard. Here, we present a machine-learning method to recover the ground states of kkk k -local Hamiltonians from just the local information, where a fully connected neural network is built to fulfill the task with up to seven qubits. In particular, we test the neural network model with a practical dataset, that in a 4-qubit nuclear magnetic resonance system our method yields global states via the 2-local information with high accuracy. Our work paves the way towards scalable state tomography in large quantum systems.
In the study of closed many-body quantum systems, one is often interested in the evolution of a subset of degrees of freedom.On many occasions it is possible to approach the problem by performing an appropriate decomposition into a bath and a system.In the simplest case the evolution of the reduced state of the system is governed by a quantum master equation with a time-independent, i.e., Markovian, generator.Such evolution is typically emerging under the assumption of a weak coupling between the system and an infinitely large bath.Here we are interested in understanding to which extent a neural network function approximator can predict open quantum dynamics-described by time-local generators-from an underlying unitary dynamics.We investigate this question using a class of spin models, which is inspired by recent experimental setups.We find that indeed time-local generators can be learned.In certain situations they are even time independent and allow to extrapolate the dynamics to unseen times.This might be useful for situations in which experiments or numerical simulations do not allow to capture long-time dynamics and for exploring thermalization occurring in closed quantum systems.
Hamiltonian learning (HL), enabling precise estimation of system parameters and underlying dynamics, plays a critical role in characterizing quantum systems. However, conventional HL methods face challenges in noise robustness and resource efficiency, especially under limited measurements. In this work, we present Inverse Physics-Informed Neural Networks for Hamiltonian Learning (iPINN-HL), an approach that incorporates the Schrödinger equation as a soft constraint via a loss function penalty into the ML procedure. This formulation allows the model to integrate both observational data and known physical laws to infer Hamiltonian parameters with greater accuracy and resource efficiency. We benchmark iPINN-HL against a deep-neural-network-based quantum state tomography method [denoted as DNN-HL (Deep Neural Network for Hamiltonian Learning)] and demonstrate its effectiveness across several different scenarios, including one-dimensional spin chains, cross-resonance gate calibration, crosstalk identification, and real-time compensation to parameter drift. Our results show that iPINN-HL can approach the Heisenberg limit and exhibits robustness to noises, while outperforming DNN-HL in accuracy and resource efficiency. Therefore, iPINN-HL is a powerful and flexible framework for quantum system characterization for practical tasks.