Online Testing for Three Fault Models in Reversible Circuits

Transcription

1 25 IEEE 45th Interntionl Smposium on MultipleVlue Logi Online Testing for Three Fult Moels in Reversile Ciruits M Asif Nshir Dept. of Mth n Computer Siene Universit of Lethrige Lethrige, AB Cn Emil: sif.nshir@uleth. Gite Gurv Bhskr Dept. of Eletril Engineering Inin Institute of Tehnolog Roorkee Roorkee, Uttrkhn, Ini Emil: gite.gurv3@gmil.om Jqueline E. Rie Dept. of Mth n Computer Siene Universit of Lethrige Lethrige, AB Cn Emil: j.rie@uleth. Astrt In this pper we propose n pproh for the esign of online testle reversile iruits. A reversile iruit ompose of Toffoli gtes n e me online testle ing two sets of CNOT gtes n single prit line. The performne of the propose pproh for eteting single it fult, rosspoint fult n the fmil of missing gte fults hs een oserve. Disussion roun the orretness of our pproh n the overhe is lso provie. I. INTRODUCTION AND MOTIVATION In reent ers reversile omputtion hs estlishe itself s promising reserh re n emerging tehnolog. This is motivte wiel supporte preition tht the onventionl omputer hrwre tehnologies re going to reh their limits in the ner future [2]. A funmentl limittion of onventionl omputing is tht eh time informtion is lost energ is issipte regrless of the unerling tehnolog. This is known s Lnuer s priniple [5]. It ws lso shown Bennett [] tht theoretil ero power issiption n onl e hieve if the iruit is logill reversile []. Reversile omputing is ijetive in nture, n efinition reversile iruits re theoretill informtionlossless. Thus using reversile omputtion, the power issiption whih results oring to Lnuer s priniple n e erese or even eliminte. In this pper we ress the re of testing for reversile iruits, n propose n online testing pproh to etet three tpes of fults in reversile iruits. The orgnition of this pper is s follows: Setion II presents the funmentls of reversile logi n onepts of testing pprohes n fult moels; Setion III esries some relte work; Setion IV introues our propose pproh; etetion of three tpes of fults using the propose pproh re presente in Setion V; Setion VI presents the shortomings of our pproh n Setion VII onlues the pper n provies future iretions. II. BACKGROUND A reversile logi iruit is n li omintionl logi iruit in whih ll gtes re reversile n re interonnete without fnout. Moreover, feek lines from the output to input re not llowe in reversile iruits [2]. In this pper we onsier three tpes of reversile gtes: NOT, CNOT (CNOT stns for Controlle NOT) n Toffoli gtes. These three gtes form the CNT (CNOT, NOT, Toffoli) gte lirr. Generell, we refer to the CNOT gte s NOT gte, to the CNOT gte s Fenmn gte n to the 2CNOT gte s Toffoli gte. The tritionl NOT gte is reversile gte, sine it is possile to restore the input of NOT gte from its output. A NOT gte (CNOT) hs no ontrol line n hene the input t the trget line is lws inverte t the output line. However in kcnot gte, there re k ontrol inputs,..., k n one trget input, t. The kcnot gte mps the vetor (,..., k,t) to the vetor (,..., k,t 2... k ). This mens the vlue t the trget input is inverte if n onl if ll the vlues t the ontrol inputs re [9]. Reversile iruits re forme sing reversile gtes. Testing is require to ensure qulit, vililit, n reliilit of iruit or evie. There re two tpes of testing: offline testing n online testing []. In offline testing iruit uner test is tken out of its norml moe of opertion. In ontrst, online testing is rrie out while the iruit is eing use for norml opertions. In this se itionl iruitr is tthe to the originl iruit to etermine whether the sstem is fult or fult free. In this pper we fous on the ltter pproh. Our pproh hs een propose to etet fults esrie three moels: single it fults [], rosspoint fults [4] n missing gte fult []. A single it fult is reflete on etl one output of gte, hnging the orret vlue of the output to fult vlue euse of the hnge in it on some line. The