Templates for Toffoli Network Synthesis

Transcription

1 Templaes for Toffoli Nework Synhesis Dmiri Maslov Faculy of ompuer Science Universiy of New Brunswick Fredericon NB EB 5A anada Gerhard W. Dueck Faculy of ompuer Science Universiy of New Brunswick Fredericon NB EB 5A anada D. Michael Miller Dep. of ompuer Science Universiy of Vicoria Vicoria B V8W P6 anada mmiller@csr.uvic.ca ABSTRAT Reversible logic funcions can be realized as neworks of Toffoli gaes. The synhesis of Toffoli neworks can be divided ino wo seps. Firs find a nework ha realizes he desired funcion is deermined. Second ransform he nework such ha i uses fewer gaes while realizing he same funcion. This paper addresses he second sep. Transformaions are accomplished via emplae maching. The basis for a emplae is a nework wih m gaes ha realizes he ideniy funcion. If a sequence in he nework o be synhesized maches more han half of a emplae hen a ransformaion reducing he gae coun can be applied. All emplaes for m» 7 are described in his paper. Keywords Reversible Logic Quanum ircuis Templaes Minimizaion. INTRODUTION Reversible logic is an emerging research area. Ineres in reversible logic is sparked by is applicaions in quanum compuing low-power MOS nanoechnology and opical compuing. The synhesis of reversible circuis differs significanly from synhesis using radiional irreversible gaes. Two resricions are added for reversible neworks namely fan-ous and back-feeds are no allowed. The only possible srucure for a reversible nework is a cascade of reversible gaes. The mos frequenly used gaes are he Toffoli gae [] and he Fredkin gae [4]. The Toffoli gae invers a single bi if he AND of a se of conrol lines is. The Fredkin gae inerchanges wo bis if he AND of a se of conrol lines is. The formal definiion is given in Secion. Only a few synhesis mehods have been proposed for reversible logic. Suggesed mehods include: using Toffoli gaes o implemen an ESOP (EXOR sum-of-producs) [0] exhausive enumeraion [] heurisic mehods ha ieraively make he funcion simpler (simpliciy is measured by IWLS 00 Laguna Beach A he Hamming disance [] or by specral means [8]) and ransformaion based synhesis [5] among ohers. Some mehods use excessive search ime ohers are no guaraneed o converge and some require many addiional oupus (garbage). We follow he wo-sep approach suggesed in [9]. Firs a nework for he given funcion is found. The algorihm for his sep is guaraneed o converge. In fac he algorihm is very fas. Improvemens on a naive algorihm are described in [9]. The second sep consiss of applying ransformaions which reduce he number of gaes. In his paper we describe he emplaes used for such ransformaions in deail.. PRELIMINARIES An n-inpu n-oupu funcion (gae) is called reversible if and only if i maps each inpu insance o a unique oupu insance. In oher words a reversible funcion (gae) permues he elemens of is domain. In pracice no all of he n! possible reversible funcions can be realized as a single reversible gae. Several reversible gaes have been proposed. However we will only deal wih Toffoli gaes in his paper. Definiion. For he se of domain variables fx ;x ; :::; x n he generalized Toffoli gae has he form TOF(; ) where = fx i ;x i ; :::; x i k g; = fxjg and [ = ; and i maps he Boolean paern fx 0 ;x 0 ; :::; x 0 ng o fx 0 ;x 0 ; :::; x 0 j ;x 0 j Φ x i x i :::x i k ;x0 j+; :::; x 0 ng. The se which conrols he change of j- h bi is called he se of conrol lines and is called he arge. In he lieraure a subse of all generalized Toffoli gaes is ypically considered. The mos popular are: he NOT gae (TOF(;;x j)) a generalized Toffoli gae which has no conrols; he NOT gae (TOF(x i;x j))[] which is also known as a Feynman gae a generalized Toffoli gae wih one conrol bi and he Toffoli gae TOF(x i +x i ;x j) (where +" denoes se union) [] a generalized Toffoli gae wih wo conrols. The hree gaes are illusraed in Figure and he gaes wih more conrols are drawn similarly. Noe ha he way he gaes are drawn is a convenion which is no relaed o he way he gaes are implemened. Gaes wih more han wo conrols are discussed in [6]. The se of generalized Toffoli gaes is known o be complee (for example see [7]) in oher words any reversible funcion can be realized as a cascade of Toffoli gaes. A regular synhesis mehod for Toffoli gae neworks is discussed in [9].

