Int. Journ. of Unconventional Computing, Vol. 4, pp. 315–340 Reprints available directly from the publisher Photocopying permitted by license only
©2008 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group
Addressing Reversibility in Quantum Devices by a Hybrid CA Approach: the JJL Case C. R. Calidonna1 and A. Naddeo2 1 CESIC-NEC
Italia, Via P. Bucci Cubo 22b – 87036 Rende (CS), Italy E-mail:
[email protected] 2 Dipartimento di Fisica “E. R. Caianiello”, Universitá degli Studi di Salerno and CNISM, Unità di Ricerca di Salerno, Via Salvator Allende, 84081 Baronissi (SA), Italy Received: April 1, 2007. Accepted: September 15, 2007.
Reversibility is a concept widely studied in several research fields. Several contributions have been provided in physics and computer science. Reversible computation is characterized by means of invertible properties [1]. Reversibility is studied in computer science also in order to design less dissipative computers. The evolution of quantum systems is described by the time evolution operator U which is unitary and invertible; therefore such systems can implement reversibility. Reversible/invertible Cellular Automata (CA) [2] are one of the most relevant reversible computational models. Here we introduce a model for a Josephson junction ladder (JJL) device addressing reversibility: it is based on a hybrid Cellular Automata Network (CAN), the CAN2 [3]. Keywords: Cellular Automata, Josephson Junction Ladder, qubit, reversibility. PACS: 07.05.T, 85.25.C, 03.67.L
1 INTRODUCTION Reversibility is a concept widely studied in several research fields. Several contributions have been provided in physics and computer science. In particular, reversible computation is characterized by means of invertible properties. Reversibility is studied in computer science also in order to design less dissipative computers [1]. Reversible logical operations in computers can reuse a fraction of energy, so improving the efficiency and performance of computing processes in terms of dissipated energy. In the past Bennett [4] theoretically
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showed how it were possible to design machines based on reversible logic. In order to make computation logically reversible, 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 performed on the connection between reversible classical functions and computation without loss of energy as a solution to the Maxwell demon paradox [4]. In particular it was shown that any classical function can be represented as a reversible function which can be computed by many elementary reversible steps; in fact the corresponding unitary matrix is decomposable into a sequence of many elementary unitary operations. The evolution of a quantum system is described by an unitary operator, the time evolution operator U , which is invertible; therefore such systems can implement logical reversibility. Reversible/invertible cellular automata [2] have been growing as one of the most relevant reversible computational models in the last thirty years. Toffoli [5] showed that it is always possible to transform an arbitrary cellular automaton in a reversible one; then Morita [6] showed that it is possible to simulate any irreversible one dimensional cellular automaton, endowed with a finite number of configurations, with a one dimensional reversible one. Recently also the concept of a reversible automaton as an historic one or, in other words, a CA with memory has been introduced [7, 8]. Finally, the idea of generalizing the classical notion of cellular automata to the quantum regime is already present in the seminal paper by Feynman [9], in which he argues that quantum computation might be more powerful than classical. Then, a lot of work on such a subject has been focused on arrays of weakly coupled quantum systems where an interaction exists only between neighboring qubits [10–13]. In the last years some specific physical systems have been proposed as candidates for quantum cellular automata, including quantum dot arrays [14] and endohedral fullerenes [15]. Recently, also the role of classical control in the context of a kind of reversible quantum cellular automata has been investigated [16]. However, although there have been several formal definitions [17], the theory is not well assessed yet. In the following we introduce a new model of JJL device which incorporates logical reversibility: a model based on a reversible hybrid CA with memory. At this high level stage of simulation we propose a classical quasi-probabilistic implementation of a quantum device, which is also reversible, a full quantum implementation based on a quantum version of our CAN2 model being the subject of a future publication. The paper is organized as follows. In Section 2 we deal with a phenomenological description of fully frustrated JJL focusing on the concept of topological order which allows for the implementation of a “protected” qubit. In Section 3 some basic concepts about reversibility in computation are given. Section 4 introduces a particular CA hybrid model, i.e. the CAN2. Section 5 deals with the description of the implemented model expressed in terms of the CAN2 formalism while in Section 6 reversibility is introduced. Finally in Section 7 some closing remarks and future perspectives of our work are presented.