rosspoint fult moel fouses on fults tht m our on the ontrol points of reversile gte. When one or more ontrol points re e erroneousl to gte then this is lle n pperne rosspoint fult. A ispperne fult ours when one or more ontrol points of gte o not work or ispper from iruit. The missing gte fult moel is pkge of four ifferent fult moels, inluing () the single missing gte fult (SMGF): fult tht is moele the ispperne of n entire gte; () the repete gte fult (RGF): n unwnte replement of gte the severl instnes of the sme gte; () the multiple missing gte fult (MMGF): when severl gtes go missing from iruit n () the prtil missing gte fult (PMGF): some of the ontrol points of gte re missing. A PMGF turns kcnot gte into k CNOT gte, where k <k. The quntit (k k ) is 95623X/5 $3. 25 IEEE DOI.9/ISMVL

2 referre to s the orer of PMGF. III. RELATED WORK In [2] the uthors propose three new reversile gtes. Two of the three gtes re use to esign n online testle lok n the other gte is use to rete heker iruit. The purpose of the heker iruit is to ompre the two prit its proue the online testle lok, whih will then etet single it fult. Similr to this pproh, the uthors in [6] propose n improve pproh for eteting single it fult. This esign oes not require n etr heker iruit to ompre the prit its. However oth of these pprohes hve ommon rwk. If single it fult ours etween se loks then the fult will go unnotie. In [8] the uthors provie n improve online single it fult testing pproh. In this pproh ll the Toffoli gtes of the iruit re hnge to Etene Toffoli gtes, n two sets of CNOT gtes n one itionl prit line re e to hieve online testilit. This pproh is effetive s long s single it fult ours in the originl portion of the iruit. If single it fult ours in the itionl iruitr (n of the CNOT loks) then the fult will go unetete. Zhong et l. propose oth the rosspoint fult moel n s well testing pproh to etet single pperne n ispperne rosspoint fults in reversile iruit [4]; however their pproh use offline testing. Authors in [] propose ll the vrints of the missing gte fult moel n lso etetion onitions for eh tpe of fult. Hes et l. propose DFT (esign for testilit) offline pproh for eteting single missing gte fults [3]. In [4] the uthors propose n online testing pproh for the etetion of single missing gte fults. In this pper we propose n online testing pproh to etet single it fults, rosspoint fults n missing gte fults. IV. ONLINE TESTING APPROACH A. Design To onvert reversile iruit to its online testle equivlent we first onvert the kcnot gtes of the iruit into Duplite Gte Bloks. We lso require the inlusion of prit line P whih is initilie with logi. For eh line in the iruit CNOT gte is inserte t the eginning n t the en of the originl iruit. The trgets of the itionl CNOT gtes re onnete to the prit line. 2 tin 2 tout Fig.. Conversion of Toffoli gte into Duplite Gte Blok A Duplite Gte Blok (DGB) onsists of two gtes. In orer to onvert Toffoli gte to Duplite Gte Blok we n itionl Toffoli gte s shown in Figure. The ontrols of the newl e gte (or uplite gte) re on the 2 tin Pin 2 tin sme lines s tht of the originl gte. However, the trget line of the uplite gte is onnete to the prit line. In the se of CNOT gte there is no ontrol line, hene the Duplite Gte Blok woul onsist of two CNOT gtes: one on the sme line s tht of the originl reversile gte n nother on the prit line. Given reversile iruit with L lines n N gtes, the first step to mke it online testle is to n etr line to the iruit. This line is the prit line, P, whih is initilie with logi. We net onvert eh gte of the iruit into its Duplite Gte Blok n se the loks in the sme orer tht the gtes pper in the originl iruit. We now hve se of Duplite Gte Bloks. The net step is to CNOT gtes to eh line t the input of the iruit. A totl of L CNOT gtes re e. The trget of eh of these gtes is onnete to the prit line. We refer to this set of CNOT gtes s the Premle Blok. Similrl, we