Topological quantum error correcting codes have emerged as leading candidates towards the goal of achieving large-scale fault-tolerant quantum computers. However, quantifying entanglement in these systems of large size in the presence of noise is a challenging task. In this paper, we provide two different prescriptions to characterize noisy stabilizer states, including the surface and the color codes, in terms of localizable entanglement over a subset of qubits. In one approach, we exploit appropriately constructed entanglement witness operators to estimate a witness-based lower bound of localizable entanglement, which is directly accessible in experiments. In the other recipe, we use graph states that are local unitary equivalent to the stabilizer state to determine a computable measurement-based lower bound of localizable entanglement. If used experimentally, this translates to a lower bound of localizable entanglement obtained from single-qubit measurements in specific bases to be performed on the qubits outside the subsystem of interest. Towards computing these lower bounds, we discuss in detail the methodology of obtaining a local unitary equivalent graph state from a stabilizer state, which includes a new and scalable geometric recipe as well as an algebraic method that applies to general stabilizer states of arbitrary size. Moreover, as a crucial step of the latter recipe, we develop a scalable graph-transformation algorithm that creates a link between two specific nodes in a graph using a sequence of local complementation operations. We develop open-source Python packages for these transformations, and illustrate the methodology by applying it to a noisy topological color code, and study how the witness and measurement-based lower bounds of localizable entanglement varies with the distance between the chosen qubits.

中文翻译：

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噪声拓扑量子代码中可定位纠缠的可扩展表征

拓扑量子纠错码已成为实现大规模容错量子计算机目标的主要候选者。然而，在存在噪声的情况下量化这些大型系统中的纠缠是一项具有挑战性的任务。在本文中，我们提供了两种不同的方法来表征嘈杂的稳定器状态，包括表面和颜色代码，根据量子位子集上的可定位纠缠。在一种方法中，我们利用适当构造的纠缠见证算子来估计可定位纠缠的基于见证的下界，这在实验中是可直接访问的。在另一个方案中，我们使用与稳定器状态局部酉等效的图状态来确定可计算的基于测量的可定位纠缠下界。如果在实验中使用，这将转化为从特定碱基中的单量子位测量获得的可定位纠缠的下限，以在感兴趣的子系统外的量子位上执行。为了计算这些下界，我们详细讨论了从稳定器状态获得局部酉等效图状态的方法，其中包括新的可扩展几何配方以及适用于任意大小的一般稳定器状态的代数方法。此外，作为后一个配方的关键步骤，我们开发了一种可扩展的图转换算法，该算法使用一系列局部互补操作在图中的两个特定节点之间创建链接。我们为这些转换开发了开源 Python 包，