Sec. \[sec:discussion\].

Detailed Derivation {#sec:analysis}
-------------------

In this section we provide the mathematical background required for the complete understanding of our technique. The educated reader may skip to the examples in Sec. \[sec:exam\].

### $\mathbb{Z}_2$-Graded Algebras and their Tensor Products {#sec:graded}

An algebra $A$ (over $\mathbb{R}$) is called $\mathbb{Z}_2$-graded [@Vela2002Central] if ${A=A^0 \oplus A^1}$ where ${a \cdot a' \in A^{i+j \pmod{2}}}$ for every ${a \in A^i}$ and ${a'\in A^j}$. In this case, $A^0$ is called “the even part" of $A$, and $A^1$ “the odd part". The elements of $A^0 \cup A^1$ are called “homogeneous", the elements of $A^0$ “even" and the elements of $A^1$ “odd". The even part forms a subalgebra, ${A^0\hookrightarrow A}$, while the odd part is not an algebra as it is not closed under multiplication.

For example, the Clifford algebra [@kitaev2009periodic; @cliff] ${A={{\mathrm{Cl}}}_{p,q}}$, which is the algebra generated over the reals by generators $x_1,\dots,x_p$ and $\gamma_1,\dots,\gamma_q$ which are anticommuting in pairs and satisfy ${x_j^2=-1}$ for every ${j \in \{1,\dots,p\}}$ and ${\gamma_j^2=1}$ for every ${j \in \{1,\dots,q\}}$ is $\mathbb{Z}_2$-graded: with $A^0$ being the usual even part of the Clifford algebra, and $A^1$ being the odd part. The same algebra can admit more than one grading: for example, the 