nother set of CNOT gtes whih egins fter the en of the se Duplite Gte Bloks. We refer to this set of CNOT gtes s the Postmle Blok. Figure 2 illustrtes the onversion. The entire iruit onsists of three loks Pin= Fig. 2. () A full er reversile iruit, n () its online testle equivlent. in sequene: the Premle Blok, Duplite Gte Blok n Postmle Blok. If the quntum ost of the originl iruit is Q n the iruit hs L lines then the quntum ost of the iruit s online testle equivlent will e 2L+2Q, sine the quntum ost of CNOT gte is [7]. B. Anlsis Figure 3 shows generlie igrm of n online testle reversile iruit. P n Q represent the prit line n the ommon lines for the orresponing level respetivel. The trget n the ontrol lines re trete s ommon lines. We n etermine the outputs t the Premle Blok s follows: Q = Q, Q 2 = Q 2,...,Q L = Q L n P = P Q Q 2 Q 3,..., Q L. The prit line is initilie to, thus P =n P = Q Q 2 Q 3,..., Q L. From the ove eqution we n s tht the Premle Blok ts like prit heker. Tht is, if the prit of the ommon lines t the input (level ) is o then fter pssing through the Premle Blok, the vlue on the prit line (P ) t level will hnge to logi. If the prit of the ommon lines t the input (level ) is even then the prit line (P ) t level will remin logi. Also, the output vlues of the () () 9

3 Premle Blok on the ommon lines will e equl to the input vlues. Thus the iruit will hve logi t the prit line when the prit of the ommon lines of tht level is o. On the other hn, the prit it will e t logi if the prit of ommon lines of tht level is even. We ll this propert the prit propert. Q Q2 QL P Premle Blok Q Q2 QL P DGB (F) (T) Q2 Q22 QL2 P2 DGB 2 (F2) (T2) Qn Q2n QLn Pn DGB n (Fn) (Tn) Q(n+) Q2(n+) QL(n+) P(n+) Fig. 3. Blok Digrm of Online Testle Reversile Ciruit Postmle Blok Q(n+2) Q2(n+2) QL(n+2) The output of the Premle Blok forms the input of the se of the Duplite Gte Blok. If there is no fult in the Premle Blok then the DGBs lso follow the Prit Propert. Let F e the output funtion of n Duplite Gte Blok. Let T n P + e the two trget lines of the originl gte n the uplite gte of Duplite Gte Blok (DGB) respetivel. The trget line of the uplite gte is lws the prit line, wheres the trget line of the originl gte is one of the ommon lines. Then T is one of the lines mongst {Q (+),Q 2(+),...,Q L(+) }. Let T = Q i(+) where i (, 2, 3,...,L); then T = F Q i n P (+) = F P. From the ove two equtions it is oserve tht if F is logi then P (+) n T will toggle the input vlue (P ) n Q i respetivel. If F is then the output of the DGB will e equl to its input n no hnge will tke ple. The hnges in T n P (+) tke ple simultneousl. In other wors, the hnge in the prit of the ommon lines n P (+) tke ple simultneousl or the o not hnge. We refer to this propert of the DGB s the Simultneous Chnge Propert. The Simultneous Chnge Propert ensures tht the Prit Propert present t the input of the DGB remins onsistent throughout the output of the iruit. Furthermore, if the input of the DGB violtes the Prit Propert then the violtion is psse to the output of the DGB. The output of the se of the Duplite Gte Blok forms the input of the Postmle Blok. If there is no fult in n of the previous loks then the input of the Postmle Blok will lso stisf the Prit Propert. Tht is, if the prit of the ommon lines is even t level (n +) in Figure 3, then the input prit (P n+ ) woul e logi or vie vers. The output equtions of the Postmle Blok re: Q (n+2) = Q (n+) ; Q 2(n+2) = Q 2(n+),...,Q L(n+2) = Q L(n+). P (n+2) = P (n+) Q (n+) Q 2(n+) Q 3(n+),...,Q L(n+) From the ove eqution it is seen tht if the prit of the ommon lines is o t level (n +) then the input prit of the Postmle Blok, P (n+) is logi. Hene the output prit P (n+2) will e logi. On the other hn, if the prit of the ommon lines t level (n +) is even then the input P(n+2) prit P (n+) is logi. Hene the output prit P (n+2) will e logi. In nutshell, in fultfree iruit opertion the