Robust and Efficient Hamiltonian Learning

Wenjun Yu1,2, Jinzhao Sun3,4, Zeyao Han5, and Xiao Yuan2

1QICI, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China
2Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
3Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, UK
4Quantum Advantage Research, Beijing 100080, China
5School of Physics, Peking University, Beijing 100871, China

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Abstract

With the fast development of quantum technology, the sizes of both digital and analog quantum systems increase drastically. In order to have better control and understanding of the quantum hardware, an important task is to characterize the interaction, i.e., to learn the Hamiltonian, which determines both static and dynamic properties of the system. Conventional Hamiltonian learning methods either require costly process tomography or adopt impractical assumptions, such as prior information on the Hamiltonian structure and the ground or thermal states of the system. In this work, we present a robust and efficient Hamiltonian learning method that circumvents these limitations based only on mild assumptions. The proposed method can efficiently learn any Hamiltonian that is sparse on the Pauli basis using only short-time dynamics and local operations without any information on the Hamiltonian or preparing any eigenstates or thermal states. The method has a scalable complexity and a vanishing failure probability regarding the qubit number. Meanwhile, it performs robustly given the presence of state preparation and measurement errors and resiliently against a certain amount of circuit and shot noise. We numerically test the scaling and the estimation accuracy of the method for transverse field Ising Hamiltonian with random interaction strengths and molecular Hamiltonians, both with varying sizes and manually added noise. All these results verify the robustness and efficacy of the method, paving the way for a systematic understanding of the dynamics of large quantum systems.

The sizes of quantum systems eventually become classically intractable as we enter the NISQ era. As quantum dynamics bring much non-trivial potential advantage, it is more desirable to figure out the intriguing quantum dynamics, specifically the Hamiltonian evolution of the given quantum system. However, there lie some obstacles to the learning of evolution. The complexity of learning a general Hamiltonian is scaling exponentially with the system sizes. Moreover, the ubiquitous noise from quantum machines would affect the learning outcomes significantly.

In this paper, we propose a method to learn the unknown Hamiltonian evolution with efficient running time scaling and robustness against quantum noise and errors. To address the exponential scaling complexity, we assume that the Hamiltonian is sparse over the Pauli basis, making the problem more manageable. Additionally, we incorporate two separate protocols to handle circuit noise and SPAM errors, ensuring the robustness of our approach.

We present numerical results obtained by simulating our method on randomly selected Hydrogen-4 molecular Hamiltonians with over 100 Pauli terms, focusing on systems of up to 8 qubits. Despite the presence of noise, our results exhibit extraordinary accuracy.

Furthermore, we discuss potential improvements for real-world execution in the appendices, such as higher-order fittings and leveraging prior knowledge.

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[2] Zheng An, Jiahui Wu, Muchun Yang, D. L. Zhou, and Bei Zeng, "Unified quantum state tomography and Hamiltonian learning: A language-translation-like approach for quantum systems", Physical Review Applied 21 1, 014037 (2024).

[3] Keerthi Kumaran, Manas Sajjan, Sangchul Oh, and Sabre Kais, "Random projection using random quantum circuits", Physical Review Research 6 1, 013010 (2024).

[4] Daniel Stilck França, Liubov A. Markovich, V. V. Dobrovitski, Albert H. Werner, and Johannes Borregaard, "Efficient and robust estimation of many-qubit Hamiltonians", Nature Communications 15 1, 311 (2024).

[5] João Augusto Sobral, Stefan Obernauer, Simon Turkel, Abhay N. Pasupathy, and Mathias S. Scheurer, "Machine learning the microscopic form of nematic order in twisted double-bilayer graphene", Nature Communications 14 1, 5012 (2023).

[6] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller, "The randomized measurement toolbox", Nature Reviews Physics 5 1, 9 (2023).

[7] Frederik Wilde, Augustine Kshetrimayum, Ingo Roth, Dominik Hangleiter, Ryan Sweke, and Jens Eisert, "Scalably learning quantum many-body Hamiltonians from dynamical data", arXiv:2209.14328, (2022).

[8] Hsin-Yuan Huang, Yunchao Liu, Michael Broughton, Isaac Kim, Anurag Anshu, Zeph Landau, and Jarrod R. McClean, "Learning shallow quantum circuits", arXiv:2401.10095, (2024).

[9] Hsin-Yuan Huang, Yu Tong, Di Fang, and Yuan Su, "Learning Many-Body Hamiltonians with Heisenberg-Limited Scaling", Physical Review Letters 130 20, 200403 (2023).

[10] Lorenzo Pastori, Tobias Olsacher, Christian Kokail, and Peter Zoller, "Characterization and Verification of Trotterized Digital Quantum Simulation Via Hamiltonian and Liouvillian Learning", PRX Quantum 3 3, 030324 (2022).

[11] Dominik Hangleiter, Ingo Roth, Jonas Fuksa, Jens Eisert, and Pedram Roushan, "Robustly learning the Hamiltonian dynamics of a superconducting quantum processor", arXiv:2108.08319, (2021).

[12] Yongtao Zhan, Andreas Elben, Hsin-Yuan Huang, and Yu Tong, "Learning Conservation Laws in Unknown Quantum Dynamics", PRX Quantum 5 1, 010350 (2024).

[13] Vicente Pina Canelles, Manuel G. Algaba, Hermanni Heimonen, Miha Papič, Mario Ponce, Jami Rönkkö, Manish J. Thapa, Inés de Vega, and Adrian Auer, "Benchmarking Digital-Analog Quantum Computation", arXiv:2307.07335, (2023).

[14] Alicja Dutkiewicz, Thomas E. O'Brien, and Thomas Schuster, "The advantage of quantum control in many-body Hamiltonian learning", arXiv:2304.07172, (2023).

[15] Matthias C. Caro, "Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix", arXiv:2212.04471, (2022).

[16] Juan Castaneda and Nathan Wiebe, "Hamiltonian Learning via Shadow Tomography of Pseudo-Choi States", arXiv:2308.13020, (2023).

[17] Bruno Murta and J. Fernández-Rossier, "One-to-one correspondence between thermal structure factors and coupling constants of general bilinear Hamiltonians", Physical Review E 105 6, L062101 (2022).

[18] Yigal Ilin and Itai Arad, "Learning a quantum channel from its steady-state", arXiv:2302.06517, (2023).

[19] Rishabh Gupta, Raja Selvarajan, Manas Sajjan, Raphael D. Levine, and Sabre Kais, "Hamiltonian Learning from Time Dynamics Using Variational Algorithms", Journal of Physical Chemistry A 127 14, 3246 (2023).

[20] Tian-Lun Zhao, Shi-Xin Hu, and Yi Zhang, "Maximum-likelihood-estimate Hamiltonian learning via efficient and robust quantum likelihood gradient", Physical Review Research 5 2, 023136 (2023).

[21] Ammar Daskin, Rishabh Gupta, and Sabre Kais, "Dimension reduction and redundancy removal through successive Schmidt decompositions", arXiv:2302.04801, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 09:48:15) and SAO/NASA ADS (last updated successfully 2024-06-22 09:48:16). The list may be incomplete as not all publishers provide suitable and complete citation data.