Cellular Automata as a Model of Physical Systems

Cellular A utomata as a Model of Ph ysical Systems D O N N Y C H E U N G 1 , C A R L O S A . P ´ E R E Z - D E L G A D O 2 , 3 1 Department of Computer Science , University of Calgary , Calgary , AB, T2N 1N4, Canada 2 Scho ol of Computer Science, University of W aterloo, W aterloo, ON, N2L 3G1, Canada 3 Department of Physics and Astron omy , University of Sheffi eld, Sheffield, S3 7RH, UK Cellular Auto mata (CA), as they are presented in the literature, are abstract mathematical models of computatio n. In this pa- per we pre s ent an altern ate ap proach: using the CA as a m odel or theory of physical systems and devices. While th is app roach abstracts away all details of the un derlying ph ysical system, it remains faithful to the fact that ther e is an un derlying physical reality wh ich it describes. This impo ses certain r estrictions on the types of computatio ns a CA can physically carry out, and the resources it need s to do so. In th is paper we explore these an d other consequ ences of o ur reformalization . 1 INTR ODUCTION In recen t years ther e has bee n an ever increasing interest in qu antum models of computation. Quantum versions of trad itional com puting mo dels such as the Turing mac hine [4, 2], finite automata[6], families of acyclic circuits [16], as well as cellula r automata [11, 14] have all been propo sed. All these quan- tizations ar e of great impor tance. At the same time, th ere is a drawback th at these quantization s are pro ne to s uffer . The pr oblem is, that n one of these are ph ysical theories. Concretely , a proper quantization takes a classical Newtonian theory of the natural world , such as Ham iltonian mech anics, and transf orms it by adding quanta , non- commutin g observables, etc. Howe ver , none of th e models mentio ned above 1 are phy s ical theories. Qu antizing these, may , or may not, lead to a sensible model of how nature works. This would depend on how p hysical the compu- tation model was to begin with. In this paper our c oncern is cellular automata. Despite CA being u sed to model natural phenom ena, the model is not a the ory of n ature. As a brie f example, consider ho w the update rule of a CA is stated to be applied to all cells instantaneously . F or this to happen, there wou ld have to be a m ethod for synchro nizing cells acro s s a potentially infinite lattice. Setting aside su perluminal commun ication for a moment, it is clear that the CA needs a method for synchroniz ation acr oss lattice cells. Ther e are se veral algorithms fo r CA synch ronization already in the liter ature [3]. In th is paper we are not concern ed wit h the alg orithms, or how the CA cells sy nchronize. Rather , we ar e conce rned with the fact that they ha ve to synchronize, s ome- how , and that this synchr onization req uires physical resources. As with any model o f compu tation (as descr ibed above), a drawback of using the traditional CA definition as a starting point for a quantized version, is that at the end, on e is left with a model that may o r may not sensibly rep- resent r eal physical system s . W ith all previously p roposed models of QCA, it is ind eed an impo rtant question when , and when not, they have analo gous physical systems. In this pape r we attempt to r ephrase CA not as a mathematical constru ct but rath er as a physical theory , or model, that descr ibes an d abstrac ts a family of r eal-w orld system s . This will allow us to p roperly form alize well-kn o w n facts about implemen tations of CA. For in s tance, the use of extra memo ry , that is the seco nd lattice that is gen erally used to implemen t CA algo rithms in trad itional compu ters, is shown not to be an “imp lementation detail”, but rather a necessary requiremen t of any physical CA device. Ultimately , this refo rmalization of CA will co nstitute th e classical fou n- dation up on wh ich we will c onstruct o ur qu antum CA mode l. A paper with the details o f this Q C A fo rmalization has b een prepare d in tandem w it h this paper, and h as now ap peared as [9]. In that paper we concentr ate fully on the quantum mo del. W e show that QCA in this fo rmalization always rep resent real quantum systems. Here we concen trate on the reformalization of the classical CA, and some of the lessons we can learn from it. 2 CLOSED CELLULAR A UTOMA T A W e recall the standa rd definition of the cellular automaton: 2 Definition 1. A cellular automaton (CA) is a 4-tuple ( L, Σ , N , f ) con si sting of a d -d imensional lattice of cells L = Z d indexed b y inte