input of the Postmle Blok will preserve the prit propert n the finl output prit P (n+2) of the iruit will e logi. V. DISCUSSION In this setion we onsier senrios for ifferent tpes of fults in the three ifferent loks. We ssume onl one tpe of fult is presente t time. Fults tht hve n effet on the output of the iruit will hnge the vlue of output prit it from to. A logi t the output prit inites tht the opertion of the iruit is fult. A. Missing Gte Fult Fmil In this setion we oserve the effet of ifferent tpes of missing gte fults in three ifferent loks in the iruit. Figure 2 shows the SMGF n PMGF. However we ssume onl one tpe of fult is presente t time. ) Single Missing Gte Fult n Repete Gte Fult: Consier rnom Duplite Gte Blok (DGB ) in the iruit. Suppose the originl gte in this Duplite Gte Blok is missing. The missing gte is reunnt if n of this gte s ontrol points re logi. Let us onsier the sitution when ll the ontrol points of the originl gte re logi. As there is no fult in the Premle Blok, so the input of this DGB will follow the Prit Propert. The output of the originl gte is onnete to T, so there will not e n hnge in the ommon lines. However, the output prit line will toggle its input it. This is euse ll the ontrol lines re logi, so the uplite gte in the DGB will toggle its trget it (the trget of the uplite gte is the prit line). Thus the Prit Propert woul e violte t the output of this DGB. Aoring to the Simultneous Chnge Propert, this violtion will e forwre to the input of the Postmle Blok. When the inputs of the Postmle Blok o not follow the prit propert then logi woul e proue t the output prit line. In this w, fult output whih is generte ue to missing gte is notifie the prit line. Pin= SMGF PMGF Fig. 4. Single Missing Gte Fult n Prtil Missing Gte Fult For emple, onsier the online testle iruit of full er s shown in Figure 4. Suppose the originl gte whih is inite s otte line is missing. When the input vetor of(,,,)is()n()then the output will e ( ) n ( ) inste of the orret output ( ) n( ) respetivel. Most importntl the prit output will e logi, whih is the inition of fult output. Now onsier the SMGF in the seon gte (uplite gte) of Duplite Gte Blok. In this se, the output of the

4 ommon lines will hnge, euse the output of the ommon lines epens on the originl gte. However, s the trget line of the fult gte is onnete to the prit line, the output prit line of the orresponing DGB will not e hnge. For instne, when the ontrol lines of the gtes re t logi then the trget line, T of the originl gte woul toggle ut the prit line woul not toggle, whih violtes the Simultneous Chnge Propert. This violtion will lso ffet the Prit Propert t the input of the Postmle Blok. As result the output prit of the iruit will e logi, whih is suffiient onition for the etetion of fult in the iruit. Now onsier the se where one of the gtes in the Premle Blok is missing. The prit lines of ll the CNOT gtes pper in the prit line, so fult in the Premle Blok will not ffet the outputs of the ommon lines. But the fult will ffet the prit output P, whih is given P = Q Q 2 Q 3,...,Q L. When fult ffets the iruit output then the fult lso hnges the P to the opposite logi vlue of wht we epet from the Prit Propert. Aoring to the Prit Propert the output prit it of the Premle Blok shoul e if the prit of the ommon lines is even n the prit output shoul e if the prit of the ommon lines is o. But s one of the CNOT gtes is missing, so the fult will violte the prit propert. This violtion will ffet the output prit line of the iruit n the high vlue t the output prit will inite the fult opertion. As fr s the Repete Gte Fult is onerne, if the numer of repetitions of gte is o then this fult oes not ffet the iruit output. However, if the numer of repetitions is even then the effet of this fult is ientil to tht of single missing gte fult []. Thus, similr fult output woul e generte for repete gte fult n the high output prit it woul inite the presene of the fult in the iruit. 