2 conrol conrol.. arge NOT NOT Toffoli (Feynman).. Figure : NOT NOT and Toffoli gaes Due o probable echnological resricions he synhesis of reversible logic is done wih no feed-back and no fan-ou []. This leaves he cascade srucure as he only model saisfying hose condiions. Thus we consider cascades of Toffoli gaes. Le he signal be propagaed from lef o righ. The picorial represenaion of a nework is shown in Figure. The cos of a funcion is defined as he number of gaes in circui realizing i (S for a nework in Figure ) line line 5. line gae S line Figure : Templaes wih or inpus. inpu... gae gae oupu ffl he conrols of he gaes are coded by ses i each of which represens a se (maybe empy) of lines; line n line n Figure : The general srucure for a nework. TEMPLATES In our previous work [9] we inroduced emplaes as a ool for nework simplificaion. In ha work a emplae consiss of wo sequences of gaes which realize he same funcion. The firs sequence of gaes is o be mached o a par of he circui being simplified and he second sequence is o be subsiued when a mach is found. The emplaes were in Figure were idenified and classified based on heir similariy. In [9] he emplae maching procedure looks for he firs se of gaes including he iniial mach o he wides gae across he enire circui. If all arge gaes are found i aemps o make hem adjacen using he moving rule: gae TOF( ; ) can be inerchanged wih gae TOF( ; ) if and only if = ; and = ;. Adjacen gaes can mach he emplae in he forward or reverse direcion. The mached gaes are replaced wih he new gaes specified by he emplae. For a reverse mach he new gaes are subsiued in reverse order. Finally if a any ime wo adjacen gaes are equal hey can be deleed (deleion rule). In his secion we give a formal classificaion of he emplaes used in [9]. However for a beer undersanding of emplae classes we inroduce he following noaion. ffl he lef hand side has a sequence of gaes ha is o be replaced wih he sequence given on he righ hand side; ffl he arge ses i each conain a single line. All ses are disjoin: i j = ;; i k = ;; l k = ;8i; j; k; l. A firs aemp o classify he emplaes resuls in he classes lised below: lass. This class unies and generalizes he emplaes (Figure ) ino a class (Figure 4a) wih he formula: TOF( + + ; )TOF( + ; ) TOF( + + ; )= =TOF( + ; ) TOF( + + ; ) () lass. This class consiss of emplaes (Figure ) and heir generalizaions. The class is illusraed in Figure 4b and can be wrien as he following formula: TOF( + ; )TOF( + + ; ) TOF( + ; )= =TOF( + + ; ) TOF( + + ; ) () lass. This class (Figure 4c) includes emplaes..-. (Figure ) and can be described by he formula: TOF( + + ; )TOF( + + ; ) TOF( + ; )= =TOF( + ; ) TOF( + + ; ) () Templae 5. can be generalized bu his generalizaion is no considered here since emplae 5. does no decrease he

3 (a) (b) (c) Figure 4: Toffoli emplaes. number of gaes in a nework. However use of a generalizaion of his emplae may be beneficial since i inroduces smaller gaes ha can be used by oher emplaes. Even if hey are no used i is beneficial o have gaes wih fewer conrols since for some echnologies heir coss are lower. For insance in quanum echnology he cos of a Toffoli gae is 7 imes higher han ha of a NOT gae []. As he number of conrols of he Toffoli gae grows he relaion beween he coss of generalized Toffoli and NOT gae grows quadraically if no addiional garbage is allowed and linearly if garbage is allowed []. The correcness of formulas ()-() is easily proven. A more ineresing quesion is wheher he se of hese hree classes of emplaes ogeher wih he wo rules (moving rule deleion rule) is a complee se of simplificaion rules for a sequence of hree generalized Toffoli gaes over n lines. To check his we ran a program which exhausively searches all sequences of hree gaes buil on hree lines o check wheher he sequence can be reduced by means of emplaes from he hree classes and he wo rules. This program found no new emplaes. Thus we conclude ha he hree classes ogeher wih moving and deleion rules form he complee simplificaion ool for any Toffoli nework wih up o hree gaes.. Unificaion of lass and lass Templaes lasses and look similar. This similariy resuls in he following descripion of he wo classes as one: ffl he firs par of he emplae has gaes of he form ABA i.e. he firs and he hird gaes are he same; ffl if