Scalar Fermionic Cellular Automata on finite Cayley graphs

Paolo Perinotti and Leopoldo PoggialiQUIT Group, Department of Physics, University of Pavia

INFN - Pavia, via Bassi 6, 27100 Pavia, Italy

Introduction Cellular Automata have been deeply studied in the last century but on the other hand Quantum Cellular Automata are still to be completely analysed. In particular, non-linear Quantum Cellular Automata (QCA) seem to be an almost completely unknown ground. We present the first result of classification of unitary non-linear QCAs in two paradigmatic cases. While the study of Cellular Automata and of QCAs might seem a pure mathematical physical exercise, we can look at a specific kind of automata, the Fermionic Cellular Automata, as challenging physical object. Indeed, it is possible to recover the free dynamics of QFT from the discrete dynamics of Fermionic Cellular Automata (FCAs), evading the well known critical issues due to the continuum problem. The study of non-linear FCAs is necessary to expand this reconstruction program to non trivially interacting QFT.

Goal and results: This analysis of two paradigmatic cases is the starting point to identify general conditions for a unitary evolution of non-linear FCAs.

Fig. 1: Margolus scheme for the Quantum Cellular Automata

Fig.3: Classification of scalar unitary FCAs. Above, the synoptic tables of the families of

solutions for the two case studies.

Fig. 2: Cayley graphs of (left) and (right), corresponding to the analysed FCAs.

a,b | a2 ,b2 ,(ab) 2

a | a5

I. Universality of Quantum Walk conditions

From preliminary considerations we found out that unitary conditions of Quantum Walks are still valid for every non-linear FCAs. Indeed, the normalization of evolved states depends solely on the linear terms of the evolutions, i.e. the Quantum Walk-like terms.

CARs preservation implies unitarity of evolution

Fermionic Cellular Automata and Homogeneity FCA is a discrete-step evolution of a global system obtained by an operational parallel composition of LFMs, or cells. In order to exploit FCAs for the program of reconstruction of the QFT dynamics we need to make three requirements to their evolution: Locality of the interaction among cells, reversibility of the evolution and homogeneity. These requirements have a twofold sake: they make the evolution of the FCA a physical evolution and allow us to connect the evolution of the single cell to a Cayley graph, see Fig. 2. With the field operator associated to the site at time , the general form of the evolution is:

which, in case of a number preserving evolution, gives an explicit evolution like (indexes i, j, k run over the neighbourhood sites):

0g,t+1 = T ( g,t) =

X

j2(N�)k

Tj j1(g),t †j2(g),t

... jk(g),t

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0g =

X

i

↵gi i + �g

ij †i i j + ⇠gijk

†i j k + �gijk

†i i

†j j k + permutations

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II. Global discriminability For every unitary FCA we found, a discrimination from the corresponding linear case is possible. For specific phase values of the eigenvalues of the transfer matrices we can make the evolution perfectly discriminable from the linear case.

References 1. B. Schumacher and R. F. Werner, Reversible quantum cellular automata (2004). 2. M. A. Nielsen, The fermionic canonical commutation relations and the Jordan-Wigner transform. 3. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Physical Review A 84, 012311 (2011). 4. G. M. D’Ariano and P. Perinotti, Phys. Rev. A 90 (2014).

5. P. Jordan and E. Wigner, Zeitschrift für Physik 47, 631 (1928), ISSN 0044-3328. 6. S. Östlund and E. Mele, Physical Review B 44, 12413 (1991). 7. K.R. Parthasarathy, in Stochastics in Finite and Infinite Dimensions, edited by Rajput et al., Trends in Mathematics, ︎Birkhauser, Boston, 2000︎, pp. 361 – 377. 8. G. M. D’Ariano, P. L. Presti, and M. G. A. Paris, Physical Review Letters 87 (2001).

Re[z]

Im[z]

Fig. 4: Eigenvalues of the evolution operator. Discriminability depends on the distance of segments from the centre of the unitary circumference.

III. No-Go theorem for -like terms

The evolution of a scalar FCA cannot present terms like those multiplied by the coefficients in the explicit expression for the evolution. These terms consist in a non-linear Fermionic operator made up by field operators related to every site in the neighbourhood.

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