块可分解矩阵积和式的有效计算
Efficient Computation of the Permanent of Block Factorizable Matrices
摘要：
我们提出了一个有效的算法来计算永久的大小为N的矩阵，可以写为一个产品的L块对角矩阵的块的大小最多为2。对于固定的L，时间和空间资源在N中线性缩放，具有在L中指数缩放的前因子。这类矩阵包含具有带状逆的带状矩阵。我们表明，这样的分解成一个产品的块对角矩阵产生一个电路作用于希尔伯特空间的张量积结构和永久的是等于过渡振幅的电路和产品的基态。在这种对应关系中，块对角矩阵产生电路的一层，其中每个块到一个门作用于单个张量分量或两个相邻的张量分量。这一观测结果使我们能够采用矩阵乘积态（一种用于模拟量子系统的凝聚态物理和量子信息论的计算方法）来评估跃迁振幅。
英文摘要：
We present an efficient algorithm for computing the permanent for matrices of size N that can written as a product of L block diagonal matrices with blocks of size at most 2. For fixed L, the time and space resources scale linearly in N, with a prefactor that scales exponentially in L. This class of matrices contains banded matrices with banded inverse. We show that such a factorization into a product of block diagonal matrices gives rise to a circuit acting on a Hilbert space with a tensor product structure and that the permanent is equal to the transition amplitude of this circuit and a product basis state. In this correspondence, a block diagonal matrix gives rise to one layer of the circuit, where each block to a gate acting either on a single tensor component or on two adjacent tensor components. This observation allows us to adopt matrix product states, a computational method from condensed matter physics and quantum information theory used to simulate quantum systems, to evaluate the transition amplitude.
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