he following algorihm produces a valid nework he emplae exiss oherwise i does no (correcness can be easily proven): Take he second gae and pu i firs in he second par of he emplae. On each line here may be a logical AND connecion (ffl) an EXOR (Φ) or no connecion wih he verical line (denoed ). We build he second gae of he righ hand side of he emplae by aking values from Table using he symbols on ha line from A and B (since he able is symmeric here is no need o specify which argumen isa and which is B). If he symbol E occurs during he building process he emplae canno be buil. I is easy o see why since if all Φ are on he same line he ffl Φ ffl Φ ffl ffl ffl Φ Φ E Table : Second gae building process moving rule is applicable and he nework can be changed o he form AAB afer which applicaion of deleion rule ransforms he nework o he form B. In oher cases for TOF( + ; ) TOF( + ; ) TOF( + ; ) for example he algorihm produces logical AND on he firs line and nohing a all on he oher lines. This makes no sense. Tha is no reducion is possible for his sequence of gaes. 4. TEMPLATES - A NEW APPROAH Alhough he emplae descripion in Secion is formal and shorer ( classes and rules in comparison o 4 emplaes wih rules as used before) i can be simplified even furher. For his we need a new undersanding of emplaes. Le a size m emplae be a sequence of m gaes (a circui) which realizes he ideniy funcion. Any emplae of size m mus be independen of emplaes of smaller size i.e. for a given emplae size m no applicaion of any se of emplaes of smaller size can decrease he number gaes. The emplae G 0 G ::: G m can be applied in wo direcions:. Forward applicaion: A piece of nework ha maches he sequence of gaes G i G (i+) mod m ::: G (i+k ) mod m of he emplae G 0 G ::: G m exacly is replaced wih he sequence G (i ) mod m G (i ) mod m::: G (i+k) mod wihou changing he nework's oupu where k N;k m.. Backward applicaion: A piece of nework ha mache he sequence of gaes G i G (i ) mod m::: G (i k+) mod m exacly is replaced wih he sequence G (i+) mod m G (i+) mod m :::G (i k)modmwihou changing he nework oupu where k N;k m. These definiions of emplae applicaion need a correcness proof he nework oupu should no be changed for each of he lised operaions. orrecness can be verified as follows. Noe ha a reversible cascade ha realizes a funcion f read in reverse (from he oupus o he inpus) realizes f is inverse.

4 Firs we prove he correcness of he forward applicaion of a emplae saring wih elemen G 0. The operaion for his case requires subsiuion of G 0 G ::: G k wih G m G m ::: G k. Since G 0 G ::: G m realizes he ideniy funcion G k G k+::: G m realizes he inverse of he funcion realized by G 0 G ::: G k. Therefore read in reverse order G k G k+::: G m realizes inverse of he inverse i.e. he funcion iself. Thus he funcion realized by G 0 G ::: G k was subsiued by iself which does no change he oupu of he nework. orrecness of he remaining forward applicaions can be proven by using Lemma. orrecness of all reverse applicaions follows from he proof above and from he observaion ha he inverse of he ideniy funcion is he ideniy funcion. Nex observe ha a emplae can be used in boh direcions forward and backward as he formulas show. Also we can sar using i from any elemen. Thus i is beer o hink of a emplae as a cyclic sequence. The correcness of viewing a emplae as a cyclic sequence is proven by he following Lemma. Lemma. If a nework G 0 G ::: G m realizes he ideniy funcion hen for any k-shif G k G (k+) mod m ::: G (k ) mod m realizes he ideniy. Proof. We prove he Lemma for -shif G G ::: G m G 0. Then all k-shifs can be proven by applying he -shif k imes. The proof for a - shif follows from: Id = G 0 G ::: G m G 0 Id = G 0 G 0 G ::: G m G 0 = G G ::: G m Id = G 0 G 0 = G G ::: G m G 0: The condiion k m is used as we don' wan o increase he number of gaes when a emplae is applied and equaliy yields a simpler classificaion scheme. The following is a classificaion of emplaes up o size 7. We use he noaion inroduced in he previous secion. ffl m=. Size emplaes do no exis since each generalized Toffoli gae produces a change of is inpu. ffl m=. There is one class of emplaes of size (Figure 5a) and i is he deleion rule which is described by he sequence (AA) TOF( ; ) TOF( ; ): ffl m=. There are no emplaes of size. ffl