近日，美国微软量子Station Q的Jeongwan Haah与美国华盛顿大学的Ewin Tang等人合作并取得一项新进展。经过不懈努力，他们从高温吉布斯态和实时演化中学习量子哈密顿量。相关研究成果已于2024年3月6日在国际知名学术期刊《自然—物理学》上发表。

该研究团队研究了在给定精度下学习局部哈密顿量H的问题，假设研究人员在已知逆温度β下得到吉布斯态ρ=e^-βH/Tr(e^-βH) 的副本，或者研究人员在已知演化时间t时能够获得统一实时演化e^-itH。在改进最近研究成果的基础上，研究人员展示了如何学习局部哈密顿量H的系数，以达到误差ε。

具体而言，这可以通过S=O(logN/(βε)²)个吉布斯态或Q=O(logN/(tε)²)次实时演化运行实现，其中N是系统中的量子比特数量，β<β_c且t<t_c，β_c和t_c分别为临界逆温度和临界演化时间。研究人员设计了一种经典的后处理算法，在这两种情况下，时间复杂度都与样本量呈线性关系，即O(NS)和O(NQ)。在吉布斯态输入情况下，研究人员证明存在一个匹配的下界，表明他们算法的样本复杂度是最优的，因此，他们的时间复杂度也是最优的。

据悉，系统的行为是由它的哈密顿量决定的。在许多情况下，精确的哈密顿量是未知的，必须通过分析测量结果来提取。

附：英文原文

Title: Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions

Author: Haah, Jeongwan, Kothari, Robin, Tang, Ewin

Issue&Volume: 2024-03-06

Abstract: The behaviour of a system is determined by its Hamiltonian. In many cases, the exact Hamiltonian is not known and has to be extracted by analysing the outcome of measurements. We study the problem of learning a local Hamiltonian H to a given precision, supposing either we are given copies of its Gibbs state ρ=e^-βH/Tr(e^-βH) at a known inverse temperature β or we have access to unitary real-time evolution e^-itH for a known evolution time t. Improving on recent results, we show how to learn the coefficients of a local Hamiltonian H to error ε with S=O(logN/(βε)²) Gibbs states or with Q=O(logN/(tε)²) runs of the real-time evolution, where N is the number of qubits in the system and if β<β_c and t<t_c for some critical inverse temperature βc and critical evolution time t_c. We design a classical post-processing algorithm with time complexity linear in the sample size in both cases, namely, O(NS) and O(NQ). In the Gibbs-state input case, we prove a matching lower bound, showing that our algorithm’s sample complexity is optimal, and hence, our time complexity is also optimal.

DOI: 10.1038/s41567-023-02376-x

Source: https://www.nature.com/articles/s41567-023-02376-x