This paper explores a new approach to fault-tolerant quantum computing, relying on quantum polar codes. We consider quantum polar codes of Calderbank-Shor-Steane type, encoding one logical qubit, which we refer to as $\mathcal{Q}_1$ codes. First, we show that a subfamily of $\mathcal{Q}_1$ codes is equivalent to the well-known family of Shor codes. Moreover, we show that $\mathcal{Q}_1$ codes significantly outperform Shor codes, of the same length and minimum distance. Second, we consider the fault-tolerant preparation of $\mathcal{Q}_1$ code states. We give a recursive procedure to prepare a $\mathcal{Q}_1$ code state, based on two-qubit Pauli measurements only. The procedure is not by itself fault-tolerant, however, the measurement operations therein provide redundant classical bits, which can be advantageously used for error detection. Fault tolerance is then achieved by combining the proposed recursive procedure with an error detection method. Finally, we consider the fault-tolerant error correction of $\mathcal{Q}_1$ codes. We use Steane's error correction technique, which incorporates the proposed fault-tolerant code state preparation procedure. We provide numerical estimates of the logical error rates for $\mathcal{Q}_1$ and Shor codes of length $16$ and $64$ qubits, assuming a circuit-level depolarizing noise model. Remarkably, the $\mathcal{Q}_1$ code of length $64$ qubits achieves a pseudothreshold value slightly below $1\%$, demonstrating the potential of polar codes for fault-tolerant quantum computing.

翻译：本文探索了一种基于量极代码的不宽容量计算新方法。 我们考虑的是卡尔德班- 肖尔- 斯蒂恩型的量极值代码, 编码一种逻辑的量极值代码, 我们称之为$\ mathcal=1$的代码。 首先, 我们显示一个亚类的 $\ mathcal=1$代码相当于众所周知的 Shor 代码组。 此外, 我们显示, $\ mathcal=1$的代码大大超出一个长度和最小距离的美元标准。 其次, 我们考虑的是 $\ mathcal=1 代码组的错误容忍度代码。 我们给出了一个循环程序, 仅以 $\ mathcal=1 代码组的美元代码为基数。 但是, 该程序本身并非容错, 其测量操作提供了多余的古典比值, 可用于检测错误。 然后通过将提议的递归回程序与一个错误识别方法实现。 最后, 我们考虑的是, 美元的错误校正值错误校正值 $ mal=l=l=1 codeal cal code code code codeal code 。 我们使用一个逻辑程序, rodeal rodeal rocreal rode rodeal rodealal rodeal rodeal rode romode 。