Dynamic Template Matching with Mixed-polarity Toffoli Gates

Transcription

1 Dynmi Templte Mthing with Mixed-polrity Toffoli Gtes Md Mzder Rhmn 1, Mthis Soeken 2,3, nd Gerhrd W. Duek 1 1 Fulty of Computer Siene, University of New Brunswik, Cnd 2 Deprtment of Mthemtis nd Computer Siene, University of Bremen, Germny 3 Cyer-Physil Systems, DFKI GmH, Bremen, Germny Astrt The Toffoli gte, s originlly proposed, hd only positive ontrols. It hs een shown tht mixed polrity ontrolled Toffoli gtes n e effiiently implemented. In ft, their quntum ost is the sme s for positive ontrolled gtes in most ses. Thus it is dvntgeous to onsider iruits with mixed polrity Toffoli gtes. Templte mthing hs een suessfully used to redue the numer of Toffoli gtes in reversile iruits. Little work on templtes with mixed polrity gtes hs een reported. Unfortuntely, the numer of potentil templtes inreses drmtilly, if mixed polrity is introdued. Here we propose dynmi templte mthing lgorithm tht tkes templtes with few lines nd dynmilly extends the lines to find mthes. Experimentl results show tht the proposed pproh hs signifint impt on reduing the totl numer of gtes (57% in the est se) in iruits. I. INTRODUCTION Applitions of reversile logi re found in quntum omputtion [1] where exponentil speed up n e hieved for some lgorithms [2]. For lssil reversile funtions ( speil se of Boolen funtions), series of synthesis nd post-synthesis methods re employed to otin quntum iruits. The most ommon method to otin quntum iruits from reversile funtions onsists of severl steps. First, iruit of Toffoli gtes [3], [4] is onstruted. The Toffoli iruit is then optimized with suh methods s desried in [5], [6]. Next, the Toffoli iruit is trnsformed into quntum iruit y deomposing [7] Toffoli gtes. Templte mthing n e used to redue the numer of gtes in Toffoli iruits [8] s well s quntum iruits [9]. When the Toffoli gte ws first proposed [3], only positive ontrols were onsidered. However, it hs een shown tht using oth positive nd negtive ontrols [10], [11] will result in iruits with lower quntum ost. This is the motivtion for onsidering templtes tht trditionlly only onsidered positive ontrols, with mixed polrity (positive nd negtive ontrols). The ojetive of our work is to investigte the use nd effet of mixed polrities in templted-sed optimiztion. We first find omplete set of 2-line templtes for NOT nd CNOT gtes with mixed ontrols. It turns out tht the numer of templtes is very lrge. This hs two disdvntges: first, lrge numer of templtes must e stored; seond, for eh templte the iruit must e snned for potentil mth. We develop dynmi templte mthing tht tkes set of templtes for few lines s input. For suessful mth on suset of lines in the iruit to e optimized it is heked whether the templte n e extended to mth the remining lines. Note tht the size of the set of mthed gtes must e more thn hlf of the size of the templte. Experiments show very good results in reduing the numer of gtes in iruits in omprison to the est known results of MCT enhmrk iruits [12]. It is oserved tht the runtime of the proposed heuristi is high sine mny possiilities re heked y dding ontrols to the gtes in templte. Previous work hs onsidered using negtive ontrol lines for reversile iruit optimiztion. Both [13] nd [14] present pprohes, lso lled templte mthing, to redue the numer of gtes in reversile iruits. However, they neither present n lgorithm to rete identities nor lssify iruit identities. Hene, the lgorithms n etter e desried s rule-sed pprohes. This is further justified y the ft tht suh rules re presented in terms of suiruit nd its optimized form, rther thn in terms of n identity iruit. In ft, the tehnique desried in [13] onsiders sequenes of gtes tht shre the sme trget line nd n therefore e effeiently redued using ESOP minimztion s demonstrted in [15]. In ft, ll optimiztion pprohes tht tke negtive ontrols into ount re instnes of rewriting ording to the elementry rules presented in [16]. This lso explins the huge numer

