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tensor products of Pauli matrices: 

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The ones in ξ$_{X}$ account for the Pauli X matrices, the ones in ξ$_{Z}$ for the Y matrices, and if ξ$_{X}$ ( x$_{0}$ ) = ξ$_{Y}$ ( x$_{0}$ ) = 1, there is a Y at x$_{0}$ : 

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The binary strings are infinitely long, but, as we always deal with localized observables, only finitely many posi- tions have non-zero entries. We omit the zeros before and after the observable to get a more convenient notation. Similarly, we will omit the identities before and after the observable. The binary strings form a vector space over $^{Z$_{2}$}$, which we call phase space. 

The addition in the classical description is commuta- tive, so we have to encode the commutation relations in an auxiliary function. Our operators fulfill the condition 

which is known from the Pauli matrices as 

where σ$_{1}$ = X , σ$_{2}$ = Y , σ$_{3}$ = Z , and ε$_{ijk}$ = 1 if ( i, j, k ) is an even permutation of (1 , 2 , 3), ε$_{ijk}$ = − 1 if ( i, j, k ) is an odd permutation of (1 , 2 , 3), and ε$_{ijk}$ = 0 otherwise. In our definition we have no i, because we chose w (1 , 1) = i Y . Therefore, the w ( ξ ) fulfill the commutation relations 

In both cases terms of the type ξ$_{X}$ η$_{Z}$ are scalar products where the addition is carried out modulo 2. 2 We call the function 

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the symplectic form. Thus, the commutation relations are encoded in the symplectic form. 

To further simplify our classical description, we trans- form the binary strings to Laurent polynomials using 

We use these polynomials as abstract objects without ever evaluating them for some value of u . Our example observable then looks like 

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We change the function w in a way that it maps vectors of Laurent polynomials to tensor products of Pauli ma- trices. By a slight abuse of notation, we will also name it w . 

Now we want to describe CQCAs in this picture. A CQCA T maps every Pauli matrix to a tensor product of those. Because it is an automorphism and determined by the local transformation T$_{0}$, we can calculate the image of any observable from the image of the Pauli matrices it can be decomposed into. Of course, the images of neigh- boring Pauli matrices in general overlap and have to be multiplied. Because we required them to commute, this multiplication is uniquely defined. In the phase-space picture with binary strings this multiplication becomes an addition. The CQCA’s action is thus a convolution of the observable’s vector of binary strings ( ξ t X ξ t Z ) with the automaton’s single cell images ( t$_{X}$$_{→}$$_{X}$ t$_{X}$$_{→}$$_{Z}$ ) and t$_{Z}$$_{→}$$_{X}$ t$_{Z}$$_{→}$$_{Z}$ ) : t +1 t 

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The transformation (5) to the Laurent polynomials has the nice property of turning convolutions into multipli- 

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