gers, a fi nite set Σ of cell states, a neigh bourhood scheme N , which is a fi nite list of lattice vectors fr o m Z d , and a local transition function f : Σ N → Σ . A con figuration is simply an assignmen t of states from Σ to each cell in L . Given a configur ation, th e tra nsition fu nction f determine s a new co nfigu- ration after a single discrete time step. For each cell x , th e ne w state is giv en by applying f to the curren t states of the cells in the list x + N . When we co nsider this defin it ion of a CA as a m athematical object, the ev o lution of a CA w it h a giv en loca l transition function f fr om a n initial configur ation is well-d efined. However , if we wish to u se this to model phys- ical systems, questions abou t sy nchronization arise: if we app ly the transition function f to a n eighbourhoo d and up date the state o f a pa rticular cell, we alter th e input d ata for the neigh bouring cells. I n order to ad dress this, some external me mory resou rce n eeds to be u sed to sto re these n e w states bef ore the cells themselves can b e upd ated. For example, w hen imple menting CA algorithm s on traditional comu puters, a stand ard way to a ddress this is to h a ve a second lattice o f memo ry cells which can store th e n e w config uration as it is comp uted. Af ter the en tire config uration is c omputed, we may up date the state of each cell of the original lattice ind ependently .[1 5 ] W e can physically implement this strategy by having, for example, a lattice of cells with two registers, u si ng cell states Σ × Σ . On e register would be u sed to store the configur ation, while the other register would be used as temporary storage. There are two main points her e th at we would like to h ighlight. First, in an actual ph ysical device, ther e m ay not be a nee d fo r distinguishing two different types of memory . While the two-register model we described above giv es us a way to see how a physical cellular automata device can b e used to intrinsically simu late the ev olution of any CA, the d e v ice itself is cap able of implementin g a more gener al set of tra nsitions. Second, if w e are to avoid synchro nization issues in these physical de vices without external inter actions with ou tside resour ces, it is necessary to adop t an interaction -update model. That is, at each time-step of the evolution of su ch a d e v ice, there has to be an interaction phase where the cells intera ct w it h each o ther accord ing to some loc al rule, f ollo wed by an upd ate phase, where each in ternal state o f each cell is indep endently updated. This requ irement is indepen dent of how the cells attempt to co-ordinate and synchron ize th eir u pdate procedure s with one another : any such method req uires this extra memo ry space, a nd tiered step approach . 3 W e now consid er a formal m odel which provid es an ab st raction fo r any ev o lution of a phy sical cellular au tomata device. First, the interaction phase will inv olve perfo rming a lo cal interaction function f : Σ N → Σ N on th e neighbo urhood of each cell. This is a g eneralization of the act of compu ting the local transition fun ction and storing the result in au xiliary memory . Note that in th is mod el, th e local interaction function is a map fro m a neighb our - hood to itself, ra ther than fr om a neig hbourhoo d to a sing le cell. In o rder to av o id the issues of sy nchronization, we must ensure th at this function can be applied to a p articular n eighbourho od in a mann er wh ich d oes not interfer e with its app lication to any oth er neighbour hood. In ou r previous illustration, this was done by lo gically par titi oning each cell in to two sep arate memo ry registers, one of w hich is never overwritten. F or a more general p icture, we introdu ce the notion of translation commutativity . Definition 2. Given a local interaction fun ction f : Σ N → Σ N for a lattice L a nd a neighbou rhood scheme N , we say that f is translation commutative if for any two c ells x , y ∈ L , f x and f y commute, wher e f x denotes the local interaction function f ap plied to the neighbourh ood N + x . As lo ng as the lo cal interaction function satisfies this p roperty , it may be applied to e ach cell of th e lattice in any or der , or even in parallel. Th e evo- lution remains deter ministic. Finally , the up date phase will simply ap ply a local update fun ction g : Σ → Σ to each cell. This o ccurs after all o f the local interaction function s have been applied . W e ca n encap sulate this into a f ormal mod el, which we ha ve named