2 of templtes when onsidering negtive ontrols nd justifies dynmi mthing pproh s presented in this pper. The reminder of the pper is strutured s follows: Setion II riefly desries the fundmentls of reversile logi nd templte mthing. In Setion III we disuss the sis for onstruting templtes with positive nd negtive ontrols. Setion IV desries the dynmi templte mthing method. The signifine of the proposed pproh is shown with experiments. All 3-line MCT iruits nd 43 enhmrks re optimized nd the results re reported in Setion V. The pper onludes with some oservtions nd diretions for future reserh in Setion VI. II. BACKGROUND To keep this pper self-ontined, this setion riefly desries the essentils for reversile logi, reversile iruits, nd iruit optimiztion using templte mthing. A Boolen logi funtion f : B n B n over set of vriles X = {x 1,..., x n } is reversile if it is ijetive. Clssil reversile funtions [1] re the speil se of multiple-output Boolen funtions. A mixed-polrity multiple-ontrolled Toffoli (MPMCT) gte g(c, t) is reversile gte onsisting of set of ontrol lines C {x, x x X} tht re literls of X nd trget line t X suh tht {t} C =. A literl n our with t most one polrity in C. A ontrol line is lled positive if it ours s positive literl in C nd negtive if it ours s negtive literl in C. The semntis of suh gte re s follows: The vlue t the trget line is inverted if ll literls evlute to true. All remining vlues re pssed through the gte unhnged. Gtes for whih C = 0 nd C = 1 re known s NOT nd CNOT, respetively. In the literture, often only the se of gtes whih hve solely positive ontrol lines hs een onsidered. Reversile iruits re relized y sding reversile gtes. An MPMCT iruit is relized y sding MPMCT gtes. A iruit for funtion is miniml, if no iruit with fewer gtes relizes the sme funtion. The size of iruit G, denoted y G, is the numer of gtes in G. Two djent gtes g 1 (C 1, t 1 ) nd g 2 (C 2, t 2 ) n e interhnged if C 1 {t 2 } = nd C 2 {t 1 } =. Tht is, gte my move within the iruit y ommuting gtes; this is known s moving rule [10]. If gtes in iruit G n e rought together y the moving rule, then they form suiruit of G. Let two gtes g 1 (C 1, t 1 ) nd g 2 (C 2, t 2 ) in iruit t on the sme line(s) nd they relize the funtions f nd f 1 respetively. If they n e mde djent y using the moving rule, then they oth n e deleted; this is known s the deletion rule. These rules origin from previous work in whih only postive ontrol lines were onsidered. By llowing negtive ontrol lines for the iruit desription, one gins more flexiility [16]. A. Templtes nd Templte Mthing A templte T is iruit tht relizes the identity funtion. If sequene of gtes in iruit mthes with sequene of gtes with size s 1 > T /2 in templte T, then the mthed sequene of gtes in the iruit n e repled with the inverse of the remining sequene of size s 2 < T /2. This is lled templte mthing. Gtes in reversile iruit n often e reordered without ffeting the funtion of the iruit. The order of gtes in templte re onsidered to e irulr nd the moving rule n lso e pplied. The proess of mthing gtes from templte to iruit n strt from nywhere, nd mthing n e extended either in forwrd or kwrd diretion of the templte. Hene, if templte n e pplied to iruit, y rerrnging the gtes in the iruit s well s in the templte, then suh mth will e found in ext templte mthing. In this ontext, in ext templte mthing, gte g 1 (C 1, t 1 ) of templte mthes with gte g 2 (C 2, t 2 ) of iruit suh tht g 1 nd g 2 relize the sme funtion. An exmple of mthing of the gtes of templte to the gtes of iruit is shown in Fig. 1. However, grph-sed lgorithm for ext templte mthing n e found in [17]. Exmple 1: Let the iruit shown in Fig. 1() e optimized with the templte shown in Fig. 1(). The gte sequene 0, 1, 2, 3, 4, 5 of the templte mthes the gte sequene 0, 1, 2, 5, 3, 6 in the iruit in ext templte mthing. Therefore, the gte sequene of optimized iruit would e the gte sequene 10, 9, 8, 7, 4 of the templte () Ciruit G () Templte T Fig. 1: () A iruit nd () A templte

3 III. BASE TEMPLATES AND RULES FOR TEMPLATE CONSTRUCTION Templtes n e onstruted in mny different wys y using rewriting rules. However, some empiril results show tht the set of templtes of more thn 2 lines n e very lrge. Moreover, pplitions of templtes depend on how iruits to e optimized re onstruted, nd mny templtes my not e useful. For exmple, if 3- line funtion is synthesized y using the trnsformtionsed lgorithm, then it is unlikely tht the templte in Fig. 2(i) is pplied to redued the otined iruit. In this setion, we first find set of templtes of 2 lines whih re referred to s se templtes. From these se templtes, set of rules is defined to redue the templte set. We lso present rules of deriving templtes with lrge numer of lines dynmilly in templte mthing. A heuristi for dynmi templte mthing is desried in the next setion. For the NOT nd CNOT gtes with oth positive nd negtive ontrols, n exhustive serh lgorithm [18] finds omplete set of 2-line templtes whih re shown in Fig. 2. () () () (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) Fig. 2: The set of ll 2-line templtes In ontrst to dynmi templte mthing, in ext templte mthing, gte g 1 (C 1, t 1 ) in templte n only mth gte g 2 (C 2, t 2 ) in iruit if C 1 = C 2 nd oth C 1 nd C 2 hve sme numer of positive nd negtive ontrols. Tht is, oth g 1 nd g 2 hve to relize the sme funtion up to vrile nmes. This hs onsequene tht lrge set of templtes must e known in dvne for ext templte mthing. However, if templtes with MPMCT gtes re onsidered, then redued set of templtes otined y the rules in susetion III-A n e used in n ext templte mthing. Moreover, templtes with gtes of lrge numer of ontrols still require for optimizing iruits with gtes of lrge numer of ontrols. On the other hnd, the dynmi templte mthing tkes templte with few lines s n input nd if the lines of gtes in the templte prtilly mth with the lines of gtes in the iruit then it will extend the lines of the templte. Tht is, gte g 1 (C 1, t 1 ) in templte mthes gte g 2 (C 2, t 2 ) in iruit if C 1 C 2 s well s oth g 1 nd g 2 do not neessrily hve to relize the sme funtion. A. Equivlene of Templtes Let T e l-line templte of size n, then the following rules n e pplied to redue the set of templtes. 1) All polrities of the ontrols of gtes in templte n e flipped. For instne, the templtes in Fig. 2(g) nd (h) n e onsidered s equivlent in templte mthing. 2) On line where no gte hs trget, ll polrities of the ontrols on tht line n e flipped. For instne, the templtes in Fig. 2() nd () n e onsidered s equivlent. Aording to these equivlene rules, the set of templtes in Fig. 2 n e redued to the set of templtes whih re shown in Fig. 2(), (), (d), (e), (f), (g), (i), (k), nd (n). B. Extension of Lines in Templtes Let T is n l-line templte of size n, then the following rules n e pplied to derived new templtes. 1) Adding line to T results in templte of l + 1 lines. 2) If the removl of suset S of m (2 m n) gtes in T results in n identity, then new ontrol of eh gte of S in T n e dded nd pled into new line. All new ontrols of gtes in S must either ll e positive or ll e negtive. This results in new templte. Aording to these rules, it follows tht the templtes in Fig. 2(), (), nd () n e onsidered s equivlent in dynmi templte mthing. In summry, we otin redued set of templtes s shown in Fig. 3. IV. DYNAMIC TEMPLATE MATCHING In this setion, we present n lgorithm for dynmi templte mthing. To illustrte the lgorithm in Fig. 4, we onsider the iruits in Fig. 5. Let the proedure DynmiTemplteMthing() tke the iruit