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2 JOSEPHSON JUNCTION LADDERS Josephson junction arrays (JJA) are a very useful tool for investigating quantum-mechanical behavior in a wide range of parameters space, from e2 EC EJ (where EC = 2C is the charging energy and EJ = 2e Ic is the Josephson coupling energy; C is the capacitance of each island and Ic is the critical current of each junction) to EJ EC . In fact there exists a couple of conjugate quantum variables, the charge and phase ones on each superconducting island, which results in two dual descriptions of the array [18]: a) through the charges (Cooper pairs) hopping between the islands, b) through the vortices hopping between the plaquettes. Furthermore, in the presence of an external magnetic field charges gain additional Aharonov-Bohm phases and, conversely, vortices moving around islands gain phases proportional to the average charges on the islands [19]. Such basic quantum interference effects found application in recent proposals for solid state qubits for quantum computing, based on charge [20] or phase [21] degree of freedom in JJAs. “Charge” devices operate in the regime EC EJ while “phase” or “flux” devices are characterized by strongly coupled junctions with EJ EC . Recently new kinds of JJAs have been proposed [22], with non-trivial topology, which allow for a novel state at low temperature characterized by a discrete topological order parameter [23]. In fact, the Hilbert space of their quantum states decomposes into mutually orthogonal sectors (“topological” sectors), each sector remaining isolated under the action of local perturbations and protected from leakage through a gapped excitation spectrum. If we choose the two qubit states from ground states in different sectors we get a 2K -dimensional subspace (K is the number of big openings in the array, in general equal to the effective number of qubits) which is “protected” from the external noise [24]. The superconducting phase is characterized by a condensate of 4e charges and gapped 2e excitations. An extra Cooper pair which is injected at the inner boundary can never escape it; therefore, two states differing by the parity of the Cooper pairs number at the boundary are indistinguishable by a local measurement. Conversely, the dual insulating phase, which can be reached by increasing the hc hc charging energy, shows condensation of 2( 2e ) phase vortices and gapped 2e excitations. Because of the hole, the system acquires a new binary degree of freedom characterized by the presence or the absence of a half vortex; the states so obtained cannot be distinguished by local measurements [22]. Once the protected qubit has been implemented, global operators must be found in order to manipulate its degenerate states, say |0 and |1. In this way the NOT operation |0 → |1,
|1 → |0
(1)
can be established together with the Hadamard transformation: 1 |0 → √ (|0 + |1), 2
1 |1 → √ (|0 − |1). 2
(2)
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Further steps are the implementation of non-trivial two-qubit operations (such as the C-NOT gate) and the construction of a register of qubits. It is evident that such arrays could be promising candidates for a physical implementation of an ideal quantum computer [25]. 2.1 Fully frustrated JJLs After such a brief account of general features of “protected” JJA qubits, we focus on the simplest physical array one can devise to meet all the above requests, that is a ladder with N plaquettes closed in a ring with a half flux hc quantum ( 12 0 = 12 2e ) threading each plaquette [26]. The number of plaquettes must correspond to the number of junctions on the vertical links of the ladder, so that each plaquette contains two junctions on the left and right link respectively. Such a condition is fulfilled when each plaquette of the ring contains an odd number of the so called π -junctions [27] (a π -junction is characterized by a π shift in the phase of the wave function on one side of the junction, so that the current-phase relation is I = Ic sin(φ + π )), one in our case, or putting the array in a transverse magnetic field. The first choice is the best in view of a feasibility study of a “protected” qubit because it avoids the switching of an external uniform magnetic field at least at the earliest stage of the definition of the ground states. Now let us describe the system we will study in the following, that is a closed ladder of Josephson junctions with periodic or Moebius boundary conditions. With each site i we associate a phase ϕi and a charge 2eni , representing a superconducting grain coupled to its neighbors by Josephson couplings; ni and ϕi are conjugate variables satisfying the usual phase-number commutation relation. The system is described by the quantum phase model (QPM) Hamiltonian [18]: EC ∂ 2 H =− − Eij cos(ϕi − ϕj − Aij ), 2 ∂ϕi i
(3)
ij
where EC is the charging energy at site i, while the second term is the Josephson coupling energy between sites i and j and the sum is over near 2π j est neighbors. Aij = A·dl is the line integral of the vector potential 0 i flux associated to an external magnetic field B and 0 is the superconducting quantum. The gauge invariant sum around a plaquette is p Aij = 2πf with e , where e is the flux threading each plaquette of the ladder. Let us f = 0 (a)
label the phase fields on the two legs with ϕi , a = 1, 2 and assume Eij = Ex and Eij = Ey for horizontal and vertical links respectively. Let us also make the gauge choice Aij = +πf for the upper links, Aij = −πf for the lower links and Aij = 0 for the vertical links, which corresponds to a vector potential parallel to the ladder and taking opposite values on upper and lower branches.
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Thus the effective quantum Hamiltonian (3) can be written as [26]: 2 2 EC ∂ ∂ −H = + (1) (2) 2 ∂ϕi ∂ϕi i (a) (a) Ex + cos ϕi+1 − ϕi + (−1)a πf i
a=1,2
+ Ey cos
(1) ϕi
(2) − ϕi (1)
(4)
. (2)
Performing the change of variables [26]: ϕi = Xi + φi , ϕi = Xi − φi , the effective quantum Hamiltonian (4) becomes:
EC ∂ 2 ∂ 2 H =− + 2 ∂Xi ∂φi i [2Ex cos(Xi+1 − Xi ) cos(φi+1 − φi − πf ) + Ey cos(2φi )], (5) − i (1)
(2)
where Xi , φi (i.e. ϕi , ϕi ) are only phase deviations of each order parameter from the commensurate phase and should not be identified with the phases of the superconducting grains [26]. When f = 12 and EC = 0 (classical limit) the ground state of the 1D frustrated quantum XY (FQXY) model displays in addition to the continuous U (1) symmetry of the phase variables - a discrete Z2 symmetry associated with an antiferromagnetic pattern of plaquette chiralities χp = ±1, measuring the two opposite directions of the supercurrent circulating in each plaquette. Performing the continuum limit of the Hamiltonian (5):
2 EC ∂ 2 ∂ −H = dx + 2 ∂X ∂φ
2 ∂X ∂φ π 2 + dx Ex + Ey cos(2φ) , (6) + − ∂x ∂x 2 we see that the X and φ fields are decoupled. In fact the X term of the above Hamiltonian is that of a free quantum field theory while the φ one coincides with the quantum sine-Gordon model. Through an imaginary-time path-integral formulation of such a model it can be shown that the 1D quantum problem maps into a 2D classical statistical mechanics system, the 2D fully 1 frustrated XY model, where the parameter α = ( EECx ) 2 plays the role of an inverse temperature [26]. For small EC there is a gap for creation of kinks in the antiferromagnetic pattern of χp and the ground state has quasi long range
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chiral order. We are now ready to introduce the modified ladder [28]. In order to do so let us first require the ϕ (a) , a = 1, 2, variables to recover the angular nature by compactification of both the up and down fields. In such a way the XY-vortices, causing the Kosterlitz-Thouless transition, are recovered. As a second step let us introduce at a point x = 0 a defect which couples the up and down edges through its interaction with the Cooper pairs of the two edges, that is let us close the ladder and impose Moebius boundary conditions. In the limit of strong coupling such an interaction gives rise to non trivial boundary conditions for the fields which add to the periodic ones and can be expressed in an unified way as [29]: (a)
(a)
ϕL (x = 0) = ∓ϕR (x = 0) − ϕ0 ,
a = 1, 2.
