10 Figure 9: Time evolution of the glider CQCA on a finite chain of 7 spins. In the left-hand illustration we can see that the gliders are reflected at the ends of the chain. In the other two illustrations we see that on a finite chain the glider CQCA mirrors the position of single-site Pauli matrices. As all Pauli products can be decomposed into single-site Pauli matrices, this holds for all observables. (All observables can be decomposed into a sum of Pauli products, which are each mirrored in the same number of time steps, so the whole observable is mirrored.) x t 0 1 2 3 4 -1 -2 -3 -4 = Z = X x t 0 1 2 3 4 -1 -2 -3 -4 0 3 4 1 2 8 7 6 5 0 1 2 3 4 5 x t 0 1 2 3 4 -1 -2 -3 -4 0 3 4 1 2 8 7 6 5 decoupling one spin (at the end of the chain) from the chain and then transporting the other one to its neigh- boring position. Then the gate is applied and the qubits are transported back. This scheme was experimentally realized in a NMR-system [12]. on their initial position on the chain. Thus, temporal control (when to apply the global gates) can be turned into spatial control and any quantum operation can be conducted on the system. An example of such behavior would be the effect of a global Pauli Y operation (a local Y on each site). It gives a sign on each X and Z tensor factor. So, in our example shown in Figure 9 the Z$_{−}$$_{2}$ would gain a phase of − 1 when the Y is applied in one of the steps { 0 , 1 , 6 , 7 , 8 } , while it would gain no phase in the other steps. The X$_{−}$$_{1}$ gains the phase in steps { 0 , 1 , 2 , 3 , 6 , 7 , 8 } . The steps where a phase is gained be- long to the contraction resp. expansion of the observables, while the steps where no phase is gained belong to the transmission of the expanded observable over the chain. So, if we e.g. run the automaton for 2 L + 2 = 16 time- steps and apply a global Y in step 3, X$_{−}$$_{1}$ gains a phase but Z$_{−}$$_{2}$ does not. Applying Y in steps { 1 , 3 , 6 , 7 , } Z$_{−}$$_{2}$ gets the phase. So, we turned temporal control into spa- tial control. To achieve universal quantum computation, we use arbitrary translation-invariant local rotations in- stead of the global Y . We have introduced CQCAs and their classical descrip- tion. It was shown that CQCAs acting on translation- invariant stabilizer states generate entanglement with the highest possible rate for translation-invariant operations. The rate only depends on the trace of the CQCA’s ma- trix. It is independent of the class of the CQCA. We furthermore commented on some applications of CQCAs in quantum computational schemes. These schemes use finite chains of qubits, while our analysis is based on in- finite spin-chains. Thus, a future task is, to complete the theory of CQCAs to include finite spin chains. This was already done for periodic boundary conditions [1], but the case of non-periodic boundaries is still an open ques- tion. Additionally, the investigation of invariant states and convergence of states under CQCA action which was started in [2] will be continued for finite chains to strengthen the connection to the applications of CQCAs. Conclusion and Outlook Acknowledgements C. The Quantum Computational Scheme by Fitzsimons and Twamley The author would like to thank the Rosa Luxemburg Foundation for support. The original publication is avail- able at www.springerlink.com (DOI: 10.1007/s00340- 009-3840-1). In [8] the same property of the glider CQCA is used. However in this case the non-Clifford operations are not translation-invariant, but only conducted at the ends of the chain. Separate control of the ends of the chain is justified by the fact that due to the missing neighbor the physical properties of the systems at the end differ from those in the middle of the chain. The CQCA is used to transport qubits to the end of the chain, which can then be manipulated. Two-qubit gates are achieved by first [1] Dirk M. Schlingemann, Holger Vogts, and Reinhard F. Werner. On the structure of clifford quantum celluar automata. Journal of Mathematical Physics , 49, 2008,