4 () () () (d) (e) (f) (g) (h) Fig. 3: The redued set of 2-line templtes G in Fig. 5() nd the templte T in Fig. 5() s input; n = 5 nd l = 2. The proedure mth() returns true nd sets the vlues of m, m l, u l, nd u if more thn hlf of the gtes of I, n e mpped into suiruit of G. For instne, for the gtes {0, 1} in the templte T, lines nd re mpped into the iruit lines x 1 nd x 4. The gtes in the inner retngle in C re the ndidtes for mthing with the gtes {0, 1} in T. Therefore, m = {3, 4}, m l = {1, 4}, u l = {0, 2, 3}, nd u = {2}. Now for the line x 0, there re ontrols of gtes in the set m = {3, 4} in G. Therefore, the proedure ddconinnewline() will dd new line to the iruit I nd lso dd ontrols with respet to the gtes set m = {3, 4} nd i = 0. The resulting iruit I is shown in Fig. 5() in whih the unmthed gte 2 of T remins unhnged. Now for the gte u = {2} in the iruit shown in Fig. 5(), the proedure ddconhekid() heks whether ny identity n e found y dding ontrols to the gte u = {2} in I in ll possile wys. In this se two possily generted iruits re shown in Fig. 5(d) nd (e). Sine the iruit in Fig. 5(e) is n identity, the proedure ddconhekid() returns true. Sine lines {2, 3} in the iruit hve no ontrols for the gtes set m = {3, 4}, the inner for loop in the lgorithm will terminte without hnging su = true. Now su = true, hene, the proedure doreplement() reples the gtes set m = {3, 4} with the gte 2 of the identity iruit of Fig. 5(e). The resulting optimized iruit is shown in Fig. 5(f). (i) V. EXPERIMENTAL RESULTS We implemented ext templte mthing nd the dynmi templte mthing heuristi using C++. Along with 2-line templtes, we hve lso generted set of 3-line templtes of size up to 6. In our first experiment, we pply totl of 228 templtes to optimize ll 3-line iruits (gtes in iruits hve only positive ontrols) otined from the trnsformtion sed synthesis lgorithm. (1) iruit& DynmiTemplteMthing(G, T ) (2) G is iruit of n lines (3) T is templte l lines (4) /* Let m e n rry for index of mthed gtes in the (5) iruit */ /* Let u e n rry for index of unmthed gtes in the templte */ (6) /* Let m l nd u l e rrys of mthed lines in the iruit where m l = l nd u l = n l */ (7) Let I = T is iruit (8) while mth(g, I, &m, &m l, &u l, u) (9) if m = I (10) if isidentity(g, m) /* whether the whole su-iruit for m in G is identity */ (11) G = doreplement(g, I, m, m l ); (12) else (13) ool su = true; (14) for eh i u l (15) if line i hs ontrols of the gtes in m (16) I = ddconinnewline(i, m, i); (17) /* numer of lines of I is inresed y only 1 (18) */ m l = m l i (19) if!ddconhekid(i, u) (20) su = flse; (21) rek; (22) if su (23) G = doreplement(g, I, m, m l ); (24) I = T ; (25) return G; Fig. 4: Algorithm for dynmi templte mthing x 0 x 0 x 1 = x 1 = x 2 x 2 x 3 x 3 x 4 = x 4 = () Ciruit G () () Templte T (d) (e) An identity x 0 x 0 x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 (f) Optimized iruit Fig. 5: Dynmi templte mthing Column 3 in Tle I shows the verge numer of gtes for ll funtions. The verge numer of gtes in iruits otined from ext templte mthing re show in olumn 4 in Tle I. The totl verge redution is 24.18%. It is oserved tht in ext templte mthing, out of 228 templtes, only 31 templtes re pplied. In our seond experiment, we hve tken enhmrk MCT iruits from RevLi [12] s shown in olumn 1 in Tle II. Column 2 shows the numer of gtes in

