  We study the code obtained by concatenating the standard single-mode
Gottesman-Kitaev-Preskill (GKP) code with the surface code. We show that the
noise tolerance of this surface-GKP code with respect to (Gaussian)
displacement errors improves when a single-mode squeezing unitary is applied to
each mode assuming that the identification of quadratures with logical Pauli
operators is suitably modified. We observe noise-tolerance thresholds of up to
$\sigma\approx 0.58$ shift-error standard deviation when the surface code is
decoded without using GKP syndrome information. In contrast, prior results by
Fukui et al. and Vuillot et al. report a threshold between $\sigma\approx 0.54$
and $\sigma\approx 0.55$ for the standard (toric-, respectively) surface-GKP
code. The modified surface-GKP code effectively renders the mode-level physical
noise asymmetric, biasing the logical-level noise on the GKP-qubits. The code
can thus benefit from the resilience of the surface code against biased noise.
We use the approximate maximum likelihood decoding algorithm of Bravyi et al.
to obtain our threshold estimates. Throughout, we consider an idealized
scenario where measurements are noiseless and GKP states are ideal. Our work
demonstrates that Gaussian encodings of individual modes can enhance
concatenated codes.