(7)
(a)
(a)
Let us now write each phase field as the sum ϕ (a) (x) = ϕL (x) + ϕR (x) of left and right moving fields defined on the half-line because of the defect located in x = 0. Then, let us define for each leg the two chiral fields (a) (a) (a) ϕe,o (x) = ϕL (x) ± ϕR (−x), each defined on the whole x-axis [30]. (a) In such a framework, the dual fields ϕo (x) are fully decoupled because the corresponding boundary interaction term in the Hamiltonian does not involve them; they are involved in the definition of the conjugate momenta (a) (a) = (∂x ϕo ) = ( ∂(a) ) present in the quantum Hamiltonian. Perform∂ϕe
(1)
(2)
(1)
ing the change of variables ϕe = X + φ, ϕe = X − φ (ϕo = X + φ, (2) ϕo = X − φ for the dual ones) we get the quantum Hamiltonian (6) but, now, all the fields are chiral. In this way the ladder, sketched in Fig. 1 for N = 10 plaquettes and the two possible boundary conditions (periodic and Moebius), has a ground state twofold degenerate with antiferromagnetic ordering as a result of currents circulating clockwise and counterclockwise in odd and even plaquettes respectively, hence it can be mapped into a linear antiferromagnetic chain of half-integer spins. Furthermore it can acquire, in addition to superconducting quasi long range order, a topological order parameter [23] due to its peculiar geometry, as it has been shown recently [28]. The equivalence between the ladder with π-junctions and the one with external full frustration is shown in Fig. 2. It is now possible to construct symmetric (s) and antisymmetric (a) linear combinations of such degenerate ground states and then to control their amplitude and relative phase: such operations are needed in order to prepare
FIGURE 1 JJL with N = 10 junctions on the vertical links (0 and π ) and N = 10 loops in a ring geometry.
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FIGURE 2 Equivalence between the Josephson junction ladder with π -junctions and the one with external full frustration (a half flux quantum threading each plaquette).
the qubit in a definite state [25]. We get 1 |s = √ (|0 + |1), 2
1 |a = √ (|0 − |1). 2
In order to manipulate the qubit so obtained we must perform an adiabatic change of local magnetic fields that drags one half vortex across the system and flips the state of the system, so lifting the degeneracy [22]. Based on the above considerations it could be possible to implement a protected qubit [22] with fully frustrated Josephson ladders arranged in a non trivial geometry [28]. It can be shown that the tunneling between the two ground states |0, |1 corresponds to the physical process of creation and annihilation of kink-antikink pairs [31], which gives rise to a sequence of double flips in the antiferromagnetic pattern of spins. A kink-antikink pair can be produced increasing the local magnetic field, that is applying a frustration single sawtooth pulse. Our qubit has N degrees of freedom and N2 double flips are needed to pass from |0 to |1. But there are in general ( N2 )! paths along which such processes can occur. Then, switching off the frustration the system relaxes on the new state and the transition is carried out. Josephson junction ladders with annular geometry have been fabricated within the trilayer Nb/Al − AlOx /Nb technology and experimentally investigated [32]. So in principle it could be simple to conceive an experimental setup for the realization of our protected qubit: the JJL, arranged in a ring geometry, should be equipped with a multi-coil Pbias which can be used to set the system in one of the two ground states |0, |1; another multi-coil Pbarrier with a current Ibarrier is needed in order to control the energy barriers of the single junctions or alternatively N coils can be used to add local frustration pulses to the system and sweep it across zero in order to maximize the mixing of the two ground states. The current Ibarrier sets in the energy scale of the system EJT = 2e Ic Ibarrier , where Ic is the critical current of each Josephson junction. Finally N read-out coils, coupled to external conventional SQUIDs, are needed in order to read out the state of the system after the quantum evolution. In this way a reliable qubit is built up, whose quantum evolution can be controlled in order to perform all possible single qubit logical operations. Then, such qubits can be antiferromagnetically coupled by means of suitable superconducting flux transformers (“eight” coils)
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which provide an inductive coupling and whose strength can be controlled within a wide range of useful values: in this way a quantum register can be realized and all multi-qubits logical operations can be performed. Summarizing, a single non-interacting qubit is described by a double well potential and the external magnetic flux controls the energy difference between the minima, the symmetric situation being for e = 0. Each logical state, |0 or |1, is represented by a wave function localized in a distinct potential well and corresponds to distinguishable flux states trapped in the plaquettes of the ladder with current flowing in opposite directions in alternating plaquettes. When the energy difference ε of the minima of the two different wells is small with respect to the oscillation frequency ω around the minima, ε ω, these states become coupled and the wave functions spread over both the wells, the coupling being maximum in resonance conditions (ε = 0) while the energy eigenstates tend to be localized in one of the wells away from resonance as ε is increasing. The coupling of the states can be described by a tunneling amplitude (ε) and the effective Hamiltonian of any qubit reduces to the regular two-state form in the basis of these logical states: Heff =
1 1 [ε(|00| − |11|) − (|01| + |10|)] = (εσ z − σ x ), (8) 2 2
where σ x , σ z are the Pauli spin matrices: 0 1 1 σx = , σz = 1 0 0
0 . −1
(9)
The diagonal elements of Heff can be easily controlled by an external magnetic field in the z-direction producing an external flux e , while the off-diagonal elements are related to the tunneling amplitude which strongly depends on the energy scale EJT = 2e Ic Ibarrier or, alternatively, are controlled by local frustration pulses. The general state vector of such a qubit is the linear combination of the basis states: α | = α|0 + β|1 = , (10) β so it is described by two complex numbers α and β. Alternatively it can be described by a density matrix ρ whose dynamics is ruled by the Liouville equation: i
∂ ρ = [Heff , ρ]. ∂t
(11)
The density matrix can be expressed as the linear combination of the SU (2) generators as ρ=
1 (1 + λx σ x + λy σ y + λz σ z ), 2
(12)
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where λa = σ a , a = x, y, z are the components of the so called “coher− → ence” vector λ . So, the two ground states |0, |1 correspond to the two − → − → coherence vectors λ 0 = [0, 0, 1]T , λ 1 = [0, 0, −1]T respectively. The coherence vector satisfies the dynamical equation: − → dλ − → − → = × λ, dt
(13)