5 the originl iruits. By using the redued set of 2-line templtes in Fig. 3 nd 214 templtes of 3 lines, the otined results from dynmi templte mthing re shown in olumn 3 in Tle II. Aording to olumn 5 in Tle II, signifint perentge of redution of gtes n e notied. More thn 20% redutions re highlighted. For 43 enhmrks, the verge redution is 23.46%. The est result is 57% redution for the enhmrk rd It n e notied tht for the enhmrk urf3 155 with 26, 468 gtes, the numer of gte redutions is 10, 732. Among 223 templtes, those templtes whih re frequently used for enhmrks optimiztion is shown in Fig. 6. It is very ler tht we only use templtes of lines up to 3 in whih trgets of gtes in templtes re pled into t most 3 lines, nd only ontrols of gtes in templtes re extended. However, templtes in whih gte trget ts on more thn 3 lines n e more enefiil for the optimiztion of iruits with lrge numer of lines. Even templtes of size more thn 6 n e investigted. We oserve tht redution of gtes in Toffoli iruits my not led to iruits with redued quntum osts when nive gte ount of quntum iruits is onsidered s quntum ost. Toffoli iruits with fewer gtes my hve higher quntum osts thn the osts of originl iruit. For instne, if the iruit shown in Figure 7() is repled with the iruit shown in Figure 7(), then the quntum ost of the resulting iruit inreses where quntum osts of Toffoli gte with 2 ontrols nd Toffoli gte with 3 ontrols re 5 nd 14, respetively. The question is tht whether Breno s [7] or lirrysed Toffoli [19] deomposition would e enefiil to otin possily low quntum ost iruits. However, we onjeture tht for templte T, if the gte sequene of size T /2 + 2 in T mthes with the gte sequene of iruit, then the resulting Toffoli iruit lwys leds to the low quntum ost iruit. In the literture, different quntum ost models re proposed [20], [21], therefore, in this work, we leve the mesurement of quntum osts of iruits for future reserh. VI. CONCLUSION We present heuristi of dynmi templte mthing in whih templtes of more thn 2 quits hve een generted during the mthing proess. The results from optimizing enhmrk funtions re promising, wheres omputtionlly, it is not fesile to otin ll templtes even with three lines efore templte mthing strts. Relting our findings in templte mthing with mixed- TABLE I: Results of optimizing 3-line MCT iruits otined from trnsformtion-sed lgorithm Size(min) #f #Avg(G)T B #Avg(G)Optz Totl #Avg(G)T B: Averge no. of gtes in originl iruits; #Avg(G)Optz: Averge no. of gtes in optimized iruits; TABLE II: Results of optimizing enhmrks Benhmrks #G(Org.) #G(Optz.) Rd Rd(%) 4gt11-v gt13-v mod5-v mod5-v mod5mils 65l mod5mils gt12-v1 89l l l deod24-v gt4-v mod5d lu-v lu-v mod mod5-v mod j-e gt gt12-v mod j-e lu-v mod rd rd hw gt4-v lu-v mod5dder hm rd rd sym hw hw hw urf urf urf urf #G(Org.): No. of gtes in originl iruits; #G(Optz.): No. of gtes in optimized iruits; Rd: No. of gtes redued; Rd(%): Perentge of redutions; polrity Toffoli gtes to the results of [16] n help to devise etter understnding of templte mthing nd estimte the numer of totl templtes, ut lso to find rewriting strtegies to guide iruit rewriting s n lterntive to optimiztion. A ost-enefit nlysis of iruits in quntum deomposition would e useful sine the gtes in the resulting iruits hve oth positive nd negtive ontrols. The heuristi of dynmi templte

