Maximum Likelihood Detection of Signal Vectors in Gaussian Noise
The probability of error of the ML detector is equal to the probability that the received vector $\mathbf{y}$ is closer to $\mathbf{x}_1$ than to $\mathbf{x}_0$, which is equal to the probability that the noise component in the direction of $\mathbf{x}_0-\mathbf{x}_1$ is greater than half of the Euclidean distance between $\mathbf{x}_0$ and $\mathbf{x}_1$. The Euclidean distance between $\mathbf{x}_0$ and $\mathbf{x}_1$ is

$$D=||\mathbf{x}_0-\mathbf{x}_1||=\sqrt{\sum_{i=1}^d(x_{0,i}-x_{1,i})^2}=\sqrt{\sum_{i=1}^d4E_s}=2\sqrt{dE_s}\tag{1}$$

The noise variance in the direction of $\mathbf{x}_0-\mathbf{x}_1$ equals $N_0/2$ (as in any other direction). The probability that a zero mean Gaussian noise variable with variance $N_0/2$ assumes a value greater than $D/2$ is given by

$$P_E=Q\left(\frac{D/2}{\sqrt{N_0/2}}\right)=Q\left(\frac{\sqrt{dE_s}}{\sqrt{N_0/2}}\right)=Q\left(\sqrt{\frac{2dE_s}{N_0}}\right)\tag{2}$$