− → where i = Tr(σ i Heff ), i = x, y, z; so, from eq. (8) we get = [− , 0, ε]T . In equation (13) the precession of the coherence vector around − → is codified, with the coherence vector describing the state of the qubit and − → describing the influence of the environment. Let us now briefly consider an elementary single-qubit rotation in the λ-space; if the barriers are low, ε, − → − → and e = 0, then ∼ = [− , 0, 0]T and the result is the precession of λ around the −x axis. In particular, a rotation by an angle π realizes the NOT operation, which corresponds to the unitary operator: 0 1 U−x,π = . (14) 1 0 When the inductive coupling among qubits is turned on, there could be a bias in, say, the j -th qubit even though ej = 0 and, as a consequence, its logical states may be asymmetric. In the approximation in which every JJL can be considered as a two level system, the system of flux linked qubits can be described by an effective Hamiltonian of the kind: H eff = εj σjz + j σjx + kj σkz σjz . (15) j
j
jk
In order to control such Hamiltonian, we should be able to modulate the tunneling amplitude of each qubit as well as to switch on and off the magnetic coupling between neighbors qubits. In the following sections we try to simulate the single JJL qubit and to realize the simple rotation which gives rise to the NOT operation, the simulation of multi-qubit logical operations will be the subject of a future publication.
3 REVERSIBILITY AND COMPUTING In 1961 Landauer [33] discussed the limitation of the efficiency of a computer imposed by physical laws. In particular he argued that, according to the second law of thermodynamics, the erasure of one bit of information requires a minimal heat generation kB T ln 2, where kB is Boltzmann’s constant and T is the temperature at which one erases. Its argument runs as follows. Since erasure is
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a logical function that does not have a single-valued inverse it must be associated with physical irreversibility and therefore requires heat dissipation. A bit has one degree of freedom and so the heat dissipation should be of order kB T . Now, since before erasure a bit can be in any of the two possible states and after erasure it can only be in one state, this implies a change in information entropy of an amount −kB ln 2. He argued that, since entropy cannot decrease, it must appear somewhere else as heat. Such an argument is based on the implicit assumption that information entropy translates into physical entropy. More clearly [34], the one-to-one dynamics of Hamiltonian systems implies that, when a bit is erased, the information which it contains has to go somewhere. If the information goes into observable degrees of freedom of the computer, such as another bit, then it has not been erased but merely moved; but if it goes into unobservable degrees of freedom such as the microscopic motion of molecules it results in an increase of entropy of at least kB ln 2. So, a crucial result of Landauer’s theory was that there is no fundamental energy dissipation limit associated with logically reversible computation, while physical computation always involves some energy dissipation; in this way a connection is established between logical reversibility and physical reversibility. Inspired by such studies, a considerable amount of work has been made on the thermodynamics of information processing, which include Maxwell’s demon problem [4], reversible computation [35], the proposal of the algorithmic entropy [36] and so on. The thermal cost of more general information processes has also been investigated [37]. Bit erasure is the simplest logically irreversible process because it requires a one bit input and always returns the null state as the output, so making impossible to recover the input value from just the output value. But it is possible to make logically reversible the erasure of a bit in a specific cell of a cellular automaton if a copy of the bit is previously made. Landauer provided a method [33] for building up a logically reversible erasure in a bistable potential by taking into account the value of the bit being erased. In the context of cellular automata, such a process involving a “demon” cell can be used to copy the bit stored in a cell and then perform a logically reversible erasure process according to the value of the stored bit. So, it is clear that a logically reversible erasure process corresponds to the sequence copy bit to the demon, then erase bit. The action of copying a bit to the demon implies the physical interaction between the cell and the demon cell in order to set the demon cell into the same state as the cell. That can be viewed also as the demon cell which reads the information in the primary cell. The role of the demon can be played by another cell or by any other read-out circuit which stores a copy of the bit. Without the demon cell the erasure process is logically irreversible as no copy of the bit is retained. We now argue on the amount of energy dissipated during the two processes of erasure and copy, then erase. Indeed erasure without a demon cell generally requires an amount of dissipated energy on the order of the signal energy and therefore larger
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than kB T ln 2. Conversely the copy, then erase process does not require any minimum amount of energy to be dissipated, given the constraint of slowing the process down. In other words, if a copy of the information is stored, the energy dissipated in erasing a bit from a cell can be made arbitrarily small if the erasure time is sufficiently long. In fact, the copy operation does still require the transfer of information between the cells of the cellular automaton with an amount of energy transferred greater than kB T ln 2 per bit, but now such a signal energy is not lost to the environment and could be recovered and employed again. The energy balance, and then the entropy change, of the copy, then erase process can be better understood if rephrased in terms of the Szilard version [38] of the Maxwell’ demon paradox and its reformulation made by Landauer and Bennett [4]. Let us suppose that there is a physical system (a box filled with a one atom gas) and a physical memory device (demon) whose state can record the results of measurements made on the system (the copy operation). In order to avoid paradoxes, entropy must be calculated for the joint system comprising both the memory and the physical system. Measurement is the step whereby information is created, so it creates a correlation between the memory and the physical system: knowing the state of the memory helps us to predict the state of the system and vice versa. During the measurement, an entropy change may occur in the joint system, of sign and magnitude determined by the specifics of measurement and memory. Of course, any entropy decrease during measurement will be accompanied by an entropy increase of equal or greater magnitude in the joint system’s environment; otherwise the second law would be violated. Landauer showed that entropy is released into the environment when Maxwell’s demon erases the memory associated with his reversible measurements. Bennett used algorithmic information theory to prove that this entropy is equal, on average, to the amount of entropy that the demon could possibly destroy in the system by virtue of the information he has regarding the system. In the following sections we will try to implement the copy, then erase sequence in order to build up reversibility in our CA model for the JJL qubit.