Closed Cellular Automata to emp hasize the fact tha t we ar e considering cellular au- tomata as clo sed physical systems, wh ich do no t interac t with external re- sources of any type. Definition 3. A Closed Cellular Automaton (CCA) is a 5 -tuple ( L, Σ , N , f , g ) consisting of a lattice of cells L = Z d , a finite state spa ce Σ , a n eighbourhood scheme N , a translation co mmutative loca l inte r action functio n f : Σ N → Σ N , and a local upda te fun ction g : Σ → Σ . The g lobal transition fu nction f or each time step con si sts of two p hases. First, the interaction ph ase co nsists of applyin g f to each cell of the lattice L . Since f is translation co mmutati ve, the transition func ti ons may be applied in any order . W e may thus define the globa l interactio n fu nction as F = Q x ∈ L f x . For some physical d e v ices, these tran s ition function s may happen simultaneou s ly and in par allel. Howe ver , this is not assumed by the d efinition of CCA, and it is n ot n ecessary for having a well-defined global interactio n 4 function F . The interaction phase is followed by the upda te ph ase, which consists o f applyin g g to each c ell. W e may a ls o define the glob al update function G = Q x ∈ L g x . In each time step, th e global transition fun ction GF is applied to the entire lattice. In our discussion ab o ve, w e have seen that a n y CA can be intrinsically simulated by a Closed CA. Howe ver , we can also v ie w the set of Closed CA as a subset o f traditional CA. T o see this, we d efine the extended neigh bour - hood of a cell to be th e union of a ll neighbo urhoods con taining th e cell. In one time step , the new state of this cell ca n be comp letely determined by th e current state of this extended neigh bourhood by applying th e lo cal interactio n function to eac h neigh bourhood, and then app lying th e single- cell local up- date fun ction. This y ields a CA transition fu nction which m aps the state of the extended neighbou rhood to a single ce ll and gives the same g lobal transition function as the orig inal Closed CA. The neighbourh ood scheme for this CA is simply the exten ded neighbou rhood of each cell o f the Closed CA. Thu s, we can th ink of these physical ce llular automata d e v ices as being b oth physical systems which implement CA, and also as CA in their own right. 3 REVERSIBILI TY Re versibility is an important consideratio n in a mod el of closed physical cel- lular a utomata devices. Since closed physical systems in general tend to un- dergo reversible ev o lution, rev ersible CA are useful fo r mod elling such sys- tems. W e con s ider a cellular automata to be rev ersible if the global transition function is a bijection on the set of co nfigurations. There are two ways to introdu ce r e versibility into the Closed CA mo del. First, we may simply co n- sider Closed CA which intrinisically simulate a reversible CA. Second, we may consid er Closed CA whose own ev olution is r e versible, having global transition functio ns GF which are b ijections on the set of co nfigurations. If we wish to explore the question of whether Closed CA can be used to mod el closed phy sical systems with hom ogeneous ev olution, where inf ormation or energy is not imp licitly leaked into the en v ironment, then we should consider the second option . In par ticular , when we co nsider cellular automata mo dels of quantu m systems in Section 5, we start with a reversible model of CCA. The Closed CA mod el gives a natural way to in troduce reversibility . W e define a re versible CCA as a C CA in which both the local interaction function f an d the local upd ate function g are reversible. In o rder to reverse this evo- lution, we co uld apply th e r e verse fu nctions in the o pposite order, th at is, to apply g − 1 to e ach cell, and th en f − 1 to e ach neighb ourhood. This im plements 5 the glo bal tr ansition fu nction F − 1 G − 1 , which is the reverse of the orig inal global transition function GF . However , we can also e xpress the re verse of a reversible CCA as another re versible CCA. In order to do this, we implement G − 1 ( GF − 1 G − 1 ) = F − 1 G − 1 , using g − 1 as th e loca l upd ate func tion, and Gf − 1 G − 1 as the local interaction fu nction. Note that Gf − 1 G − 1 acts tr i v- ially on any cell not in the neighbo urhood wh ich f acts up on. Th is allo ws us to express Gf − 1 G − 1 as a reversible functio n acting only on a loc al neigh- bourh ood. Note that since a n y n on-rev ersible local interactio n function f or local up date function g will implicitly erase infor