m=4. There is one class of emplaes (Figure 5b) he moving rule from he previous secion which can be wrien as follows (ABAB): TOF( + ; 4 + 5)TOF( + ; 4 + 6) TOF( + ; 4 + 5) TOF( + ; 4 + 6): The se noaion is used o describe he arges since hey may inersec or no which is impossible o describe in one formula using he i noaion for he arges. The upper emplae in Figure 5b has j 4j =0 which resuls in j 5j = and j 6j = when he lower has j 4j = resuling in j 5j = 0 and j 6j =0. ffl m=5. Surprisingly here is only class of emplae of size 5 (Figure 5c) which unies he hree earlier classes ()-() and includes emplaes and from Figure. The class can be wrien as (ABAB): TOF( + + ; ) TOF( + ; ) TOF( + + ; ) TOF( + ; ) TOF( + + ; ): ffl m=6. There are wo classes here (Figure 5d) and hey are described by formulas (ABAB) TOF( + ; ) TOF( ; ) TOF( + ; )TOF( + + ; ) TOF( ; )TOF( + + ; ) and (ABAD) TOF( + ; )TOF( ; ) TOF( + ; ) TOF( + + ; ) TOF( ; )TOF( + + ; ): Noe he wo formulas for he classes look very similar and in fac using Fredkin gaes hey can be generalized o form one very simple emplae FRE( + + ; + ) FRE( + + ; + ) (where FRE(; + ) is a gae which swaps values of bis and if and only if se has all ones on is lines) bu i is bu we do no pursue his here as we are resricing our aenion o generalized Toffoli gaes. ffl m=7. There are no emplaes of size 7. For m>7 he number of emplaes is expeced o grow very fas (exponenially). One way o reduce he number of emplaes is o allow Fredkin gaes. To verify he correcness of he above classificaion we mus show no emplae of larger size can be reduced o a emplae of smaller size. ffl The size 4 emplae is independen of he size emplae since no adjacen gaes are equal. ffl The size 5 emplae is independen of he size emplae since no adjacen gaes are equal. The size 4 emplae can be applied o move gae anywhere in a emplae bu i does no allow any simplificaion of size a 5 emplae by smaller emplaes. ffl Size 6 emplaes are independen of he size emplae since no adjacen gaes are equal. A size 4 emplae can be applied o inerchange gaes A and of emplae ABAB only and does no lead o any simplificaion.

5 Figure 5: All emplaes for m» 7. The size 5 emplae maches a mos gaes of emplae ABAB and herefore can no be applied. 4. ompleeness Firs of all we wroe a program which builds all he 4- inpu 4-oupu circuis of size 7 ha realize he ideniy funcion and ries o apply he emplaes. The program resul shows ha he se of our 5 emplaes (AA ABAB ABAB ABAB ABAD) is he complee se of emplaes of size 7 or less for 4 inpus and less. The mahemaical proof of compleeness of his se for any number of inpus is harder. For emplaes of size i can be done by hand since here are no so many choices o look a. For emplaes of size 4 and 5 he following lemma is useful. Lemma. A size m emplae has a mos b m c differen lines wih EXOR signs. Proof. Prove by conradicion. Suppose here are b n c+ or more lines which conain EXOR sign. Then by he pigeon hole principle here will be one line wih one EXOR sign only. u he cycle so ha he gae wih his EXOR TOF(; ) comes firs. Now if we assign o all x j he value of changes o μ as he signal is propagaed in he emplae. Thus he emplae does no realize he ideniy funcion which conradics is definiion. Use of his Lemma allows us o say ha all he emplaes of size 4 have EXOR signs on eiher wo lines (wo signs on one and wo on he oher) or line (all 4 on line). Thus an exhausive search proof becomes reasonable. For he size 5 emplaes we can guaranee ha hey all will have only wo lines wih EXOR. 5. EXPERIMENTAL RESULTS We wroe programs o verify he correcness of our resuls build he new emplaes and apply hem. The resuls of he verificaion program were discussed in above. The program which simplifies he neworks works as follows. Firs we found ha i is convenien o sore emplae ABAB as a separae rule which helps o bring he gaes ogeher o mach a emplae. Then he circui is simplified as follows. For he hierarchy of emplaes AA χ ABAB χ ABAB χ ABAD ry o mach asmany gaes of a emplae as possible by looking ahead in he nework and using he moving rule. If a emplae can be applied apply i for he greaes applicaion parameer k possible. Afer applying any emplae sar rying o apply he emplaes in hierarchical order from he very beginning. If none of he emplaes can be applied he simplificaion process is finished. Example. We ook a nework for hree bi adder produced by he synhesis algorihm presened in [9] (Figure 6) and applied our program o simplify i. As expeced he program used a size 5 emplae and mached gaes. Thus hey were subsiued by he remaining gaes of he emplae read in reverse order. This circui is opimal since no furher reducion is possible. Suppose an adder can be realized wih gaes or less. Then addiion of hese gaes o he end of he buil size 4 cascade resuls in a new emplae which was proven (by enumeraion) no o exis for size 7 and less and four inpus. 