4 CELLULAR AUTOMATA AND A HYBRID PARADIGM FOR DISCRETE COMPLEX SYSTEMS SIMULATION The Cellular Automata Network version 2 (CAN2) model [3] extends the standard CA model introducing the possibility to have a hybrid network of standard cellular automata components and global operators. Each automaton of the network represents, for instance, a component of the physical system to be simulated and the connections among the network automata represent a disjoinable evolving law which characterizes the evolution of the physical system.
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4.1 The Cellular Automata Network Model version 2 (CAN2) The CAN2 model [3] provides the possibility to simulate a multistage evolutionary process. It allows the following facilities: • to simplify the modelling of a complex system, and • to codify and classify the kind of the adopted simulation model. In fact we decided to adopt a hybrid model between a functional model, in which there are directional flows of signals among transfer functions, and a spatial model. Intuitively our model allow us to determine all the components of a complex system, which are ruled by a defined transfer function. Furthermore signals are also determined, which follow a well defined directional flux: they represent properties of the components which act as input and output of each transfer function. In this way the CAN2 model is a hybrid vision and finally the whole evolution of a complex system can be divided into more than a stage. In conclusion, such a model offers global informationprocessing capabilities which are not explicitly represented in the elementary components of the network or in their interconnections. In the CAN2 model an automaton is denoted by a name and its behavior is described by a set (possibly empty) of properties (the automaton grids), by a neighborhood type, by a boundary condition and finally by a transition function. A property corresponds either to a physical property of the system to be simulated, such as the temperature, the volume and so on, or to some other feature of the system, such as the probability for a particle to move. Furthermore each property corresponds to a computational grid of a standard CA automaton. In this scheme a cell of an automaton consists of an array of values, each one given by the corresponding value of the property’s cell, as sketched in Fig. 3. According to our model, it is not always possible to represent all the system components as properties of a single automaton. This is why it could be necessary to represent a physical system as composed of more than one automaton. In the CAN2 model this is done through the use of the cellular automata network abstraction. At each time step of the simulation a cellular automata network can be represented as an acyclic directed graph which is called the CAN dependence graph, where each node represents an automaton and each edge a dependence relation. In a simplified view, given a network of two automata, A and B, where AδB (which means that A depends on B), the transition function of the automaton B is a functional of the transition function of the automaton A and the transition function of the whole network is the functional composition of the transition functions of the two automata composing the network. From an execution point of view, the network execution consists of executing, at each time step, all the transition functions of the cellular automata
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FIGURE 3 An automaton cell and its correspondence with more properties.
composing the network, where the execution of the automaton A must precede the execution of the automaton B because of their dependence relation. Therefore we need to employ a network of cellular automata in all the cases where different values of the properties must be used at the same time step evolution. According to the component decomposition method, see ref. [3] for details, it is possible to model a complex system in terms of its components; these could be cellular automata together with global operators. The last ones could account for phase transitions (stages) or global behaviors respectively. The application of the transition function and/or the global operators must follow a well defined order. Once stages are determined through the substitution relation, the subsequent stages can employ the output signal flows (properties), from their previous stages, by means of the composition relation. A precedence relation between a global operator and a cellular automaton occurs if a global operator accesses a property of the cellular automaton; or, in an opposite direction, a cellular automaton accesses a global variable defined through a global operator. 4.2 The CAN2 model: a more formal definition According to the considerations made in the previous subsection we are now ready to give a more formal definition of a CAN2 network in one dimension cellular space, which can be trivially generalized to higher dimensions.
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Let L be a cellular space; a CAN2 network is the following tuple: L, X, S, G, P , Pvar , f, g, F
(16)
where • L = {x : x ∈ N , 0 ≤ x ≤ lx } is the set of points with integer coordinates in the finite region where the system evolves, N is the set of natural numbers, and lx is the upper bound of the set of points, i.e. it determines the bounds of the region of the system evolution. • X#Ntot , where #Ntot is the cardinality of the neighborhood set1 , or X = [i − r, i + r] ⊂ I, is the set which identifies the geometrical pattern of cells which influences the cell state change, i.e. the neighborhood set for each cell i, where r is the radius. ms • S is a finite set of states, where S = Xi=1 Si is the Cartesian product of all the sets of sub-states and ms is the total number of states.
