arXiv 2404.187512024-04quant-ph

基于Clifford增强矩阵乘积态的量子态设计

Quantum State Designs with Clifford Enhanced Matrix Product States

Guglielmo Lami, Tobias Haug, Jacopo De Nardis

AI总结：

本文研究随机矩阵乘积态的平均非稳定子性，证明其2-稳定子Rényi熵以N/χ²标度收敛到Haar随机态，并提出Clifford增强矩阵乘积态（CMPS），证明其能以χ⁻²标度逼近4-球面设计，表明Clifford酉算符与张量网络态结合可生成高度非平凡的量子态。

摘要：

非稳定子性（或称“魔力”）是一种关键的量子资源，它与纠缠一起刻画了量子态的非经典复杂性。在此，我们处理量化随机矩阵乘积态（RMPS）平均非稳定子性的问题。RMPS代表随机乘积态的一种推广，其特征是有界纠缠随键维数χ呈对数增长。我们证明，2-稳定子Rényi熵以N/χ²的标度收敛到Haar随机态的值，其中N为系统尺寸。这表明具有适中键维数的MPS与一般态一样具有魔力。随后，我们引入Clifford增强矩阵乘积态（CMPS）系综，它由Clifford酉算符作用在RMPS上构建而成。利用我们之前的结果，我们证明CMPS可以以任意精度逼近4-球面设计。具体而言，对于常数N，CMPS以χ⁻²的标度接近4-设计。我们的发现表明，将Clifford酉算符与多项式复杂度的张量网络态相结合，可以生成高度非平凡的量子态。

英文摘要：

Nonstabilizerness, or `magic', is a critical quantum resource that, together with entanglement, characterizes the non-classical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random Matrix Product States (RMPS). RMPS represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension $χ$. We demonstrate that the $2$-Stabilizer Rényi Entropy converges to that of Haar random states as $N/χ^2$, where $N$ is the system size. This indicates that MPS with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford enhanced Matrix Product States ($\mathcal{C}$MPS), built by the action of Clifford unitaries on RMPS. Leveraging our previous result, we show that $\mathcal{C}$MPS can approximate $4$-spherical designs with arbitrary accuracy. Specifically, for a constant $N$, $\mathcal{C}$MPS become close to $4$-designs with a scaling as $χ^{-2}$. Our findings indicate that combining Clifford unitaries with polynomially complex tensor network states can generate highly non-trivial quantum states.

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