This paper explores a new approach to fault-tolerant quantum computing (FTQC), 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 error correction, 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 logical error rate very close to $10^{-6}$ for the physical error rate $p = 10^{-3}$, therefore, demonstrating the potential of the proposed polar codes based approach to FTQC.

翻译：本文探索了一种基于量子极化码的容错量子计算（FTQC）新方法。我们考虑Calderbank-Shor-Steane型量子极化码，其编码一个逻辑量子比特，我们称之为$\mathcal{Q}_1$码。首先，我们证明$\mathcal{Q}_1$码的一个子族等价于著名的Shor码族。此外，我们表明在相同长度和最小距离下，$\mathcal{Q}_1$码显著优于Shor码。其次，我们考虑$\mathcal{Q}_1$码态的容错制备。我们给出了一种仅基于双量子比特Pauli测量的递归过程来制备$\mathcal{Q}_1$码态。该过程本身不具备容错性，但其测量操作提供了冗余的经典比特，可用于错误检测。然后，通过将所提出的递归过程与错误检测方法相结合，实现了容错性。最后，我们考虑$\mathcal{Q}_1$码的容错纠错。我们使用Steane纠错方法，该方法整合了所提出的容错码态制备过程。我们假设电路级退极化噪声模型，对长度分别为$16$和$64$量子比特的$\mathcal{Q}_1$码和Shor码的逻辑错误率进行了数值估计。值得注意的是，当物理错误率$p = 10^{-3}$时，长度为$64$量子比特的$\mathcal{Q}_1$码实现的逻辑错误率非常接近$10^{-6}$，从而展示了所提出的基于极化码的FTQC方法的潜力。