Holographic Algorithms - Equivalence of Bases
Tensors (in this sense) are just arrays of numbers, so I don't think you'll find tensor free surveys unless they're completely free of calculations.

The "$T^{\otimes k}$" operation formalizes both the operations of changing basis and attaching gadgets to each output node (in fact I like to think of a change of basis as a sort of gadget operation). Let $\Gamma$ be a generator matchgate with standard signature $M_{i_1i_2\cdots i_k}=u(\Gamma)$. This an array of $2^k$ numbers. The signature in a new basis is given by

$(T^{\otimes k}M)_{i_1i_2\cdots i_k}:=\sum_{i_1&#39;,\cdots,i_k&#39;} T_{i_1i_1&#39;} \cdots T_{i_ki_k&#39;} M_{i_1&#39;i_2&#39;\cdots i_k&#39;}$

where $T$ is some two-by-two matrix descring the new basis.

Let $\Gamma&#39;$ be the matchgate formed by adding $k$ new vertices to $\Gamma$, taking these to be the new outputs, and connecting them to the old outputs by an edge of weight $x$. Then the new signature is $C^{\otimes k}M$ where $C_{ij}$ is the matrix $\begin{pmatrix}0&amp;x\\1&amp;0\end{pmatrix}$. If you then perform the change of basis $TC^{-1}$ you get the signature $T^{\otimes k}M$.