6. ONLUSION Several auhors considered nework ransformaions. Shende e. al [] used several 4-bi circui equivalencies o be able o rewrie gaes in a differen order. Their circui equivalence rules were no proposed for circui simplificaion. In our work we covered and classified all he emplaes hey had generalized he noion of emplae and showed how o use hem o simplify neworks. Iwama Kambayashi and Yamashia [5] inroduced some circui ransformaion rules which mainly served o bring a nework o a canonical form and hus saing ha he se of ransforms is complee. However heir approach uses unlimied garbage whereas in our approach no garbage is allowed. One of he ransforms in [5] was proposed for circui simplificaion bu he acual applicaion procedure was no described. Our work generalizes and classifies he emplaes used by [5] and adds new classes. We also show a way of using emplaes for nework simplificaion and have implemened i.

6 a b c d (consan 0) A B garbage propogae sum carry Figure 6: Opimal circui for a full adder. The larger he se of emplaes he more reducions can be done. For insance if for some naural number k k- opimaliy is defined as he impossibiliy of simplifying a nework wih size k and less emplaes hen all he emplaes of size n Λ n+ and less form he complee simplificaion ool for he synhesis mehod provided in [9]. The heoreical algorihm from [9] produces a valid nework wih a mos n Λ n gaes herefore if his nework is no opimal and was no simplified by all emplaes wih size n Λ n+ and less no all he emplaes are lised. Thus we come o a conradicion which proves he saemen. In his work we buil he se of emplaes and showed a procedure allowing us o creae 4-opimal circuis for neworks wih number of inpus less han or equal o 4. We generalized hese emplaes and proposed hem as he se of rules which produces a 4-opimal nework ou of hose given. The emplae ool was generalized and shown in a readily usable form. 7. REFERENES [] A. Barenco. H. Benne R. leve D. P. DiVinchenzo N. Margolus P. Shor T. Sleaor J. A. Smolin and H. Weinfurer. Elemenary gaes for quanum compuaion. The American Physical Sociey 995. [] G. W. Dueck and D. Maslov. Reversible funcion synhesis wih minimum garbage oupus. In Inernaional Symposium on Represenaions and Mehodology of Fuure ompuing Technologies March 00. [] R. Feynman. Quanum mechanical compuers. Opic News pages [4] E. Fredkin and T. Toffoli. onservaive logic. Inernaional Journal of Theoreical Physics pages [5] K. Iwama Y. Kambayashi and S. Yamashia. Transformaion rules for designing cno-based quanum circuis. In Proceedings of he Design Auomaion onference New Orleans Louisiana USA June [6] D. Maslov and G. W. Dueck. Asympoically opimal regular synhesis of quanum neworks. Submied o Inernaional Workshop on Logic Sysnhesis May 00. [7] D. Maslov and G. W. Dueck. Garbage in reversible design of muliple oupu funcions. In 6h Inernaional Symposium on Represenaions and Mehodology of Fuure ompuing Technologies March 00. [8] D. M. Miller and G. W. Dueck. Specral echniques for reversible logic synhesis. In 6h Inernaional Symposium on Represenaions and Mehodology of Fuure ompuing Technologies March 00. [9] D. M. Miller D. Maslov and G. W. Dueck. A ransformaion based algorihm for reversible logic synhesis. In Proceedings of he Design Auomaion onference 00. [0] A. Mishchenko and M. Perkowski. Logic synhesis of reversible wave cascades. In Inernaional Workshop on Logic Sysnhesis June 00. [] M. Nielsen and I. huang. Quanum ompuaion and Quanum Informaion. ambridge Universiy Press 000. [] V. V. Shende A. K. Prasad I. L. Markov and J. P. Hayes. Reversible logic circui synhesis. In IAD San Jose alifornia USA Nov [] T. Toffoli. Reversible compuing. Tech memo MIT/LS/TM-5 MIT Lab for omp. Sci 980.