TY - JOUR

T1 - Tomography of time-dependent quantum Hamiltonians with machine learning

AU - Han, Chen Di

AU - Glaz, Bryan

AU - Haile, Mulugeta

AU - Lai, Ying Cheng

N1 - Funding Information:
This work was supported by the Air Force Office of Scientific Research through Grant No. FA9550-21-1-0438 and by the Army Research Office through Grant No. W911NF-21-2-0055.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/12

Y1 - 2021/12

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevA.104.062404

DO - 10.1103/PhysRevA.104.062404

M3 - Article

AN - SCOPUS:85120609150

VL - 104

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 6

M1 - 062404

ER -