Computational Limits of Neural Quantum State Learning from Local Indistinguishability

          @misc{ pirsa_PIRSA:25100189,
            doi = {10.48660/25100189},
            url = {https://pirsa.org/25100189},
            author = {Kumar, Tarun Advaith},
            keywords = {},
            language = {en},
            title = {Computational Limits of Neural Quantum State Learning from Local Indistinguishability},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {oct},
            note = {PIRSA:25100189 see, \url{https://pirsa.org}}
          }
          

Abstract

Neural quantum states have emerged as a powerful framework for representing classical probability distributions derived from quantum many-body systems. However, fundamental questions remain about their computational learnability, particularly for physically relevant quantum states. We identify and analyze a fundamental obstacle to the efficient learning of locally indistinguishable quantum states using neural network architectures. We introduce a restricted statistical query (SQ) learning model that captures the essential features of noisy gradient descent training in autoregressive models such as recurrent neural networks networks. Within this framework, we prove that locally indistinguishable states cannot be learned in polynomial time, establishing an inherent computational barrier. Conversely, we show that distributions with finite Markov length remain efficiently learnable in the restricted SQ model. We demonstrate the practical implications through numerical experiments on paradigmatic examples of strong-to-weak symmetry breaking, including ferromagnetic ground states and syndrome distributions of quantum error-correcting codes beyond their decoding thresholds. Our findings suggest that the hardness of neural quantum state learning could serve as a novel computational probe for identifying mixed-state phase transitions and decoding thresholds in both numerical simulations and experiments.