标题： 经典与量子计算$2+1D$SU(2) 规范理论的剪切粘度 显示英文标题

标题： Classical and Quantum Computing of Shear Viscosity for $2+1D$ SU(2) Gauge Theory

摘要： 我们利用格点哈密顿量形式对$(2+1)$维 SU(2) 规范理论的剪切粘度进行了非微扰计算。通过局域希尔伯特空间截断对格点哈密顿量进行精确对角化，并从实时演化中计算出能量-动量张量的-retarded Green 函数，然后借助 Kubo 公式得到剪切粘度。在取连续极限时，我们考虑了耦合常数的重正化群流，但没有引入额外算符的重正化。我们在一个$4\times4$蜂窝状格点上，在局域电荷表示被截断到$j_{\rm max}=\frac{1}{2}$的情况下，发现在几个温度下剪切粘度与熵密度之比$\frac{\eta}{s}$与一个已知的全息结果$\frac{1}{4\pi}$一致。我们还发现，当频率较小时，谱函数与频率$\frac{\rho^{xy}(\omega)}{\omega}$的比值呈现出峰结构。无论是精确对角化方法还是超越$j_{\rm max}=\frac{1}{2}$的更大格点上的简单矩阵乘积态经典模拟方法，都需要指数增长的资源。 因此，我们开发了一种量子计算方法来计算 retarded Green 函数，并分析了计算的各种系统性问题，包括$j_{\rm max}$截断和有限尺寸效应、Trotter 误差以及热态准备效率。 我们的热态准备方法仍然需要随着晶格尺寸呈指数增长的资源，但在高温下有一个非常小的前因子。 我们在较小的晶格上对 Quantinuum 模拟器和 IBM 模拟器上的量子电路进行了测试，并得到了与经典计算一致的结果。 显示英文摘要

摘要： We perform a nonperturbative calculation of the shear viscosity for $(2+1)$-dimensional SU(2) gauge theory by using the lattice Hamiltonian formulation. The retarded Green's function of the stress-energy tensor is calculated from real time evolution via exact diagonalization of the lattice Hamiltonian with a local Hilbert space truncation, and the shear viscosity is obtained via the Kubo formula. When taking the continuum limit, we account for the renormalization group flow of the coupling but no additional operator renormalization. We find the ratio of the shear viscosity and the entropy density $\frac{\eta}{s}$ is consistent with a well-known holographic result $\frac{1}{4\pi}$ at several temperatures on a $4\times4$ honeycomb lattice with the local electric representation truncated at $j_{\rm max}=\frac{1}{2}$. We also find the ratio of the spectral function and frequency $\frac{\rho^{xy}(\omega)}{\omega}$ exhibits a peak structure when the frequency is small. Both the exact diagonalization method and simple matrix product state classical simulation method beyond $j_{\rm max}=\frac{1}{2}$ on bigger lattices require exponentially growing resources. So we develop a quantum computing method to calculate the retarded Green's function and analyze various systematics of the calculation including $j_{\rm max}$ truncation and finite size effects, Trotter errors and the thermal state preparation efficiency. Our thermal state preparation method still requires resources that grow exponentially with the lattice size, but with a very small prefactor at high temperature. We test our quantum circuit on both the Quantinuum emulator and the IBM simulator for a small lattice and obtain results consistent with the classical computing ones.