[论文解读] Efficient learning of ground & thermal states within phases of matter

Emilio Onorati, Cambyse Rouzé+2 more|arXiv (Cornell University)|Jan 30, 2023

Quantum many-body systems被引用 8

一句话总结

本论文开发样本高效的算法，以学习 Gibbs/基态并在物质相内预测局部观测量，通过 Wasserstein（W1）连续性、 Lipschitz 可观测量和相聚焦的层析，获得在精度和系统规模上的指数级改进。

ABSTRACT

We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; and (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we wish to minimise the number of samples we use to learn these properties to a given precision. For the first task, we develop new techniques to learn parameterisations of classes of systems, including quantum Gibbs states of non-commuting Hamiltonians with exponential decay of correlations and the approximate Markov property. We show it is possible to infer the expectation values of all extensive properties of the state from a number of copies that not only scales polylogarithmically with the system size, but polynomially in the observable's locality -- an exponential improvement. This set of properties includes expected values of quasi-local observables and entropies. For the second task, we develop efficient algorithms for learning observables in a phase of matter of a quantum system. By exploiting the locality of the Hamiltonian, we show that $M$ local observables can be learned with probability $1-δ$ to precision $ε$ with using only $N=O\big(\log\big(\frac{M}δ\big)e^{polylog(ε^{-1})}\big)$ samples -- an exponential improvement on the precision over previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponential decay of correlations. In addition, our sample complexity applies to the worse case setting whereas previous results only applied on average. Furthermore, we develop tools of independent interest, such as robust shadow tomography algorithms, Gibbs approximations to ground states, and generalisations of transportation cost inequalities for Gibbs states.

研究动机与目标

• Motivate and enable efficient learning of unknown Gibbs states and their expectation values for Lipschitz observables.
• Extend state-learning and phase-learning methods to thermal phases with exponential decay of correlations and generalized local indistinguishability (GALI).
• Reduce sample complexity for learning all extensive properties and quasi-local observables from limited copies across phases of matter.

提出的方法

• Develop tomography algorithms for decaying Gibbs states that estimate parameters until Wasserstein distance is nε with logarithmic/ polylog sample complexity.
• Utilize Lipschitz observables and quantum Wasserstein distance W1 to connect state recovery to extensive property estimation.
• Prove bounds showing W1-distance control over expectation values of extensive and non-linear functionals (e.g., entropies) via regional reductions and concentration arguments.
• Introduce GALI (generalized approximate local indistinguishability) to extend learning guarantees to gapped ground states and certain thermal phases beyond exponential decay, enabling worst-case-type guarantees.
• Provide learning algorithms for families of parametrized Hamiltonians that, from samples of states at random parameters, predict local observables across the whole parameter space with bounded error.
• Leverage belief propagation, classical shadow tomography, and anti-concentration in parameter space to construct tractable estimators.]
• research_questions[" Can we learn the parameters of an unknown Gibbs state and predict all extensive properties from a polylogarithmic number of samples in system size?","Can we extend Gibbs state tomography to non-commuting Hamiltonians, high-temperature, and uniform clustering/Markov conditions with feasible sample complexity?","Can we learn expectation values of local observables across a phase of matter (thermal or ground-state) with guarantees that hold point-wise rather than in expectation?","What are the minimal assumptions (e.g., exponential decay of correlations, GALI) that ensure efficient learning for ground and Gibbs states?","How does one extend phase learning from gapped ground states to thermal phases and beyond, with robust bounds on sample complexity?"]
• key_findings ["Gibbs state tomography is possible with N = O(log(1/δ) polylog(n) ε^-2) samples for commuting cases, and extends to non-commuting cases under high temperature or uniform clustering/Markov conditions.","Under exponential decay of correlations, one can infer the full set of extensive properties from O(polylog(n)) copies, achieving exponential improvement over prior bounds.","Local observables in a phase can be learned from samples drawn from the phase with N = O(log(M/δ) log(n/δ) e^{polylog(ε^-1)}) while ensuring uniform error over M local terms.","The GALI framework extends efficient learning to gapped ground states and certain thermal states, yielding guarantees with N = O(log(M/δ) log(n/δ) e^{polylog(ε^-1)}) samples and uniform error bounds.","The approach provides point-wise (not just average) recovery guarantees and connects learning performance to Wasserstein-continuity bounds of Gibbs states.","The methodology combines robust shadow tomography, Gibbs-approximations, and transport-inequality generalizations for non-commuting Hamiltonians."]
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实验结果

研究问题

• RQ1 可以在系统规模的 polylog 次方样本内学习未知 Gibbs 状态的参数并预测所有广义性质吗？
• RQ2 是否可以将 Gibbs 状态层析扩展到非对易哈密顿量、高温以及统一聚簇/马尔科夫条件，并具有可行的样本复杂度？
• RQ3 是否能够对一个相的局部观测值的期望在点上而非期望上提供保证地学习（包括热相或基态相）？
• RQ4 确保对基态和 Gibbs 状态有效学习的最低假设是什么（例如协相关的指数衰减、GALI）？
• RQ5 如何将相的学习从带隙基态扩展到热相及其之外，并获得对样本复杂度的鲁棒界？

主要发现

• Gibbs 状态层析在可对易情形下的样本数为 N = O(log(1/δ) polylog(n) ε^-2)，在高温或一致聚簇/马尔科夫条件下可扩展到非对易情形。
• 在协相关性指数衰减的情形，可以从 O(polylog(n)) 的拷贝中推断出广义性质的完整集合，相较于先前界限实现指数级改进。
• 相内的局部观测量可以通过来自相的样本学习，样本数为 N = O(log(M/δ) log(n/δ) e^{polylog(ε^-1)})，同时对 M 个局部项提供统一误差。
• GALI 框架将高效学习扩展到带隙基态和某些热状态，给出 N = O(log(M/δ) log(n/δ) e^{polylog(ε^-1)}) 拷贝的保证及统一误差界。
• 该方法提供点对点（而非仅平均）恢复保证，并将学习性能与 Gibbs 状态的 Wasserstein 连续性界联系起来。
• 该方法学结合鲁棒影子层析、Gibbs 近似以及非对易哈密顿量的传输不等式广义化。