Exponential quantum advantages in learning quantum observables from classical data
Riccardo Molteni, Casper Gyurik, Vedran Dunjko
备注：
In this revised version, we present a new theoretical result establishing the classical hardness of our learning problems based on a more widely studied complexity-theoretic assumption. Additionally, we include a Background section and a Discussion section, where we provide a detailed analysis of the practical relevance and applicability of our findings to significant physical tasks
人们相信量子计算机在模拟量子多体系统方面能带来计算优势。然而，近期的研究表明，经典机器学习算法能够利用经典数据预测量子系统的众多性质。尽管已有许多具有可证明量子优势的学习任务示例被提出，但它们都涉及密码学函数，并不代表实验室环境中遇到的任何物理场景。在本文中，我们证明了在从经典（测量输出的）数据中学习量子可观测量的物理相关任务中存在量子优势。我们考虑了两类可观测量：首先我们证明了 Pauli 字符串线性组合的学习优势，然后我们将结果扩展到酉参数化可观测量的更广泛情况。对于每种类型的可观测量，我们划定了边界，将经典计算机能够利用量子测量数据求解的物理相关任务，与仍然需要量子计算机进行数据分析的任务区分开来。与以往的工作不同，我们将经典困难性结果建立在较弱的假设之上，即 $\mathsf{BQP}$ 困难过程无法由多项式规模的经典电路模拟，并提供了一种非平凡的量子学习算法。我们的研究结果阐明了量子计算机在量子多体物理领域机器学习问题中的效用，从而暗示了量子学习改进可能出现的新方向。
英文摘要：
Quantum computers are believed to bring computational advantages in simulating quantum many body systems. However, recent works have shown that classical machine learning algorithms are able to predict numerous properties of quantum systems with classical data. Despite various examples of learning tasks with provable quantum advantages being proposed, they all involve cryptographic functions and do not represent any physical scenarios encountered in laboratory settings. In this paper we prove quantum advantages for the physically relevant task of learning quantum observables from classical (measured out) data. We consider two types of observables: first we prove a learning advantage for linear combinations of Pauli strings, then we extend the result for a broader case of unitarily parametrized observables. For each type of observable we delineate the boundaries that separate physically relevant tasks which classical computers can solve using data from quantum measurements, from those where a quantum computer is still necessary for data analysis. Differently from previous works, we base our classical hardness results on the weaker assumption that $\mathsf{BQP}$ hard processes cannot be simulated by polynomial-size classical circuits and provide a non-trivial quantum learning algorithm. Our results shed light on the utility of quantum computers for machine learning problems in the domain of quantum many body physics, thereby suggesting new directions where quantum learning improvements may emerge.
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