• G is a finite set of global variables, where G ⊆ R (R is the set of real numbers). • P is a finite set of parameters, where P ⊆ R. • Pvar is the set of global parameters. • f is the set of the cellular automata transition functions. • g is the set of global operator functions. • F : S #Ntot → S is the transition for all the cells in L. Complex systems, which are made of different components, give rise to different transition functions, i.e. different cellular automata nodes for the network. In such a case all the components can be enumerated according to two main driving procedures: 1. dependence relation constraints; 2. consequent sequential transition function application. Sometimes, when a direct dependence is not present a different sequence of application of the transition function between components can be adopted; this last issue will be addressed later. In order to simplify the definition of the total transition function for the whole system, it could be useful to separately define the CA component and the global operator components as follows. 1 If our model is made by M components, let us denote with i each of them, we can define the #Ni . total neighborhood set as X#Ntot = M i=1 X
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Let f1 · · · fM be the transition functions for a system made of M components; the total transition function of the network, denoted with F , can be expressed in terms of a composition law such as: F = f 1 ◦ f 2 ◦ · · · ◦ fM
(17)
where ◦ is the composition operator. From an execution point of view the following precedence relations hold: f1 ≺ (f2 · · · fM ) f2 ≺ (f3 · · · fM ) . fM−1 ≺ fM This implies that any transition function must be executed before the M-th, ≺ being the precedence relation. Let g1 · · · gK be global operators for the system made of K components, then the global function g of the network can be expressed in terms of a composition law as follows: g = g 1 ◦ g 2 ◦ · · · ◦ gK
(18)
where ◦ is the composition operator. From an execution point of view, if relation (18) holds the following precedence relations must be satisfied: g1 ≺ (g2 · · · gK ) g2 ≺ (g3 · · · gK ) , gK−1 ≺ gK which implies that any global operator, but K-th, must be executed before the last one. Let A be a cellular automaton, whose property is denoted by p and transition function by f, and GO a global operator, whose global variable is denoted by gv , and whose function is denoted by g. If A needs to know the value/s of a global variable gv at the same macro-step, defined as the whole network time step evolution, in order to evolve at each micro-step, defined as the time step evolution for each component cellular automaton or global operator node, the execution of the g function must precede the execution of the A transition function f. So, merging the previous definitions we obtain the following F :g◦f
(19)
F : M,K i,j =1 [gj ] ◦ [fi ]
(20)
where the [gj ] ([fi ]) symbol implies that its presence could be optional and the corresponding precedence relations must be expressed according to it.
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4.3 The hybrid components and related computational issues The requirement of introducing components and their composition mechanism is needed in order to have the possibility to represent a complex system as a composition of more than one cellular automaton or global operators. In the CAN2 model this is made possible through the abstraction of a network of cellular automata. The introduction of a precedence relation between the network components is entirely due to the owner-rule which obeys to the following requirements: • each cellular automaton is the owner of its properties; • each global operator is the owner of its global variables; • only owners can update their properties/global variables; • non-owners can access, in the same macro-step, properties/global variables only after they were updated. A network of cellular automata can be represented as a graph, the CAN precedence graph, where: • nodes represent cellular automata or global operators components, • edges represent the precedence relations between nodes. Summarizing, the introduction of the CAN2 model implies that: 1. from a functional point of view, the transition functions and/or the global operator functions could be built up as parts or phases of the whole function describing the network function evolutionary law; 2. from an execution point of view, the network execution consists, at each macro-step, of executing all cellular automata/global operator nodes belonging to the network; the executions must obey to the precedence relations between nodes.
5 A CAN2 QUALITATIVE MODEL FOR THE JJL QUBIT Let us now recall our qualitative model for the JJL system introduced in [3]. This is the starting point in order to introduce reversibility in our generalized JJL model. In our model we identified respectively: • a grid, • a boundary condition, • each plaquette with each cell grid, • the phase differences values at each cell side for the device initialization,
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• a set of parameters, such as the current, the capacitance and so on, • a frustration single sawtooth magnetic pulse B. Given the considerations above, see refs. [3] for details, let us now formalize a definition of the CAN2 model for the JJL system under study as follows: L, X, S, G, P , Pvar, f, g
(21)
• L ⊆ N is the set of integer points in the finite region, the array, where the system evolves; each point identifies a cell. The lattice grid is a linear array with, for example, #L = 10 cells. • X is the neighborhood set x − 2, x + 2 for each cell x. • S = S1 × S2 × S3 × S4 is the set of state values; it is specified by the cartesian product of the finite sets of values of the four following sub–states: 1. Pseudo_S, pseudospin assuming (−1, 1) values, 2. Mp, magnetic pulse for each cell, fixed and invariant for each macro time step, 3. LABEL, cell label in order to identify the cell, corresponding to a monotonic enumeration for all the cells, itself invariant for each time step. 4. FLIP is the flip state in order to register if pseudospin flipping has been taken. • G is the set of global variables: 1. Btot is the total applied magnetic pulse, and 2. Start c is the number of the corresponding starting cell. • P is the finite set of global constant parameters: − I, current inside the cell − C, capacitance • Pvar is the finite set of the CA component variables: STEP is the step iterator which allows to trigger the evolution. • f : S → S is the deterministic state transition for the determination of the pseudospin state and values. • g : S2 → G expresses the global operator which controls the total magnetic pulse applied on the system.
