High-accuracy Hamiltonian learning via delocalized quantum state evolutions

Davide Rattacaso1, Gianluca Passarelli2, and Procolo Lucignano1

1Dipartimento di Fisica ``E. Pancini'', Università di Napoli Federico II, Complesso di Monte Sant'Angelo, via Cinthia, Napoli 80126, Italy
2CNR-SPIN, c/o Complesso di Monte Sant'Angelo, via Cinthia, Napoli 80126, Italy

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Learning the unknown Hamiltonian governing the dynamics of a quantum many-body system is a challenging task. In this manuscript, we propose a possible strategy based on repeated measurements on a single time-dependent state. We prove that the accuracy of the learning process is maximized for states that are delocalized in the Hamiltonian eigenbasis. This implies that delocalization is a quantum resource for Hamiltonian learning, that can be exploited to select optimal initial states for learning algorithms. We investigate the error scaling of our reconstruction with respect to the number of measurements, and we provide examples of our learning algorithm on simulated quantum systems.

The state of a system of qubits in a quantum device evolves according to its unknown Hamiltonian. Yet, the precise knowledge of the system's Hamiltonian is a cornerstone in the race toward quantum advantage, since it allows one to engineer accurate unitary gates for quantum algorithms. This motivates the ongoing collective effort to devise fast Hamiltonian reconstruction methods, which promise to learn the unknown Hamiltonian by repeatedly querying the quantum device along the trajectory of a quantum evolution. In our contribution, we establish a clear connection between the performance of Hamiltonian learning and ergodicity. When the system is in a uniform superposition of all the energy eigenstates, it efficiently samples the configuration space during its evolution, minimizing the reconstruction error of its driving Hamiltonian.

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