mation, th e y cannot yie ld reversible CCA. Thus, th is descrip tion encapsulates all CCA with reversible ev o lution. W e know that any CA can be intrinsically simulated by a CCA, as d e- scribed in Section 2. Th us, we can also simulate any reversible CA using a CCA using this techniqu e. However , we can also simulate any re versible C A using a reversible CCA by adap ting a technique describ ed in [12]. Gi ven a reversible CA with an alphab et Σ over a lattice L , let σ ( x, t ) denote the state of the ce ll x ∈ L at time step t ∈ Z , and let C ( t ) denote the con figuration of the en tire lattice. Let us sup pose that we have local transition ru les for bo th the f orward and reverse versions of the CA, so that σ ( x, t ) can be co mputed from th e configu ration of x + N a t either time step t − 1 o r t + 1 . W e ca n simulate such a reversible CA using a rev ersible CCA over th e lattice L using the alphabet Σ × Σ . First, we need an addition r elation o n the origina l alphabet, + : Σ → Σ , which ca n be g i ven by using any bijection from Σ to Z | Σ | . No w , we may view a configuratio n of the rev ersible CCA as two configur ations of the orig - inal rev ersible CA over the alphabe t Σ . At time t , the first co nfiguration contains the current configur ation C ( t ) , while the secon d configur ation con- tains pre vious configuratio n C ( t − 1) . For a giv en cell x , the lo cal in teraction function f x reads the current con figuration of its neighb ourhood fro m the fi rst configur ation, and adds σ ( x, t + 1) − σ ( x, t − 1) to the state of the second con - figuration . Note that becau s e each f x reads input fr om the first configuration and adds to the seco nd, it is a translatio n co mmutati ve func ti on. Since th e second configur ation initially contains C ( t − 1) , it w il l contain C ( t + 1) af ter the interaction phase. The local update function simp ly swaps the contents of the two configu rations, so th at the first con figuration now contains C ( t + 1 ) and the second contain s C ( t ) . No te that this co nstruction req uires both the forward and re verse l ocal transition rules of the origin al r e versible CA. Since we are pr oposing tha t reversible CCA be thou ght of as a m odel for closed ph ysical implementations of cellular au tomata devices, we should 6 also conside r the question of whether reversible CA can be also gen erally though t of as closed p hysical systems. Ho wever , th e shift- right CA over a one-dim ensional lattice, in which σ ( x, t + 1) = σ ( x − 1 , t ) , cannot b e ex- pressed in term s of a r e versible CCA or , in fact, any system of local op era- tions. T o see this, consider any circuit, over an infinite o ne-dimensional lat- tice, which does ef fect the shift-right operation using only local operations— ones w hich act o n a finite neig hbourhoo d. W e may suppose that at any time step, an in finite n umber o f g ates may be o perating thro ughout th e lattice in parallel, but w e may insist th at the depth of th e circu it be fin ite, so tha t the shift-right g lobal operatio n F can be deco mposed as F = f n f n − 1 . . . f 2 f 1 , where eac h f k is the pro duct of potentially infin it ely many local gates on disjoint local neighbo urhoods, and where n is the depth of the circuit. Now , co nsider an individual cell, x 0 . By examining the depen dencies o f the in di vidual local operations which make u p F , we can find a range of cells, P = { x : a ≤ x ≤ b } fo r some a, b ∈ Z such that the new state of the cell x 0 depend s only on th e cur rent state of the ce ll s in P . This value m ust be the value o f the qu antum state in the cell to the imm ediate left of x 0 before F is a pplied. W e may also find a minim al set of local operators f rom F such that the new value o f the qu antum state at cell x 0 is co mputed witho ut violating any of the dep endency rela ti onships betwee n the local operato rs. W ith out loss o f g enerality , we can implemen t F by applying these gates fir s t, to obtain the new state of x 0 , an d then applying the rest of the gates. Howe ver , by c onstruction, n ote th at none o f the o perations remaining can affect cells on both sid es o f x 0 , since th ese would have alr eady bee n perfo rmed. This means that the rema ining ope rations can b e d i v ided into two operation s acting indepen dently on eith er sid e of x 0 . Since, a t this point, o nly cells in the set P h a ve be en affected at all, an d the cells to the lef t and rig ht of x 0 may no longer interact with each other, the cells to the right of x 0 must