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• gs : G → G expresses the global operator which chooses randomly the driving cell. In view of the implementation of a protected qubit (see Section 2) the boundary condition topology is annular. In order to get a transition between the two ground states the magnetic pulse period must be related to the CA time step and it is equal to the pulse period, in order to capture the maximum pulse value. The flipping procedure between the input and output states implements a tunneling between the two ground states which corresponds to the physical processes of creation and annihilation of kink-antikink pairs [31]. In general, for N cells, (N/2) double flips are needed to switch from |0 to |1. In the quantum regime such tunneling processes display (N/2)! paths along which they can occur. Our model will select out one particular path, so giving rise to a high level description of such tunneling processes: in fact, at this preliminary stage we are interested only in the net result, i.e. the NOT operation. For this reason we choose to use a double step for the CA transition component, with each time step equal to the half of the single sawtooth magnetic pulse period. The CA component has, as initial condition, the pseudospin configuration obtained in the precedent stage since it must obey to an antiferromagnetic arrangement and the flipping state is zero. At the initial time, our device is in a steady state, in one of the two possible ground states. Each parameter is fixed and the LABEL values are fixed for all transition steps, but the variable STEP is initialized to each macro-step T. The studied model reproduces the NOT operator in a more detailed model adopted in the past [3] in which, together with the basic behavior, one of the possible flipping procedures is taken into account. The general transition function takes into account the coupling factor, adding an external frustration, as a single sawtooth magnetic pulse acting on each lattice cell. The system transition is assumed to be given by the possible simultaneous application of the two global operators followed by the transition function. The evolution of the model obeys to the following function F : S_f ×G×P ×Pvar → S_f ×G×Pvar. The transition function scheme shows two global operators, the pulse application and the random starting cell chooser, and the transition function applies repeatedly the cellular automata component according to the multiplicity related to the double flips. The quasiprobabilistic behavior is due to the chooser operator which randomly at each network time step chooses the driving cell for the flipping procedure. The system transition, expressed in pseudocode, is sketched in the boxes of Fig. 4. Let us explain some implementation choices shown in Fig. 4. The Pulse Operator assures that the device is under frustration and the transition could occur; otherwise an error will occur and the device is not active as a NOT gate. As this is a classical simulation of a quantum device, a probability (or a random choice) for the driving cell is adopted. In the case of a full quantum
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FIGURE 4 The JJL system components according to a CAN2 simulation.
simulation, we will not need any driving cell, as all of them are equi-probable a priori. In this sense here we can choose to start the flipping procedure with even (odd) cells to be updated, and due to the physical process of creation and annihilation of kink-antikink pairs, our model modifies the pseudospin of the i-th and the (i + 1)-th cell. Concerning the model execution, as no relation exists between the two operators the global operators components could be executed concurrently: such a consequence is a very interesting one from a high performance computing point of view.
6 INTRODUCING AND EXPLOITING REVERSIBILITY IN THE JJL QUALITATIVE MODEL In order to build up a reversible computer we could use: • AND, OR and NOT gates, • or new logic gates.
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c
x1
x2
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
→
cout
y1
y2
0 0 0 0 1 1 1 1
0 1 0 1 0 0 1 1
0 0 1 1 0 1 0 1
TABLE 1 Petri-Fredkin gate logic
In the last case reversible logic gates could be be adopted in order to fulfill such a request. Reversible gates are such that the input of the gate can be reconstructed starting from its output.2 The possibility of introducing such devices was considered for the first time by Toffoli about thirty years ago [1]: he showed how to implement in a reversible way the AND and XOR gates, in particular he built up a three-bit gate which could implement both the functions so as to be considered the universal reversible computation gate. Then, on the basis of the seminal paper of Landauer [33] Toffoli and Fredkin [35] built up the celebrated Conservative Logic, that is a mathematical model of computation, in order to improve the efficiency and the performance of computing processes in terms of dissipated energy. The model is based on the Fredkin Gate, a universal three-input/threeoutput Boolean gate which is both conservative and reversible, as showed in Table 1; in particular it treats signals as unalterable objects which can be moved during the course of computation but never created or destroyed. As a matter of fact such a gate was introduced by Petri [39] some years before Fredkin and thus in the following we will refer to it as to the Petri-Fredkin Gate. In Fig. 5 we sketch how a Petri-Fredkin gate works, in particular schemes depict gate input and output signals according to the two possible values of the control signal, c = 0, 1. Now the idea is to implement a copy, then erase sequence in order to obtain a reversible qualitative qubit. In order to build up such a device literature tells us that reversibility could be addressed avoiding the bit erasure. The evolution of the Toffoli and Petri-Fredkin gate may be viewed as the foundations of the actual reversible logic and computation. Deutsch [40] showed how to obtain a universal quantum computation, defined as an arbitrary unitary transformation on a discrete Hilbert space 2 It is trivial to show that conventional AND gates are not reversible.
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FIGURE 5 Operations in the Petri-Fredkin gate.