6 () () () (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) Fig. 6: The set of templtes suessfully used in enhmrks optimiztion () Fig. 7: Reversile iruits with () #gte = 3 nd quntum ost 15 = nd () #gte = 2 nd quntum ost 19 = mthing n e improved y finding the est mthes for given templtes. Moreover, n in-depth nlysis of the use of the 13 templtes shown in Fig. 6 my led to the development of lgorithms for expnsion of gtes in templtes, whih is our future reserh. () REFERENCES [1] M. A. Nielsen nd I. L. Chung, Quntum Computtion nd Quntum Informtion. Cmridge University Press, [2] P. W. Shor, Polynomil-time lgorithms for prime ftoriztion nd disrete logrithms on quntum omputer, SIAM J. Comput., vol. 26, no. 5, pp , [3] T. Toffoli, Reversile omputing, Teh memo MIT/LCS/TM- 151, MIT L for Comp. Si, [4] E. Fredkin nd T. Toffoli, Conservtive logi, Interntionl Journl of Theoretil Physis, vol. 21, pp , [5] D. M. Miller, D. Mslov, nd G. W. Duek, A trnsformtion sed lgorithm for reversile logi synthesis, in Design Automtion Conferene, June [6] D. Mslov nd G. W. Duek, Reversile sdes with miniml grge, IEEE Trns. on CAD of Integrted Ciruits nd Systems, vol. 23, no. 11, pp , [7] A. Breno, C. H. Bennett, R. Cleve, D. DiVinhenzo, N. Mrgolus, P. Shor, T. Sletor, J. Smolin, nd H. Weinfurter, Elementry gtes for quntum omputtion, The Amerin Physil Soiety, vol. 52, pp , [8] D. Mslov, C. Young, G. W. Duek, nd D. M. Miller, Quntum iruit simplifition using templtes, in DATE - Design, Automtion nd Test in Europe, 2005, pp [9] M. M. Rhmn nd G. W. Duek, Templte mthing in quntum iruits optimiztion, in Reed-Muller Workshop, 2013, pp [10] D. Mslov, G. W. Duek, D. M. Miller, nd C. Negrevergne, Quntum iruit simplifition nd level omption, Computer-Aided Design of Integrted Ciruits nd Systems, IEEE Trnstions on, vol. 27, no. 3, pp , [11] R. Wille, M. Soeken, N. Przigod, nd R. Drehsler, Ext synthesis of toffoli gte iruits with negtive ontrol lines, in Multiple-Vlued Logi (ISMVL), nd IEEE Interntionl Symposium on. IEEE, 2012, pp [12] R. Wille, D. Große, L. Teuer, G. W. Duek, nd R. Drehsler, RevLi: An online resoure for reversile funtions nd reversile iruits, in Workshop on Reversile Computtion, July 2010, RevLi is ville t [13] K. Dtt, G. Rthi, R. Wille, I. Sengupt, H. Rhmn, nd R. Drehsler, Exploiting negtive ontrol lines in the optimiztion of reversile iruits, in Reversile Computtion - 5th Interntionl Conferene, RC 2013, Vitori, BC, Cnd, July 4-5, Proeedings, 2013, pp [14] M. Z. Rhmn nd J. E. Rie, Templtes for positive nd negtive ontrol Toffoli networks, in Reversile Computtion - 6th Interntionl Conferene, RC 2014, Kyoto, Jpn, July 10-11, Proeedings, 2014, pp [15] M. Soeken, Z. Ssnin, R. Wille, D. M. Miller, nd R. Drehsler, Optimizing the mpping of reversile iruits to fourvlued quntum gte iruits, in 42nd IEEE Interntionl Symposium on Multiple-Vlued Logi, ISMVL 2012, Vitori, BC, Cnd, My 14-16, 2012, 2012, pp [16] M. Soeken nd M. K. Thomsen, White dots do mtter: Rewriting reversile logi iruits, in Reversile Computtion - 5th Interntionl Conferene, RC 2013, Vitori, BC, Cnd, July 4-5, Proeedings, 2013, pp [17] M. M. Rhmn, G. W. Duek, nd J. D. Horton, An lgorithm for quntum templte mthing, J. Emerg. Tehnol. Comput. Syst., vol. 11, no. 3, pp. 31:1 31:20, De [18] M. M. Rhmn nd G. W. Duek, An lgorithm to find quntum templtes, in IEEE Congress on Evolutionry Computtion, 2012, pp [19] D. M. Miller, R. Wille, nd Z. Ssnin, Elementry quntum gte reliztions for multiple-ontrol Toffoli gtes, in Proeedings of the Interntionl Symposium on Multiple-Vlued Logi, 2011, pp [20] J. Smolin nd D. DiVinenzo, Five two-it quntum gtes re suffiient to implement the quntum Fredkin gte, Physil review. A, vol. 53, no. 4, pp , [21] D. Große, R. Wille, G. W. Duek, nd R. Drehsler, Ext synthesis of elementry quntum gte iruits for reversile funtions with don t res, in Interntionl Symposium on Multiple Vlued Logi, 2008, pp