We present a quantum algorithm to compute the entanglement spectrum of arbitrary quantum states. The interesting universal part of the entanglement spectrum is typically contained in the largest eigenvalues of the density matrix which can be obtained from the lower Renyi entropies through the Newton-Girard method. Obtaining the $p$ largest eigenvalues (${\lambda }_1>{\lambda }_2\cdots >{\lambda }_p$) requires a parallel circuit depth of $O\left[p{({\lambda }_1/{\lambda }_p)}^p\right]$ and $O[plog(N\left)\right]$ qubits where up to $p$ copies of the quantum state defined on a Hilbert space of size $N$ are needed as the input. We validate this procedure for the entanglement spectrum of the topologically ordered Laughlin wave function corresponding to the quantum Hall state at filling factor $\nu =1/3$. Our scaling analysis exposes the tradeoffs between time and number of qubits for obtaining the entanglement spectrum in the thermodynamic limit using finite-size digital quantum computers. We also illustrate the utility of the second Renyi entropy in predicting a topological phase transition and in extracting the localization length in a many-body localized system.

中文翻译：

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量子计算机上的纠缠光谱

我们提出一种量子算法来计算任意量子态的纠缠谱。纠缠谱的有趣的通用部分通常包含在密度矩阵的最大特征值中，该最大特征值可以通过牛顿－吉拉德方法从较低的Renyi熵获得。获得$p$ 最大特征值${\lambda }_{\mathrm{1个}}>{\lambda }_{\mathrm{2个}}\cdots >{\lambda }_p$）需要的并联电路深度为 $Ø\left[p{（{\lambda }_{\mathrm{1个}}/{\lambda }_p）}^p\right]$ 和 $Ø[p日志（ñ）]$ 量子比特达到 $p$ 在大小为希尔伯特空间上定义的量子态的副本 $ñ$需要作为输入。我们针对填充因子下对应于量子霍尔态的拓扑有序Laughlin波函数的纠缠谱验证了该程序$\nu =\mathrm{1个}/3$。我们的缩放分析揭示了时间和量子位数量之间的权衡，以便使用有限大小的数字量子计算机获得热力学极限中的纠缠谱。我们还说明了第二Renyi熵在预测拓扑相变和提取多体定位系统中的定位长度方面的实用性。