spanned by the set of all the states of a collection of qubits, by means of a simple generalization of the scheme for building a reversible classical network. In fact there exists a close connection between classical reversible computation and quantum computation because all unitary quantum operations need to be reversible: in particular, classical gates can be implemented in a quantum way by making the computation reversible. He generalized the operation of a three-bit gate from one which performs transformations on the 8 = 23 possible states of three bits to one which performs unitary transformations within the 23 -dimensional complex vector space spanned by the states of the three qubits. Deutsch proved that all unitary transformations could be obtained from one operating upon three qubits, which is a straightforward generalization of the Toffoli gate. But we must recall that the difficulty in building up a quantum gate increases with the number of inputs to the gate. So DiVincenzo [41] showed that two input gates are universal for quantum circuits by proving that all Deutsch’s three-bit gates could be realized in terms of a set of one- and two-bit gates: that makes more real the possibility of building a quantum computer. Later it was shown that almost any two-bit gate is universal [42] and that the classical controlled not gate, together with all one-qubit gates, make a universal set as well [43]. All the above considerations are our starting point for the introduction of reversibility in our device. 6.1 CA with memory According to the classical definition CA are memoryless: i.e. the new state of a cell depends on the neighborhood configuration only at the preceding time step. Some authors, such as Wolf-Gladrow [44], consider the CA rules as depending on the state of the cell to be updated. The memory concept introduced in refs. [7, 8] is very different: it can be viewed as a historic one, i.e. the CA state is updated, being regularly augmented with the cell state at the previous time step. Within this conception, historic memory of all past iterations can be incorporated into CA by featuring each cell by a weighted mean of all its past states: indeed, while the update rules of the CA remain the same, each cell remembers a weighted mean of all its past states [7]. The historic weighting is defined by a geometric series of coefficients based on a memory factor α [8]. The choice of the memory
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factor 0 ≤ α ≤ 1 simulates the long term or remnant memory effect: the limit case α = 1 gives rise to a memory with equally weighted records (full memory), whereas α 1 intensifies the contribution of the most recent states and diminishes the contribution of the past ones (short type memory). Finally, the choice α = 0 reproduces the memoryless model. For further details on this approach see refs. [7, 8]. 6.2 Memory and reversibility According to ref. [8], in order to reverse the state of a Petri-Fredkin gate the memory concept should be introduced; this implies that we need to consider the system at the t − 1 and t evolution time steps. In fact, in order to recover backward and forward the state we must remember the immediate previous state and the actual state. Following the approach sketched above it could be feasible to use a double address space in order to store the state value at the previous time step and at the actual one. On the other side it could be useful to store the values of the global variable representing the driving cell at the actual as well as at the previous time step in order to be able to fulfill the reversibility requests3 . This is enough to mimick the high level unitary transformation effect between two subsequent system transitions requested by the reversibility constraint. In fact, we recover the NOT device behavior (at high level due to the classical simulation scheme adopted) and the reversibility properties, once enough information on the recovery of the previous states is provided. It is worth to note that this implies the doubling of the requested memory space, as shown in Fig. 6 where for simplicity only the evolution of the boundary cells is depicted.
FIGURE 6 Memory expansion in 4 time steps. 3 Considerations regarding its implementation will be given later.
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FIGURE 7 Modified Chooser Operator for reversibility.
Within our model, in order to simplify the above scheme and the memory space requests, a reliable alternative could be to specify a condition without considering the above state addition. The solution consists in storing the sequence of the driving flipping cells in order to capture the computing properties of the model at each time step. According to Toffoli [1] a Turing Machine was supposed to implement reversibility with its infinite tape. Here we suppose that the vector storing the driving cell should have the same length of the number of the macro-steps of the simulated device. Let us now explain in more detail our approach. In our case the Chooser Operator has a crucial role and it must be modified in the following way in order to achieve our reversibility goal: a further global variable must be considered in order to store the sequence of chosen cells at each time step for the whole system evolution. This is enough for our model in order to recover all the states since the starting point and not only the previous one. The new procedure can be depicted for clarity sake as sketched in Fig. 7. The Chooser Operator is equipped with a global vector (a global variable which is able to store more values) which stores at each time step the starting cell for the evolution 4 . In this way it is much simpler to recover the previous values as, for each time step directly accessible, the first evolution cell can be recovered. This approach has two main advantages: 1. we are able to recover all the previous states as each starting cell is stored and, in this case, the previous system state is exactly the negation; 2. we can avoid to store all the cells values, what could be very cumbersome in the case of a device with a high number of cells; instead we restrict to the storage of one value for each time step evolution. In order to consider a possible qubit implementation we define two different input and output wires according to the regular and reversed behavior per each 4 In fact, the vector length coincides with the number of iterations.
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time step: • regular behavior: input (1 wire) (St−1 ) - output (St ,Vector_StartC) (2 wire) • reversible behavior: input (2 wire) (St ,Vector_StartC) - output (St−1 ) (1 wire) Both wires should coexist in order to guarantee the I/O device properties. Furthermore, to simulate the reversible CA component this task is trivially accomplished by decreasing STEP as well as by reversing the cyclic function. The CAN2 network remains unchanged and it could takes as input both configurations, where the second shows a reversed input–output. The new operator does not modify the precedence relations and the components within the CAN2 network. So, the previous considerations about a possible parallelism scheme are still valid and could yield to a computing scheme with a very good performance. 7 CONCLUSIONS AND OUTLOOKS In this paper we addressed the logical reversibility issue in the context of a generalized JJL qubit device. It was shown how it is possible to view reversibility, in whole generality, by considering a quasi-probabilistic model behavior. The advantage in considering such a scheme relies strongly in avoiding the memory waste and the consequent energy dissipation. We need to store only one variable per step instead of a set of values (which depends strongly on the number of the array cells). Our simulation of the JJL qubit is quasi-probabilistic and takes into account only the nearest neighbors interactions, ignoring the spreading of entanglement which is a strongly non local feature. Further reversibility issues could be addressed, in particular the adoption of a mechanism such that right values for the device could be chosen selectively and the adoption of a doubling memory scheme as simplified in our paper. In this case, due to the high level model representation and simulation together with the finite number of cells, our NOT gate requires only two possible configurations. The situation in a lower representation level model is very different, as the state selection could not be reduced to a simple combinatorial computation of the selection of the single cell value. In fact, in the last case a full quantum simulation of the proposed device could be feasible, which requires a generalization of the CAN2 model to the quantum regime. Such interesting issues are still under study and will be addressed in a future publication. ACKNOWLEDGEMENTS We thank an anonymous referee for useful suggestions which allowed us to improve the presentation.
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