论文标题
表面代码，量子电路和纠缠阶段
Surface codes, quantum circuits, and entanglement phases
论文作者
论文摘要
表面代码$ \ unicode {x2014} $用于量子错误校正（QEC）$ \ unicode {x2014} $和纠缠阶段$ \ unicode {x2014} $的主要量子量子动力学$ \ unicode $ \ unicode $ \ unicode {x2014} $ hereTodoRe hereTofeOre hereToceOre。在这里，我们建立了两者之间的链接。我们将二维（2D）的表面代码映射在一类不一致或相干误差（位或单轴旋转）的类别下，以$（1+1）$ d自由行量子电路通过ISING模型。我们表明，错误纠正阶段意味着该电路1D长时间状态$ |ψ_\ infty \ rangle $的拓扑上的非平凡区域定律。在误差阈值之上，我们在连贯的情况下找到了一个拓扑区域定律，以实现不连贯的错误和对数纠缠。在建立结果时，我们通过链接的ISING模型和2D散射网络为$ |ψ_\ infty \ rangle $制定了1D父级汉密尔顿人，后者通过其本地化长度和拓扑结构来展示各自的绝缘和金属相。我们预计我们的结果将概括为（$ d+1 $）D拓扑代码和$ d $二维区域法之间的二元性；这可以促进在各种错误下评估代码性能。可以将ISING模型，散射网络和父母汉密尔顿人组合的方法推广到其他费米子电路，并且可能具有独立的利益。
Surface codes$\unicode{x2014}$leading candidates for quantum error correction (QEC)$\unicode{x2014}$and entanglement phases$\unicode{x2014}$a key notion for many-body quantum dynamics$\unicode{x2014}$have heretofore been unrelated. Here, we establish a link between the two. We map two-dimensional (2D) surface codes under a class of incoherent or coherent errors (bit flips or uniaxial rotations) to $(1+1)$D free-fermion quantum circuits via Ising models. We show that the error-correcting phase implies a topologically nontrivial area law for the circuit's 1D long-time state $|Ψ_\infty\rangle$. Above the error threshold, we find a topologically trivial area law for incoherent errors and logarithmic entanglement in the coherent case. In establishing our results, we formulate 1D parent Hamiltonians for $|Ψ_\infty\rangle$ via linking Ising models and 2D scattering networks, the latter displaying respective insulating and metallic phases and setting the 1D fermion gap and topology via their localization length and topological invariant. We expect our results to generalize to a duality between the error-correcting phase of ($d+1$)D topological codes and $d$-dimensional area laws; this can facilitate assessing code performance under various errors. The approach of combining Ising models, scattering networks, and parent Hamiltonians can be generalized to other fermionic circuits and may be of independent interest.
