AI 中文总结
研究在固定哈密顿量下通过时间演化生成态\(k -\)设计，推导高斯酉系综下演化系综与初始系综框架势关系，证明简单非可积哈密顿量能高精度生成近似态\(k -\)设计，分析有限温度修正并提出淬火协议减少演化时间，阐明相关机制。
AI 中文摘要
我们研究了在固定哈密顿量下通过时间演化生成态\(k -\)设计。具体而言，考虑系综\(\mathcal{E}=\left\{e^{-iHt}|\psi_0\rangle | \ t\sim \mathrm{Unif}[0,T],\, |\psi_0\rangle\sim \mathcal{E}'\right\}\)，其中初始态从系综\(\mathcal{E}'\)中采样。对于从高斯酉系综抽取的哈密顿量，我们在大演化时间极限下推导出演化后系综\(\mathcal{E}\)的框架势与初始系综\(\mathcal{E}'\)的框架势之间的简单关系。此关系表明，只要\(\mathcal{E}'\)形成态\(1 -\)设计，\(\mathcal{E}\)在热力学极限下就形成精确的态\(k -\)设计。此外，我们通过解析和数值方法表明，在简单的非可积混合场伊辛哈密顿量下的时间演化，从适当选择的泡利基中的积态开始，可以高精度地生成近似态\(k -\)设计。我们还分析了有限温度修正，发现其缩放比例为\(O(1/T)\)。为减少演化时间，我们提出了一种\(M\)步淬火协议，将此修正抑制到\(O(1/T^M)\)，并通过数值验证。然后我们将分析扩展到酉系综，推导出酉框架势的类似递归关系。我们的结果以统一的方式阐明了最近通过顺序量子淬火生成酉\(k -\)设计的提议背后的机制。
英文摘要
We study the generation of state $k$-designs from time evolution under a fixed Hamiltonian. Specifically, we consider the ensemble $\mathcal{E}=\left\{e^{-iHt}|ψ_0\rangle | \ t\sim \mathrm{Unif}[0,T],\, |ψ_0\rangle\sim \mathcal{E}'\right\}$, where the initial states are sampled from an ensemble $\mathcal{E}'$. For Hamiltonians drawn from the Gaussian unitary ensemble, we derive a simple relation between the frame potential of the evolved ensemble $\mathcal{E}$ and that of the initial ensemble $\mathcal{E}'$ in the large evolution time limit. This relation shows that $\mathcal{E}$ forms an exact state $k$-design in the thermodynamic limit as long as $\mathcal{E}'$ forms a state 1-design. Remarkably, we further show, both analytically and numerically, that time evolution under a simple nonintegrable mixed-field Ising Hamiltonian can generate approximate state $k$-designs with high precision, starting from product states in an appropriately chosen Pauli basis. We also analyze the finite-$T$ correction and find it scales as $O(1/T)$. To reduce the evolution time, we propose an $M$-step quench protocol that suppresses this correction to $O(1/T^M)$, which is also verified numerically. We then extend our analysis to unitary ensembles, deriving an analogous recursion relation for the unitary frame potential. Our results elucidate the mechanisms underlying recent proposals for generating unitary $k$-designs through sequential quantum quenches in a unified manner.
Comments4.5 + 15 pages, 4 figures
