Physics Letters A 358 (2006) 463–469 www.elsevier.com/locate/pla Towards reversibility in a JJL qubit qualitative model by means of CAN2 paradigm C.R. Calidonna a , A. Naddeo b,∗ a CNR-Istituto di Cibernetica “E. Caianiello”, Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy b Universitá degli Studi di Salerno and CNISM, Unità di Ricerca di Salerno, Via Salvator Allende, 84081 Baronissi (SA), Italy Received 13 January 2006; accepted 20 May 2006 Available online 2 June 2006 Communicated by R. Wu Abstract Reversibility is a concept widely studied in physics and computer science. Quantum systems evolution is described by the time evolution operator U , which is unitary and then invertible. Likewise, reversible computation is characterized by means of invertible properties. Re- versible/invertible cellular automata (CA) are one of the most relevant reversible computational models. Here we introduce a model for a Josephson junction ladder (JJL) device which turns out to be reversible: it is based on a hybrid cellular automata network (CAN), the CAN2 one.  2006 Elsevier B.V. All rights reserved. PACS: 07.05.T; 85.25.C; 03.67.L Keywords: Cellular automata; Josephson junction ladder; Qubit; Reversibility 1. Introduction Reversibility is a concept widely studied in physics and computer science. In particular, reversible computation is char- acterized by means of invertible properties. Reversible logical operations in computers can reuse a fraction of energy, so giv- ing another chance to the high performance computer at a given level of power dissipation [1]. In the past Bennett [2] theoret- ically showed how it were possible to design computing ma- chines based on reversible logic: each state of the machine must have only one possible predecessor state that could be reached during the computation. In the past several studies were per- formed on the connection between reversible classical functions and computation without loss of energy as a solution to the Maxwell demon paradox [2]. In particular, it was shown that any classical function can be represented as a reversible func- tion which can be computed by many elementary reversible steps; in fact the corresponding unitary matrix is decomposable * Corresponding author. E-mail address: naddeo@sa.infn.it (A. Naddeo). into a sequence of many elementary unitary operations. The evolution of a quantum system is described by the time evolu- tion operator U , which is unitary and then invertible; therefore such systems can implement reversibility. Reversible/invertible cellular automata [3] have been growing as one of the most rel- evant reversible computational models in the last thirty years. Toffoli [4] showed that it is always possible to transform an ar- bitrary cellular automaton in a reversible one. Furthermore it is possible to simulate any irreversible one-dimensional cellu- lar automaton, endowed with a finite number of configurations, with a one-dimensional reversible one [5]. Recently the concept of a reversible automaton as a CA with memory has been intro- duced [6]. Here we introduce a new model of JJL device which incorporates reversibility: it is based on a hybrid CA with mem- ory mimicking reversibility. The Letter is organized as follows. In Section 2 we deal with a phenomenological description of fully frustrated JJL focusing on the concept of topological order. In Section 3 basic con- cepts about reversibility in computation are given. Section 4 introduces a particular CA hybrid model, the CAN2 one [7]. Section 5 deals with the implementation of the model in terms of the CAN2 formalism while in Section 6 reversibility is intro- 0375-9601/$ – see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.05.057