  Quantum error correction enables the preservation of logical qubits with a
lower logical error rate than the physical error rate, with performance
depending on the decoding method. Traditional error decoding approaches,
relying on the binarization (`hardening') of readout data, often ignore
valuable information embedded in the analog (`soft') readout signal. We present
experimental results showcasing the advantages of incorporating soft
information into the decoding process of a distance-three ($d=3$) bit-flip
surface code with transmons. To this end, we use the $3\times3$ data-qubit
array to encode each of the $16$ computational states that make up the logical
state $\ket{0_{\mathrm{L}}}$, and protect them against bit-flip errors by
performing repeated $Z$-basis stabilizer measurements. To infer the logical
fidelity for the $\ket{0_{\mathrm{L}}}$ state, we average across the $16$
computational states and employ two decoding strategies: minimum weight perfect
matching and a recurrent neural network. Our results show a reduction of up to
$6.8\%$ in the extracted logical error rate with the use of soft information.
Decoding with soft information is widely applicable, independent of the
physical qubit platform, and could reduce the readout duration, further
minimizing logical error rates.