contain all of the in formation nee ded to update no t only their own states, b ut the state of the cell x b +1 , directly to the right of P . Th is is not possible. This is not to say that a shift-right CA cannot be intrinsically simulated by a reversible CCA u si ng a larger state space. Howe ver, it reflects th e fact that it is n ot phy s ically possible to implemen t a closed ph ysical system in wh ich all d ata is shifted to th e right on a o ne-dimensional lattice. This means that there e xist re versible CA whose physical evolution does implicitly depend on interaction with an external environment. In o ther words, reversibility is a necessary condition, b ut not sufficient, to ha ving a truly closed e volution. 7 4 CONSTRUCT ION TECHNIQUES W e have seen h o w CCA an d r e versible CCA can be con s tructed as devices which simulate a c orresponding CA or reversible CA. In this section, we present two altern ati ve techniqu es for constructing translatio n commu tati ve local interaction function s. The first con st ruction is a gener alization of the idea of partition ed cellular automata, du e to Margolu s[12 ]. W e consider a CCA in which each cell is divided into two registers, with Σ = Σ 1 × Σ 2 . W e will co nsider the second register , with states Σ 2 , as the con tr ol re gister . T he idea is to use the con trol register to determ ine wh ich no n-tri vial op erations which ar e bein g ap plied to various neighbo urhoods. W e m ust also en sure that these neighbo urhoods do not intersect, so that the resulting loc al inter action functio n is translation commutative. W e ca n forma lize this id ea by consider ing con figurations of the con trol register . Define a r e gion of the lattice to be a finite sub set R ⊂ L , an d define a con fi guratio n of a region to b e an assignmen t of states in Σ 2 to the cells of R . Note th at we may also co nsider arb it ary tr anslations x + R of any region, for all x ∈ L . Finally , we say that two regions R 1 and R 2 intersect non-tr i vially if R 1 6 = R 2 and R 1 ∩ R 2 is non -empty . Definition 4. Supp ose we ar e given two finite r e gions R 1 , R 2 , and r espective config ur ations C 1 , C 2 . W e say tha t C 1 and C 2 ar e intersectab le if for some x ∈ L , the R 1 and x + R 2 intersect non-trivially an d there exists and the r e exis ts a co nfiguration C ′ of R 1 ∪ ( x + R 2 ) which is co nsistent with both C 1 on R 1 and C 2 on x + R 2 . W ith this defin ition, we define a set of configu rations to be compa tible if no two configu rations are intersectable. Giv en a set S o f compatib le con- figuration s , and a configu ration of the entire lattice, we can find the set of all r e gions of the lattice w hich are consistent with a con figuration fr om S . Furthermo re, these distinct regions will not intersect each other . For each configuratio n C i over a re gion R i in th is set S , we may consider a function f i which applied to the Σ 1 register of each ce ll in R i when the corre- sponding configuration in t he Σ 2 register matches C i . Because no non-trivial operation s are ever app lied to two intersection r e gions, this local in teraction function is translation commutative. The correspon ding local update fu nction may be any single-ce ll operatio n over Σ . I n particu lar , it may later th e state of the Σ 2 register of a cell. As an examp le, we may consider the partition ed cellular automato n intro- 8 duced by Margolus[12 ]. In this partitioned CA ov er a o ne-dimensional lattice L , we are given two distinct partitionings of the lattice into pairs of cells. On ev en time steps, a two-cell oper ation U is app lied according to one partitio n- ing. On odd time steps, ano ther tw o-cell ope ration V is applied acco rding to the o ther partitionin g. W e may imp lement this by using by addin g a register to each cell with alphabet Σ 2 = { 0 , 1 , 2 , 3 } , an d by using { 01 , 2 3 } as the set of compatib le co nfigurations. For each consecutive pair of cells, the local interaction function applies U to the Σ 1 register whe n the Σ 2 register is in the configur ation 0 1 , and V respectively for the configuration 23 . By initializing the Σ 2 registers of the lattice with alternating 0 and 1 states , and using a local update func tion which maps x ∈ Σ 2 to 4 − x , th e ev olution of the r esulting CCA simulates the evolution of the given partitioned CA. Note th at wh ile it is possible to con s truct initial co nfigurations which do not c orrespond to the ev o lution of a proper partitioned CA, the e volution of the C CA itself remains well-defined. Another