The University of Queensland
Department of Mathematics
Mathematical Physics Seminar
Quantum Soliton Cellular Automata
Dr Demosthenes Ellinas
Technical University of Crete
Friday August 25, 4.00pm
Room 112, Priestley Building
Soliton cellular automata (SCA) form a class of cellular automata operating on binary sequences, with an updating rule function for each cell that depends on past and present time cells, the number of which determines the radius r of the automaton. Contrary to usual CA ( that evolve their cells by means of past time cells only ), SCA exhibit a variety of evolution patterns that are mainly known to characterize the temporal behaviour of solutions of non linear DPSs, namely: periodic evolution of {\it particles} (i.e localized groups of binary cells ), or {\it solitonic} type of scattering of digital particles, or even {\it breathing} modes of oscillations between particles. For all these effects sufficient numerical evidence and analytic proofs are available over the last decade together with proposed applications for a new kind of computational architecture that will utilize these evolution patterns of SCA in order to provide a ``gateless" implementation of logical operations. Towards a physical microscopic realization of these suggestions, envisaged in the context of the new paradigm of {\it Quantum Computing}, it is plausible to formulate SCA in terms of Quantum Mechanics and to investigate the possible quantum effects in their time evolution.
To this end we put forward here the quantization of classical SCA (cSCA). This is achieved by substituting the binary strings of cSCA by strings $|\Psi>$ of tensor products of vectors from two-dimensional Hilbert space $H$, i.e we assign to each cell with values $\{0,1\}$ the quantum bits (qubits) $\{|0 >, |1>\}\in H$. To every updating-rule function $f$ of a cSCA we associate an operator $U_f$ acting on $|\Psi>$, which is unitary if $f$ is bijective, and implements the action of $f$ on the binary labels of $|\Psi >$. For each of the two available descriptions of cSCA, namely the so called parity rule and the Fast Rule Theorem (FRT) descriptions, we form the respective operators $U_f$, and study their structure. Concretely, by employing mathematical tools and concepts from the field of Quantum Computing, we construct a quantum network that implements $U_f$ by succesive actions on products of qubits of the universal two-bit control-not gate and of the Hadamard one-bit gate. Equivalent forms and the gate complexity of the proposed networks are discussed, and genuine quantum effects of these quantum SCA (qSCA), are further investigated.
Explicitly, we obtain that: i) instead of simple binaries, with qSCA we can evolve quantum mechanical distribution functions that determine the probabilities for the qubit of the a cell to be along $|0 >$ of $|1 >$ ii) qSCA can be seen as a channel transmiting classical ( factorized state vectors ) and quantum information ( i.e superposition of particle-states but also entangled i.e non-factorizable particle-states; we provide sufficient conditions that such transmission of states can take place during the time evolution of a qSCA. Towards a realization of qSCA as a physical process that could be realized experimentally and so would potentially provide a novel quantum computational machine, we construct an optical circuit made of elementary passive optical elements i.e light beam splitters and phase shifters that are assembled so that they implement the evolution operator of qSCA.
All interested are invited to attend.
Enquires to Yao-Zhong Zhang on 3365 2309 or yzz@maths.uq.edu.au