Empirical Sample Complexity of Neural Network Mixed State Reconstruction

Haimeng Zhao1,2,3, Giuseppe Carleo1,2, and Filippo Vicentini1,2,4,5

1Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
2Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3Zhili College, Tsinghua University, Beijing 100084, China
4CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
5Collège de France, Université PSL, 11 place Marcelin Berthelot, 75005 Paris, France

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Quantum state reconstruction using Neural Quantum States has been proposed as a viable tool to reduce quantum shot complexity in practical applications, and its advantage over competing techniques has been shown in numerical experiments focusing mainly on the noiseless case. In this work, we numerically investigate the performance of different quantum state reconstruction techniques for mixed states: the finite-temperature Ising model. We show how to systematically reduce the quantum resource requirement of the algorithms by applying variance reduction techniques. Then, we compare the two leading neural quantum state encodings of the state, namely, the Neural Density Operator and the positive operator-valued measurement representation, and illustrate their different performance as the mixedness of the target state varies. We find that certain encodings are more efficient in different regimes of mixedness and point out the need for designing more efficient encodings in terms of both classical and quantum resources.

Quantum state reconstruction using Neural Quantum States (NQS) is an efficient way to characterize quantum systems based on measurement data collected from experiments. It has found wide applications in quantum metrology, bechmarking, verification, and error mitigation. In this work, we numerically investigate how its performance depends on the mixedness of the target states. We propose a variance reduction technique that can reduce the quantum resource requirement of these algorithms. Through this investigation, we shed some light on how different classes of mixed NQS ansatze perform, which is of interest not only for state reconstruction but also for finite-temperature and non-equilibrium simulations. Our work paves the way towards the design of more efficient encodings in terms of both classical and quantum resources.

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[1] Adriano Macarone-Palmieri, Leonardo Zambrano, Maciej Lewenstein, Antonio Acin, and Donato Farina, "Deep Neural Network-assisted improvement of quantum compressed sensing tomography", arXiv:2405.10052, (2024).

[2] Hannah Lange, Anka Van de Walle, Atiye Abedinnia, and Annabelle Bohrdt, "From Architectures to Applications: A Review of Neural Quantum States", arXiv:2402.09402, (2024).

[3] Adriano Macarone Palmieri, Guillem Müller-Rigat, Anubhav Kumar Srivastava, Maciej Lewenstein, Grzegorz Rajchel-Mieldzioć, and Marcin Płodzień, "Enhancing quantum state tomography via resource-efficient attention-based neural networks", arXiv:2309.10616, (2023).

[4] Hsin-Yuan Huang, John Preskill, and Mehdi Soleimanifar, "Certifying almost all quantum states with few single-qubit measurements", arXiv:2404.07281, (2024).

[5] Johannes Mellak, Enrico Arrigoni, and Wolfgang von der Linden, "Deep Neural Networks as Variational Solutions for Correlated Open Quantum Systems", arXiv:2401.14179, (2024).

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