# Matrix Index and multiplication rules for Hermitian Pauli group products

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 $$\Omega_{a,b} = \sum_{k=0}^{2^n-1}\sum_{l=0}^{2^n-1}A_{a,b,k,l}\left|k\right>\left to find the complex coefficients $$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 Oct 21 '19 at 16:45

If you write $$Z^aX^b$$, there's an implicit "add a phase $$i$$ to make it Hermitian if necessary", although I guess there are a couple of different conventions you might use the determine the sign used. So long as you're clear about the convention it doesn't really matter because you've got the extra $$\pm 1$$ freedom to add in to adjust for it.
As for an explicit matrix index representation, $$A_{a,b,k,l}=i^{a\cdot b}(-1)^{a\cdot k}\delta_{b\oplus l=k}$$ where $$\delta$$ is the kronecker delta. Here I've chosen a slightly different convention for introducing the $$i$$ which makes the representation easier to write down: wherever there's a pair $$ZX$$, assume it's $$iZX=Y$$.
If you want to work out the effect of the product of two terms, you could do it long-hand, but I think it's easier to work it our with the Pauli operators first and translate it back: $$Z^aX^bZ^gX^h=(-1)^{b\cdot g}Z^aZ^gX^bX^h=(-1)^{b\cdot g}Z^{a\oplus g}X^{b\oplus h}.$$
• Thanks for the answer. Is there a reference for $A_{a,b,k,l}$ and $f_{a,b,k,l,u,v}$ or is it simply easy to derive and is common knowledge? Also, computing $f_{a,b,k,l,u,v}$ looks more complicated now, because the hamming weight has to be evaluated over modulo 4 instead of modulo 2 arithmetic, e.i. $(i)^{a\cdot b}$ versus $(-1)^{a\cdot b}$. If you could provide $f_{a,b,k,l,u,v}$ using $(i)^{a\cdot b}$, that would be welcomed. – linuxfreebird Oct 21 '19 at 13:30