Understanding PEPS Tensor Network for Surface Code

Question

I am reading this article on page 2, and trying to understand the meaning of building of $W(C)$, and the reason for analogy. Why there are 2 indices $i$ and $i'$ for the physical indices? what is the meaning of each $\alpha$? what is the contracted in this case? And, why is it analogous to the last term with the $Q$?:

Answer 1

$i$ represents the qubit before stabilizing (before act with $W$ / the input to $W$) and $i'$ is the output (after $W$).

We can think of e.g. $XXXX$ stabilizer as creating a superposition of 2 cases: 1 is making $XXXX$ and the other one is not making $XXXX$. (The same for any other stabilizer).

Notice that $W$ is non-zero only in 2 cases:

1. All $\alpha$ are zero, in this case $W$ is identity matrix.
2. All $\alpha$ are one, in this case $W$ is $C$ matrix.

So the total contraction is summing only those 2 cases which is exactly as I defined the stabilizer above.

It is also possible to calculate this expression and to see that it is equal to $(I+A_f)$ which is projector up to normalization:

$$\sum_{\alpha_1 \alpha_2 \alpha_3} Q^+_{i_1 i_1' \alpha_1} Q^+_{i_2 i_2' \alpha_1 \alpha_2} Q^+_{i_3 i_3' \alpha_2 \alpha_3} Q^+_{i_4 i_4' \alpha_3} = $$

$$Q^+_{i_1 i_1', \alpha_1=0} Q^+_{i_2 i_2', \alpha_1=0 ,\alpha_2=0} Q^+_{i_3 i_3' ,\alpha_2=0 ,\alpha_3=0} Q^+_{i_4 i_4' ,\alpha_3=0} + Q^+_{i_1 i_1', \alpha_1=1} Q^+_{i_2 i_2', \alpha_1=1 ,\alpha_2=1} Q^+_{i_3 i_3' ,\alpha_2=1 ,\alpha_3=1} Q^+_{i_4 i_4' ,\alpha_3=1} =$$

$$\delta_{i_1 i_1'} \delta_{i_2 i_2'} \delta_{i_3 i_3'} \delta_{i_4 i_4'} + X_{i_1 i_1'} X_{i_2 i_2'} X_{i_3 i_3'} X_{i_4 i_4'} = $$

$$\langle i_1 i_2 i_3 i_4 | I+XXXX | i_1' i_2' i_3' i_4' \rangle$$