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This paper explores a new approach to fault-tolerant quantum computing(FTQC), relying on quantum polar codes. We consider quantum polar codes ofCalderbank-Shor-Steane type, encoding one logical qubit, which we refer to as$\mathcal{Q}_1$ codes. First, we show that a subfamily of $\mathcal{Q}_1$ codesis equivalent to the well-known family of Shor codes. Moreover, we show that$\mathcal{Q}_1$ codes significantly outperform Shor codes, of the same lengthand 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. Theprocedure is not by itself fault-tolerant, however, the measurement operationstherein provide redundant classical bits, which can be advantageously used forerror detection. Fault-tolerance is then achieved by combining the proposedrecursive procedure with an error detection method. Finally, we consider thefault-tolerant error correction of $\mathcal{Q}_1$ codes. We use Steane errorcorrection, which incorporates the proposed fault-tolerant code statepreparation procedure. We provide numerical estimates of the logical errorrates 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 veryclose to $10^{-6}$ for the physical error rate $p = 10^{-3}$, therefore,demonstrating the potential of the proposed polar codes based approach to FTQC.

Fault-Tolerant Preparation of Quantum Polar Codes Encoding One Logical Qubit