2) Prtil Missing Gte Fult: If prtil missing gte fult ours in the originl gte then some of ontrol points of the gte will e missing. For the fult to e etete t lest one of the missing ontrol points shoul e logi n the rest of the ontrol points of the fult gte shoul e logi []. Thus when the missing ontrol point is logi n ll the nonmissing ontrol points re t logi then the fult gte woul toggle the trget line (T ) of the DGB, whih woul give inorret output n the prit of the ommon lines woul e hnge. However, ll the ontrol points re not t logi for the uplite gte in DGB. Therefore, the prit line (P + ) of the DGB will not hnge. Hene, the Prit Propert woul e violte t the output of this DGB. Aoring to the Simultneous Chnge Propert, this violtion is trnsferre throughout the se of DGBs to the input of the Postmle Blok. Fult input t the Postmle Blok proues logi on the output prit line. Thus, the output of the iruit woul e erroneous, whih is inite logi on the output prit line. For instne, ssume tht the ontrol point s shown in Figure 4 is missing. Now, when the input is ( ) n ( ) for n input vetor (,,, ) then the output woul e ( ) n ( ) inste of the fultfree vlue ( ) n ( ) respetivel. For these two prtiulr input vetors the vlue of the output prit woul go high, whih inites tht the opertion is fult. In this w prtil missing gte fult whih ppers in n of the originl gtes n e etete. Now onsier the se where the ontrol points of the uplite gte re missing. When the nonmissing ontrol points re logi n one of the missing ontrol points is logi then the prit line (where the trget of the uplite gte eists) woul proue n output toggling its input. However, the ommon lines simpl pss the inputs of the originl gte to the output. This is ue to the ft tht there is no PMGF in the originl gte n ll the input its re not logi. So the trget line of the originl gte woul not toggle its input it. Thus, there is hnge in the vlue of the prit line ut prit of the ommon lines remins the sme, whih violtes the simultneous hnge propert n lso the prit propert. As onsequene the output prit of the iruit woul e logi, whih ientifies the fult. If PMGF ours in the Premle Blok then CNOT gte woul eome CNOT gte. A CNOT gte will hnge ever it on its input. Thus the output prit it of the Premle Blok will lws e the opposite of the tul true vlue. As in previous ses this fult prit will propgte to the suessive loks of the iruit n the effet will pper t the iruit output. B. Crosspoint Fult The ispperne fult is ientil to the prtil missing gte fult, thus the effet of ispperne fult n its etetion mehnism is the sme s tht of PMGF. If n pperne fult ours in the originl gte of the iruit then one or more etr ontrol points re e to the gte. The fult is etetle if t lest one of the etr ontrol points hve logi while the other ontrol points re logi. In this se the trget line of the fult gte will not toggle. However the trget of the uplite gte in the DGB will toggle its input it. Therefore, fult output is generte n the output of DGB fils to stisf the Prit Propert. When the pperne fult ours in the uplite gte of the DGB then the fult woul e etetle if n of the etr e ontrol points is t logi n ll other ontrol points re t logi. In this se the originl gte will toggle the output of the trget line T. However the uplite gte woul not hve ll its ontrol points t logi, so the prit line output woul e the sme s its input. Consequentl, hnge in the prit of ommon lines n no hnge in the vlue of the prit line woul violte the Simultneous Chnge Propert n lso the Prit Propert woul e lost. If n etr ontrol point ppers on CNOT gte of the Premle Blok then the fult will hve n effet on the iruit output onl when new ontrol point hs logi n the ol ontrol point is t logi. Beuse of the presene of the fult the gte woul not toggle. Thus we get the wrong prit t the output of the Premle Blok, whih oes not stisf the Prit Propert.