u s eful techn ique fo r co nstructing valid tr anslation commu tati ve local transition func ti ons for CCA is that of cell colo uring. This tec hnique begins with a periodic labelling of the lattice cells with colours drawn from a finite set K . The colouring must satisfy the additional prop erty that th e co lour of each cell must be different f rom that of each of its neigh bouring cells. The Colour ed CA mode l can be used to construct autom ata which simulate physical phenome na such as spin chains [7, 1, 9]. At each time step, a par ticular colour is chosen, and we give a c olour up- date r ule which alters only cells o f the ch osen colour, using the states of its neighbo uring cells. This seq uence of colour update ru les must e ventually repeat. By u sing only reversible colo ur update r ules, we may a ls o make re- versible Coloured CA. As an example, we may consider a two-colo ur , black and white, one- dimensional CA, with a symme tric n eighbourhoo d of radius one. At o dd time steps white ce lls are upda ted wi th a rule that depen ds only on its cu rrent state, and the state of its two black neighb ours. At ev en time steps, black cells get similarly updated. From the con struction, we can see that the sequence of colour update rules of a Coloured CA can b e impleme nted as a translation com mutati ve interac- tion fun ction by addin g a clock register to each cell whic h keeps track of the current colou r upd ate r ule. I t is also p ossible to show that any CCA can b e simulated by an appropr iate Colour ed CA. 9 5 QU ANTUM CELLULAR A UTOMA T A The ideas lead ing to the constru ction of Closed CA o r , more sp ecifically , re- versible CCA ha ve been used as the basis of a new mode l of quantum cellular automata[9]. Since quantum mechanics deals with the phsyical ef fects which are significa nt at the nanoscale, it is imp ortant to co nsider qu antum effects when lookin g into building nano s cale devices[10 ]. Furthermo re, it open s t he possibility of q uantum co mputation using suc h a quan tum cellular au tomata device, applyin g only global co ntrol. While an introd uction to qu antum in- formation is ou ts ide of the scope of this pap er , we wish to summarize so me of the known r es ults about quantum CA from [9] which ar is e from the Closed CA model introduced in this paper . W e should note that other models of quantum cellular automata have been propo sed, in cluding some ear lier m odels by Zeilin ger[5 ], W atrous[14], and Meyers[8] and more r ecent o nes, such as th at o f Schum acher and W erner [11 ]. These mod els a re not without merit. Ho wev er , they all do shar e a commo n drawback allud ed to in the intro duction. They all u se the trad itional math- ematical model o f CA—which, as we have shown, is not an accurate de- scription of closed p hysical systems—as the starting po int fo r quan tization. The com mon result is, under standably , that these mo dels that do not, in fact, represent actual quan tum systems. In particular, all of these mo dels permit a “Shift-Right” Q C A, which we h a ve shown is not possible in a clo s ed physical system (as proven above for classical closed CA, and in [9] for quan tum sys- tems). There are oth er examples of non-physical beha viour within the earlier models, which have been pointed out before [11, 9]. By comparison, the quantization of the CCA presented her e can be proven to mode l an y and all ap propriate quantum phenomen a, and nothing more . In other words, any f ormal QCA in this axiom atic system can b e implemen ted as a re al phy s ical d e v ice [9]. Ano ther advantage o f using a model of quantum CA which is based on the CCA mode l, is that it allows constructio n tech- niques analogou s to those in Section 4 to be used. Briefly , a quantu m CA can be described as fo llo ws. First, the d - s tate classical alphab et Σ of a CCA is replaced with quantu m d -level system d e- scribed by a Hilbert spac e. E ach ce ll o f th e lattice contains a p hysical sys- tem, called a qudit , which is an instance o f this d -dimension al Hilbert space. This leads to the definition of the Local Unitary Quantum Cellular A utomata (LUQCA), which is a qu antum-mechanica l analo g of th e r e versible CCA. Since clo sed quantum systems evolve according to un itary operations in dis- crete time steps, unitary op erations are used in th e ev olution of the q uantum 10 CA, corresp onding to reversible oper ations in the r e versible CCA. W e can giv e an analogo us d efinition of tran s lation commu tati vity for un itary opera- tors. Definition 5 . A Local Unitary Quan tum Cellular Automaton is a 5-tuple ( L, Σ , N , f 0 , g 0 ) consisting o f a d -dimen s ional ce ll lattice L = Z d , a finite set Σ of d o rthogonal basis states, a n eighborhood scheme N , a un itary trans- lation commutative lo cal interaction operator f : ( H Σ ) ⊗N → ( H Σ ) ⊗N , and a unitary local update operator g : H Σ → H Σ . Here, H Σ denotes th e complex inner product space spanned by the o r - thonor mal basis gi ven by the states o f Σ . No te that while there do exist math- ematical techniques to handle the uncountable-d imensional Hilbert space that emerges f rom th is defin ition[13 ], for the purposes of simulation the e volution of an y fi nite subregion using a q uantum cir cuit, these need n ot b e con s idered. In [ 9 ] , we sho w th at there exists a un i versal L UQCA that can efficiently an d exactly simulate any qua ntum comp utation in the quantum circuit model. W e also show that it is possible to b u ild an ef ficient qu antum circu it that simulates the evolution of any finite subregion of the lattice for any finite p eriod of time. Finally , and most importan tly , we show th at for any clo sed quan tum physical system con s isting of a lattice of identical quantu m s ubsystems with a (Hamiltonian) evolution that his homo geneous in both time and spac e, there exists a LUQCA t hat ef ficiently simulates t his system inexactly , b ut t o within an arbitrary precision . This final result allows us to co nsider L UQC A as discretized versions of homogeneou s ph ysical systems. 6 CONCLUSION The main goal of this paper h as be en to rephrase cellular automata as a model of nature, rather an abstract model of computation . W e showed that the more traditional d efinition of CA is not well suited for such a role. In particular, in Section 3 w e showed that there exist b eha viour th at is allowed by the tr a- ditional mo del of CA ( e ven traditional r e versible CA) that is impossible in a proper closed physical system. The mod el we introdu ce, closed cellular au tomata or CCA, do es not have this drawback. Hence, it is suitable for th e goal we have set o ut to accomplish. W e also discussed important results pertaining to the relationship between CCA and the tr aditional formalism of CA. In particular , we show that there is an effecti ve proced ure to tu rn any tradition al CA into a CCA. Like wise, any reversible CA can be tu rned into a r e versible CCA. Th e tran sformations are 11 efficient, but may incur a cost in re s ources. This was to be expected, since with CCA all r esources a re always a ccounted for, while in traditional CA th e y are not. W e also discuss the easier result th at CCA are a sub s et of tra ditional CA. These results clo s e the gap, and allow the wealth of results pe rtaining to CA to apply to our new model. W e also discussed techn iques f or co nstructing instances o f CCA. These are useful in b oth conv erting tradition al CA algorithm s to this new mod el, and for creating algorithms in the CCA formalism directly . Finally , we briefly discussed one of the main justifications for this body of work: building a foundatio n upon which a sensible quantization of CA can be built. The quantization proper is the subject of a dif ferent paper [9], howe ver , and we point the readers there for a complete discussion of it. In sh ort, we have p rovided a model that a bstracts p hysical ph enomena— natural occu rring and oth erwise—that appeals to our intuition as behaving like ce llular automata; furthermore, we ha ve discussed the relation ship of this model to th e tradition al mathematical forma li sm. W e use it, also, to p rov e a result ab out what cann ot be don e by an actual phy si cal system with CA be- havior . Finally , we discu s s how we can use this form alism as a starting po int for a quantu m mod el of CA. Although we have discussed CA in th is p aper , this appr oach can also be used on other tra ditional models of computatio n. It remains an o pen question whether doing so ca n produ ce r esults interesting results, as it has done in this case. The authors would like to thank the following agencies for suppor t durin g the pr eparation of this man uscript: DTO-AR O, CFI, CIF AR, and MIT AC S. C. A. P ´ erez-Delgad o would also like to thank QIPIRC. REFERENCES [1] S. C. Benjamin. (Jan 2000). Schemes for para llel quantum computati on without local control of qubits. Physical Revie w A , 61(2):020301 (R). [2] E t han Bernstein and Umesh V azirani. (1993). Quantum complexity theory . 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