4

Given the Hermitian Pauli group products $$ \Omega_{a,b}=\{\pm 1,\pm i\}_{a,b}\cdot \{I,X,Y,Z\}_{a,b}^{\otimes n} $$ composed of $n$ 2x2 pauli matrices $(I,X,Y,Z)$ in tensor product, such that they span a $2^n$ by $2^n$ matrix space, and $a,b=0,1,\cdots , 2^n-2, 2^n-1$, does there exist a reference for their matrix index representation $$ \Omega_{a,b} = \sum_{k=0}^{2^n-1}\sum_{l=0}^{2^n-1}A_{a,b,k,l}\left|k\right>\left<l\right| $$ to find the complex coefficients $A_{a,b,k,l}$, where I write them using bras and kets, and matrix multiplication $$ \Omega_{a,b}\Omega_{g,h}=f_{a,b,g,h,u,v}\Omega_{u,v} $$ to find the complex structure constants $f_{a,b,g,h,u,v}$?

I have seen references, such as ZX, that use $\Omega_{a,b}\approx Z^aX^b$ notation, but those representations are never scaled by complex values to ensure they are Hermitian, nor have I seen an explicit matrix index representation for them.

Why do you describe the reference to quantum error correction as 'ZX'? – Niel de Beaudrap – 2019-10-21T16:45:54.883