5 C. Single Bit Fult In single it fult moel, etl one output of iruit is fult euse of the hnge in it on some line. Thus if n single it fult ours in our moel then the output of the ommon lines will not follow the Prit Propert. Due to the nture of Simultneous Chnge Propert of the DGBs this violtion will e propgte to the input of the Postmle Blok. The input of the Postmle Blok in turn woul not follow the Prit Propert n hene the prit output woul e logi, whih woul inite tht there is fult in the iruit. Pin= Fig. 5. Presene of Single Bit Fult Consier single it fult ouring etween the seon n thir DGB of the iruit in Figure 5. When the input vetor ( )ispplie to the iruit then the presene of the fult uses the vlue on line to hnge from to, whih violtes the Prit Propert. The epete/orret output of the iruit shoul e ( ); however the tul output, refleting the fult, is ( ). The violtion of the Prit Propert is rrie through the iruit n the vlue of the output prit line eomes high, initing the presene of the fult. If there is no fult in the Premle Blok n the Duplite Gte Blok then the input of the Postmle Blok will stisf the Prit Propert. The Postmle Blok is the sme in rhiteture s the Premle Blok. If n fult ours in the Postmle Blok then the effet of this fult will e the sme s tht of the Premle Blok. If n fult ours in the Postmle Blok the prit output will go high. A logi t the output prit inites the presene of the fult. To summrie, if there eists fult with the originl iruit then the output of the orresponing DGB nnot stisf the Prit Propert. Moreover, ue to the Simultneous Chnge Propert of the DGBs, the violtion of the Prit Propert will e propgte to the input of the Postmle Blok. When the input of the Postmle Blok oes not follow the Prit Propert then the output of the Postmle Blok proues logi on the prit line, initing tht fult eists. On the other hn, if n error ours in n of the itionl iruitr n ffets the prit line then the prit line output will go high. The other outputs will not hnge. Thus if the prit line is high n the ommon line outputs re the sme s epete then it inites tht the fult hs ourre in the etr iruitr. VI. COMPARISON AND LIMITATIONS A. Comprison In this setion we ompre our propose pproh with two other online testing pprohes. In [8], the uthors propose n online testing strteg for etetion of single it fults. TABLE I OVERHEAD FOR SELECTED BENCHMARK CIRCUITS Ciruits Originl Ciruit Testle Ciruit Overhe (%) Funtions Quits GC QC GC QC GC QC r % 2% 45g % 7% hm % 29% r % 6% hw % 5% hw % 5% hw % % frg % 2% GC = Gte Count QC = Quntum Cost The use two sets of CNOT gtes n single prit line to mke reversile iruit online testle. In their pproh ll the ktoffoli gtes (or kcnot gtes) of the originl iruit re hnge to (k +)Etene Toffoli Gtes (ETG). We oserve tht using their pproh to implement the full er iruit presente in Figure 2 the resulting testle iruit hs gte ount of 2 n quntum ost of 28. Using our propose pproh (presente in Figure 2) the testle iruit hs gte ount of 6 n quntum ost of 32. The quntum ost n the gte ount of our pproh re slightl higher s ompre to their pproh. However, the previous pproh onl onsiere single it fults n our pproh n etet three tpes of fults. We net ompre our pproh with the online testing pproh presente in [4]. Their pproh requires single prit line n eh kcnot gte of the originl iruit will e trnsforme to its orresponing Augmente Reversile Gte (ARG). An ARG ontins four gtes: three itionl gtes n the originl gte. Thus with their strteg, four gtes re require to represent single gte. Therefore, in orer to implement the full er iruit in Figure 2, their pproh requires testle iruit with gte ount 6, whih is the sme s tht of our pproh. The quntum ost of their testle iruit is 32, whih is lso the sme s tht of our pproh. However, their pproh ws esigne to etet onl single missing gte fults. Our pproh is well suite for even iruit with lrge numer of gtes. In Tle I we present the gte ount n the quntum ost of the testle iruit fter ppling our pproh to selete enhmrk iruits [3] [7]. From this tle we n see tht for iruits with lrger numer of gtes our propose pproh tull results in lower overhe (in terms of perentge of the originl sie). If we oserve the first two enhmrk iruits (with sme numer of inputs) from the tle then we fin tht the iruit overhe is signifintl lower for the iruit (45g) with higher gte ount. The reson ehin this reution is tht the numer of itionl gtes in the premle n the postmle loks oes not epen on the numer of gtes of the originl iruit, rther this numer epens on the numer of quits (inputs) of the 2

