[论文解读] Learning quantum many-body systems from a few copies

Cambyse Rouzé, Daniel Stilck França|arXiv (Cornell University)|Jul 7, 2021

Quantum many-body systems参考文献 61被引用 12

一句话总结

该论文提出了一种新颖的量子态估计算法，能够从与系统尺寸呈多对数关系、与可观测量局域性呈多项式关系的副本数中学习准局域可观测量的期望值。通过结合最大熵推理、量子最优传输和经典阴影技术，该方法在标准层析技术基础上实现了样本复杂度的指数级提升，从而在关联衰减条件下高效估计物理上相关的可观测量。

ABSTRACT

Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state from a number of copies that scales polylogarithmically with the system's size and polynomially on the locality of the target observables. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols. We achieve our results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter allows us to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well we approximate the expectation value of a fixed set of few-body observables. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity that are of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states.

研究动机与目标

• 开发一种量子态估计算法，其所需副本数远少于标准层析技术，以学习多体系统的物理特性。
• 识别在何种条件下，可以仅从少量副本中高效估计准局域可观测量的期望值。
• 用更符合物理可区分性的Wasserstein距离替代迹距离作为误差度量。
• 将经典阴影与最大熵原理结合，以规避可观测量局域性带来的指数级复杂度。
• 建立样本复杂度界限，其在系统尺寸上为对数级，在局域性上为多项式级，优于现有协议。

提出的方法

• 该方法利用最大熵原理，推断与测量的少体可观测量一致的密度矩阵，同时最小化相对于真实态的相对熵。
• 结合经典阴影技术，从少量副本中高效估计局域约化密度矩阵。
• 利用量子最优传输技术，以局域性和近似质量为参数，界定准局域可观测量期望值估计的误差。
• 采用Wasserstein距离作为主要可区分性度量，相较于迹距离，其对广延性、局域可观测量更具物理合理性。
• 在关联衰减条件下分析该框架，包括一维热态和浅层量子电路输出态。
• 推导出关联矩阵条件数的新界，从而在某些参数范围内实现高效的后处理。

Figure 1: example of observable $O=\sum_{i}O_{i}$ for $2D$ lattice system of size $n$ . Each $O_{i}$ is supported on a $L\times L$ square ( $L=3$ in the figure). We have $\|O\|_{\operatorname{Lip}}=\mathcal{O}(\sqrt{n})$ and $\|O\|=n/L^{2}$ . Thus our methods require $\textrm{poly}(L,\log(n),\epsilo

实验结果

研究问题

• RQ1 我们能否实现对准局域可观测量期望值的估计，且副本数在系统尺寸上呈多对数增长？
• RQ2 是否可能实现副本数在目标可观测量局域性上为多项式增长的样本复杂度？
• RQ3 能否用Wasserstein距离等更符合物理意义的度量替代迹距离，以实现此类改进？
• RQ4 在何种量子态条件下，该方法能提供可证明的样本复杂度优势？
• RQ5 最大熵方法能否通过结合经典阴影与最优传输技术，规避局域性带来的指数级复杂度？

主要发现

• 该协议在Wasserstein距离下估计期望值的样本复杂度为O(ε⁻² log n)，相比标准层析技术实现了指数级改进。
• 对于具有关联衰减特性的态——如一维热态及局域阿贝尔哈密顿量的高温吉布斯态——该方法可提供可证明的高效估计。
• 该方法适用于浅层量子电路输出态及其他表现出关联衰减的态类，证实了其广泛适用性。
• 使用Wasserstein距离而非迹距离至关重要，因为迹距离即使在乘积态下也要求Ω(nε⁻²)个样本，违背了多对数复杂度目标。
• 该框架在某些参数范围内实现了高效的后处理，且对多体态关联矩阵的条件数给出了新界。
• 通过利用局域性结构与最优传输，该方法成功规避了经典阴影固有的局域性指数级复杂度。

Figure 2: Performance on a Gibbs state from the family of Eq. ( 14 ) and the 8-local observable in Eq. ( 15 ). We have set the number of qubits to $100$ , $\beta=1.1$ and the $\lambda_{i}$ unformly at random between $0.5$ and $0.9$ . The $x$ -axis denotes the logarithm of the number of samples in ba