通过攀登链映射层级用qLDPC码进行计算
Computing with qLDPC Codes by Climbing the Chain Map Hierarchy
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- Harvard University（哈佛大学）
- Stanford University（斯坦福大学）
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中文总结 AI 辅助
该研究开发了用于qLDPC码逻辑计算的链映射层级框架，实现了Clifford和非Clifford逻辑门的常深度幺正实现，为qLDPC码计算提供了统一语言。
中文摘要 AI 辅助
我们开发了一个用于qLDPC码逻辑计算的框架，该框架将逻辑Pauli、Clifford和非Clifford操作置于同等地位。这将Pauli逻辑的简单同调描述引入了逻辑Clifford和非Clifford操作的拼贴结构中，为发现新的逻辑门提供了工具。特别地，我们定义了链映射层级：这是一族链复形，其同调类编码Clifford层级任意层级的逻辑幺正操作和码手术操作，与Pauli逻辑的链复形描述完全类似。因此，可利用Pauli逻辑的直觉来发现qLDPC码上的新逻辑操作。例如，利用稳定子（即链复形的边界）变形Pauli逻辑的熟知能力，成为了搜索（非）Clifford逻辑门的常深度幺正实现的一种方式。采用该策略，我们在二维环面码的任意数量块上（包括单个块内）发现了完整逻辑Clifford群的常深度幺正实现，还在三维环面码的任意数量块上的任意三个逻辑量子比特上实现了可寻址的逻辑CCZ门。除流形码外，我们在具有多个编码量子比特的码上也发现了可寻址的非Clifford门。链映射层级自然包含并扩展了其他逻辑小工具的构造，例如为杯积提供了通用参数化。因此，我们的工作为qLDPC码的计算提供了一种统一、有用且直观的语言。
英文摘要
We develop a framework for logical computation with qLDPC codes that places logical Pauli, Clifford, and non-Clifford operations on the same footing. This brings the simple homological description of Pauli logicals to the patchwork landscape of logical Clifford and non-Clifford operations, providing a tool for the discovery of new logical gates. In particular, we define the chain map hierarchy: a family of chain complexes whose homology classes encode logical unitary and code surgery operations at any level of the Clifford hierarchy, precisely analogous to the chain complex description of Pauli logicals. Consequently, intuition for Pauli logicals can be leveraged to discover new logical operations on qLDPC codes. For instance, the familiar ability to deform Pauli logicals with stabilizers---i.e. boundaries of the chain complex---becomes a way to search for constant-depth unitary implementations of (non-)Clifford logical gates. Using this strategy, we discover constant-depth unitary implementations of the full logical Clifford group on any number of blocks of the 2D toric code, including within a single block, and addressable logical CCZ gates on any triple of logical qubits on any number of blocks of the 3D toric code. Beyond manifold codes, we find addressable non-Clifford gates on codes with many encoded qubits. The chain map hierarchy naturally encompasses and extends other constructions of logical gadgets, for instance providing a universal parameterization of cup products. As such, our work provides a unified, useful, and intuitive language for computing with qLDPC codes.
