标题： 矩阵乘积幺正的量子电路复杂度 显示英文标题

标题： Quantum Circuit Complexity of Matrix-Product Unitaries

摘要： 矩阵乘积幺正算符（MPUs）是多体幺正算符，由于其张量网络结构，它们在1D系统中保持纠缠面积定律。 然而，由于描述MPU的单个张量不是幺正的，因此尚不清楚如何将MPU实现为量子电路。 在本文中，我们表明一大类MPU可以使用多项式深度的量子电路来实现。 对于由重复的体张量和开放边界构建的$N$位MPU，我们显式构造了一个多项式深度为$T = O(N^{\alpha})$的量子电路，该电路实现了MPU，其中常数$\alpha$仅取决于体张量和边界张量，而与系统大小$N$无关。 我们证明这个类包括生成长程纠缠的非平凡幺正算符，并且特别地，包含从$C^*$-弱霍普夫代数表示构建的一大类幺正算符。 此外，我们还将我们的构造适应于非均匀平移变化的MPU，并表明它们可以通过深度为$O(N^{\beta} \, \mathrm{poly}\, D)$的电路实现，其中$\beta \le 1 + \log_2 \sqrt{D}/ s_{\min}$，其中$D$是键维数，$s_{\min}$是与MPU对应的归一化乔伊态的最小非零施密特值。 显示英文摘要

摘要： Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this paper, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an $N$-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth $T = O(N^{\alpha})$ realizing the MPU, where the constant $\alpha$ depends only on the bulk and boundary tensor and not the system size $N$. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of $C^*$-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally-varying MPUs and show that they can be implemented by a circuit of depth $O(N^{\beta} \, \mathrm{poly}\, D)$ where $\beta \le 1 + \log_2 \sqrt{D}/ s_{\min}$, with $D$ being the bond dimension and $s_{\min}$ is the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.