  Quantum error correction is necessary to perform large-scale quantum
computations in the presence of noise and decoherence. As a result, several
aspects of quantum error correction have already been explored. These have been
primarily studies of quantum memory[1, 2], an important first step towards
quantum computation, where the objective is to increase the lifetime of the
encoded quantum information. Additionally, several works have explored the
implementation of logical gates[3-5]. In this work we study a next step -
fault-tolerantly implementing quantum circuits. We choose the $[[4, 1, 2]]$
Bacon-Shor subsystem code, which has a particularly simple error-detection
circuit. Through both numerics and site-counting arguments, we compute
pseudo-thresholds for the Pauli error rate $p$ in a depolarizing noise model,
below which the encoded circuits outperform the unencoded circuits. These
pseudo-threshold values are shown to be as high as $p=3\%$ for short circuits,
and $p=0.6\%$ for circuits of moderate depth. Additionally, we see that
multiple rounds of stabilizer measurements give an improvement over performing
a single round at the end. This provides a concrete suggestion for a
small-scale fault-tolerant demonstration of a quantum algorithm that could be
accessible with existing hardware.