6 iruit. Ciruitsr32 n 45g hve the sme numer of quits, however 45g hs lmost four times more gtes. Compre with [4], our pproh lso oes etter for iruits with higher gte ount when onsiering quntum ost. For instne, if we single CNOT gte to n originl iruit presente in Figure 2 then for our pproh the quntum ost inreses 2 (sine the quntum ost of CNOT gte is, n we uplite the gte). However, using the pproh in [4], the quntum ost woul inrese 4. This is euse s long s the numer of quits in iruit oes not inrese, our pproh onl inlues uplite gte for eh originl gte. B. Limittions Firstl, the propose online pproh onsiers onl the gtes from the CNT gte lirr. So it is not urrentl pplile for other reversile gtes. The propose pproh n not etet single it fult if the fult ours in the Premle Blok. A single it fult in the Premle Blok uses reversile iruit to proue fult output. However in this se the output prit line will e logi. Thus oserving the output prit it we n not etet the single it fult. In ition, our pproh fils to etet prtiulr se when eling with multiple missing gte fult. Multiple missing gte fults our when severl onseutive gtes go missing in iruit []. Suppose tht N onseutive gtes in the originl gtes re missing. Here N might e even or o. For ifferent omintion of inputs, ifferent gtes mongst the missing gtes woul e irreunnt. If we onsier se where n even numer of missing gtes re irreunnt, then the Prit Propert woul e violte for n even numer of times. When the prit propert is violte n even numer of times then the fult is reunnt. Thus in those ses when the numer of missing gtes is even then the output will e inorret ut the Prit Propert will e preserve. So the prit output will e logi whih oes not inite the error even though the output is inorret. Now onsier n input omintion when the irreunnt missing gtes re o. The Prit Propert is violte n o numer times. Hene the prit output vlue onverts to high whih properl inites the error in the output. Therefore, some ut not ll the possile MMGF fults re etetle the our propose online moel. VII. CONCLUSION AND FUTURE WORKS This pper presents n online testing pproh for reversile iruits se on the CNT gte lirr. With this pproh reversile iruit n e onverte to its online testle version ing set of CNOT gtes n single prit line in well efine mnner. With these smll moifitions we rete iruit tht omputes its originl funtionlit n in ition the iruit will etet single it fult, missing gte fult or rosspoint fult. The propose pproh requires 2(L + N) itionl gtes. This inreses the iruit overhe in terms of gte ount n quntum ost, however in return, the iruit eomes online testle for three fult moels. Our pproh n lso etet fult even if the fult ours in the itionl iruitr, unlike other pprohes in the literture. In ition, sine the numer of itionl CNOT gtes epens on the numer of quits of reversile iruit, the perentge of overhe in terms of gte ount n quntum ost will e reue for iruit with lrger gte ount. We onsiere ifferent fult senrios in reversile iruit n oserve the output. If fult ours in the originl gte of iruit then the output will e inorret n the prit line will go high. We lso oserve tht if fult ours in n of the etr iruitr then the originl output of the iruit will not e ffete. However the prit line will go high whih lerl inites the presene of fult in the iruit. Therefore, if the prit line is high n the output is sme s epete then we n ssume tht the fult hs ourre in the itionl iruitr. Etension of this pproh to etet ll the possiilities of single it fults n multiple missing gte fults is the re of further reserh. REFERENCES [] C. H. Bennett. Logil reversiilit of omputtion. IBM Journl of Reserh n Development, 6: , 973. [2] M. P. Frnk. Introution to reversile omputing: Motivtion, progress, n hllenges. In Proeeings of the 2n Conferene on Computing Frontiers, pges , Ishi, Itl, 25. ACM Press. [3] J. P. Hes, I. Polin, n B. Beker. Testing for missinggte fults in reversile iruits. 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