相关矩阵的压缩及一种形成费米子高斯态矩阵乘积态的有效方法
Compression of Correlation Matrices and an Efficient Method for Forming Matrix Product States of Fermionic Gaussian States
摘要：
在这里，我们提出了一个有效的和数值稳定的程序压缩相关矩阵到一组本地酉单粒子门，这导致了一个非常有效的方式形成的矩阵乘积态（MPS）近似的纯费米子高斯态，如基态的二次哈密顿量。该过程涉及到连续对角化的相关矩阵的子块，以隔离纯粹占用或未占用的局部状态。少量的最近邻酉门隔离每个局部状态。该状态的MPS通过将这些门的多体版本应用于产品状态而形成。我们处理的简单情况下，压缩的相关矩阵的无自旋自由费米子与确定的粒子数的详细信息，虽然该程序很容易扩展到费米子自旋和更一般的BCS状态（利用形式主义的马约拉纳模式）。我们还提出了一个类似DMRG的算法，直接从跳跃哈密顿量获得压缩相关矩阵。此外，我们讨论了一个轻微的变化的过程，这导致一个简单的构造的多尺度纠缠重整化的量子态（MERA）的费米子高斯态，并提出了一个简单的图片的正交小波变换的门结构，我们在本文中提出。作为一个简单的演示，我们分析了苏-施里弗-希格模型（自由费米子的一维晶格与交错跳跃振幅）。
英文摘要：
Here we present an efficient and numerically stable procedure for compressing a correlation matrix into a set of local unitary single-particle gates, which leads to a very efficient way of forming the matrix product state (MPS) approximation of a pure fermionic Gaussian state, such as the ground state of a quadratic Hamiltonian. The procedure involves successively diagonalizing subblocks of the correlation matrix to isolate local states which are purely occupied or unoccupied. A small number of nearest neighbor unitary gates isolates each local state. The MPS of this state is formed by applying the many-body version of these gates to a product state. We treat the simple case of compressing the correlation matrix of spinless free fermions with definite particle number in detail, though the procedure is easily extended to fermions with spin and more general BCS states (utilizing the formalism of Majorana modes). We also present a DMRG-like algorithm to obtain the compressed correlation matrix directly from a hopping Hamiltonian. In addition, we discuss a slight variation of the procedure which leads to a simple construction of the multiscale entanglement renormalization ansatz (MERA) of a fermionic Gaussian state, and present a simple picture of orthogonal wavelet transforms in terms of the gate structure we present in this paper. As a simple demonstration we analyze the Su-Schrieffer-Heeger model (free fermions on a 1D lattice with staggered hopping amplitudes).
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