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arXiv:2306.03935 · Physical Review Applied

Inferring interpretable dynamical generators of local quantum observables from projective measurements through machine learning

Giovanni Cemin, Francesco Carnazza, Sabine Andergassen, Georg Martius, Federico Carollo, Igor Lesanovsky

Published Apr 3, 2024• 8 citations
arXivHTMLPDF

Abstract

Reconstructing the effective equation of motion for the time evolution of a subset of degrees of freedom of a larger system remains a problem of interest in quantum physics. Many methods have been developed, but they either rely on an ad hoc ansatz, demand data that is not experimentally accessible, or lack physical interpretability. The authors employ machine-learning methods to infer the effective dynamical generator from a noisy, finite set of local measurements. Their method yields interpretable results that may be used to infer noise models on quantum simulators, or to study thermalization dynamics in quantum many-body systems.

DOI
10.1103/physrevapplied.21.l041001

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Reconstructing the effective equation of motion for the time evolution of a subset of degrees of freedom of a larger system remains a problem of interest in quantum physics. Many methods have been developed, but they either rely on an ad hoc ansatz, demand data that is not experimentally accessible, or lack physical interpretability. The authors employ machine-learning methods to infer the effective dynamical generator from a noisy, finite set of local measurements. Their method yields interpretable results that may be used to infer noise models on quantum simulators, or to study thermalization dynamics in quantum many-body 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.
Inferring the dynamical generator of a many-body quantum system from measurement data is essential for the verification, calibration, and control of quantum processors. When the system is open, this task becomes considerably harder than in the purely unitary case, because coherent and dissipative mechanisms can produce similar measurement statistics and long-time data can be insensitive to coherent couplings. Here we tackle this so-called Lindbladian learning problem of open-system characterisation with maximum-likelihood on Pauli measurements at multiple experimentally friendly \emph{transient} times, exploiting the richer information content of transient dynamics. To navigate the resulting non-convex likelihood loss-landscape, we augment the physical model neural differential-equation term, which is progressively removed during training to distil an interpretable Lindbladian solution. Our method reliably learns open-system dynamics across neutral-atom (with 2D connectivity) and superconducting Hamiltonians, as well as the Heisenberg XYZ, and PXP models on a spin-1/2 chain. For the dissipative part, we show robustness over phase noise, thermal noise, and their combination. Our algorithm can robustly infer these dissipative systems over noise-to-signal ratios spanning four orders of magnitude, and system sizes up to N=6N=6N=6 qubits with fewer than 5×1055 \times 10^55×105 shots.
Abstract Full information about a many-body quantum system is usually out-of-reach due to the exponential growth—with the size of the system—of the number of parameters needed to encode its state. Nonetheless, in order to understand the complex phenomenology that can be observed in these systems, it is often sufficient to consider dynamical or stationary properties of local observables or, at most, of few-body correlation functions. These quantities are typically studied by singling out a specific subsystem of interest and regarding the remainder of the many-body system as an effective bath. In the simplest scenario, the subsystem dynamics, which is in fact an open quantum dynamics, can be approximated through Markovian quantum master equations. Here, we formulate the problem of finding the generator of the subsystem dynamics as a variational problem, which we solve using the standard toolbox of machine learning for optimization. This dynamical or ‘Lindblad’ generator provides the relevant dynamical parameters for the subsystem of interest. Importantly, the algorithm we develop is constructed such that the learned generator implements a physically consistent open quantum time-evolution. We exploit this to learn the generator of the dynamics of a subsystem of a many-body system subject to a unitary quantum dynamics. We explore the capability of our method to recover the time-evolution of a two-body subsystem and exploit the physical consistency of the generator to make predictions on the stationary state of the subsystem dynamics.
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.
Coupling between a quantum system and its environment causes decoherence by transferring information from the system to environmental degrees of freedom. When discretized in time, such interactions can be interpreted as sequences of weak measurements that provide an effective model of noisy quantum dynamics. Motivated by this picture, we propose an AI-assisted error-mitigation framework for quantum diffusion processes generated by sequential local weak measurements. The forward process progressively erases information from the input state through weak measurements performed in randomly selected Pauli bases, producing basis-dependent local dephasing and locally depolarizing dynamics on average. Machine-learning models are trained on exact synthetic density matrices to learn a channel- and distribution-specific denoising map and estimate the corresponding pre-noise state. We benchmark the approach on single-qubit states and separable and entangled multi-qubit registers. We also study distribution-dependent local-to-global reconstruction, in which local reduced density matrices are used to reconstruct the global state. This experimentally motivated setting relies on locally accessible information and is therefore compatible with noisy and distributed quantum systems. More broadly, the framework provides a hybrid classical-quantum approach for approximating non-unitary dynamics and mitigating coherence loss.
We introduce a machine-learning approach for identifying hidden structural features of open quantum dynamics under restricted experimental access. Unlike most existing data-driven methods which focus on detection or prediction of dynamical behavior, our framework targets the inference of invariant algebraic structures underlying the effective Markovian evolution. Measurement limitations, symmetries, and superselection rules are incorporated through a *-algebraic description of accessible observables. The learning problem is formulated as maximum-likelihood estimation from multi-time measurement sequences, where the algebraic type of an invariant subalgebra, particularly a decoherence-free subalgebra, is treated as a discrete structural hypothesis. The feasibility of the approach is illustrated on multiple synthetic models and a waveguide quantum electrodynamics system, where nontrivial intermediate algebraic structures are identified directly from measurement data.