  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.
