A Fmily of Logil Fult Moels for Reversile Ciruits Ili Polin John P. Hyes Thoms Fiehn Bern Beker Alert-Luwigs-University Georges-Köhler-Allee 5 79 Freiurg i. Br., Germny {polin fiehn eker}@informtik.uni-freiurg.e Avne Computer Arhiteture L University of Mihign Ann Aror, MI 489-222, USA jhyes@ees.umih.eu Astrt Reversiility is of interest in hieving extremely low power issiption; it is lso n inherent esign requirement of quntum omputtion. Logil fult moels for onventionl iruits suh s stuk-t moels re not wellsuite to quntum iruits. We erive fmily of logil fult moels for reversile iruits ompose of k- CNOT (k-input ontrolle-not) gtes n implementle y mny tehnologies. The moels re extensions of the previously propose single missing-gte fult (MGF) moel, n inlue multiple n prtil MGFs. We stuy the si etetion requirements of the new fult types n erive ouns on the size of their test sets. We lso present optiml test sets ompute vi integer liner progrmming for vrious enhmrk iruits. These results inite tht, lthough the test sets re generlly very smll, prtil MGFs my nee signifintly lrger test sets thn single MGFs. Keywors: ATPG, fult moels, reversile iruits, quntum iruits Introution The reversiility of omputtion hs long een stuie s mens to reue or even eliminte the power onsume y omputtion [, 2]. Of prtiulr interest is type of reversile omputing known s quntum omputing [3] [4], whih funmentlly hnges the nture of omputtion y sing it on quntum mehnis rther thn lssil physis. It hs ttrte onsierle reserh ttention euse some importnt prolems suh s prime ftoriztion n e solve exponentilly fster y quntum methos thn y ny known non-quntum (lssil) pprohes. Quntum iruits store informtion in mirosopi sttes n proess it using quntum mehnil opertions or gtes tht moify these sttes. The unit of quntum informtion is lle quit. Like lssil it, quit n e in zero or one stte, onventionlly enote y n, respetively. However, it n lso e in superposition of these sttes, i.e., α + α, where α n α re omplex numers lle mplitues. A set of n > quits n store 2 n inry wors onurrently. For instne, two-quit stte Ψ = α + α + α 2 + α 3 stores four wors. These 2 n wors n e proesse simultneously y quntum iruit, proviing kin of mssive prllelism. However, quntum iruits re sujet to some severe restritions on their ehvior. The oservility of sttes is very limite. Mesurement of quntum stte yiels just one of its 2 n superimpose vlues, with proility given y the squre of the vlue s mplitue. Hene, ll mesure outputs re proilisti. In ition, quntum gtes n iruits must e reversile, i.e., informtionlossless, n so must hve the sme numer of inputs n outputs. Moreover, quntum sttes nnot e opie (the no-loning property), so fn-out is not llowe. Quntum lgorithms hve to e very refully esigne in orer to ount for the foregoing non-lssil ehvior. In quntum omputtion, liner lger reples the fmilir Boolen lger of lssil logi. A quntum gte is efine y liner opertion over Hilert spe, n is represente y unitry mtrix. Figure shows ontrolle-not or CNOT gte, whih is si 2-quit quntum gte, in stnr nottion. It hs two inputs (ontrol) n t (trget) n two outputs. The mtrix U CN esriing the CNOT gte is U CN = () Applying the CNOT gte to 2-quit stte Ψ = α + α + α 2 + α 3 yiels new 2-quit stte Ψ = α + α + α 2 + α 3, where (α,α,α 2,α 3) T = U CN (α,α,α 2,α 3 ) T in mtrix-vetor nottion. For the four sis sttes (those with one mplitue α i equl to n ll other mplitues equl to ), the funtion of CNOT gte n e written s the Boolen funtion t ( t) ; see Figure. Quntum iruits n e ompose of vrious types of gtes [3], inluing CNOT gtes n their generliztions, k-cnot gtes with k ontrol inputs. Note tht k-cnot gte hs k + inputs n outputs n is esrie y 2 k+ 2 k+ mtrix.
t Figure : A CNOT gte n its tion on the four 2-quit sis sttes Mny physil implementtions of quntum iruits hve een suggeste, lthough prtil quntum omputer hs yet to e uilt [5]. Quntum stte representtions inlue photon polriztion n eletron spin. Suh sttes re frgile n error-prone ue to their nnosle imensions, extremely low energy levels, n teneny to intert with the environment (eoherene). Hene, it is expete tht effiient testing n fult-tolernt esign methos will e essentil for the suessful implementtion of quntum iruits. Beuse of the omplexity of their norml n fulty ehvior moes, the testing prolems pose y generl quntum iruits re very hllenging [3, 6, 7]. It is therefore useful t this stge to onsier speil ses tht pture some key hrteristis of quntum iruits, for exmple, reversile iruits ompose of k-cnot gtes. Ptel et l. [8] onsier testing for stuk-t n ell fult moels in these iruits. They onlue tht the omplexity of test genertion is lower for reversile iruits thn for onventionl irreversile ones; for exmple, ll multiple stuk-t fults re overe y omplete test set for single stuk-t fults. However, Hyes et l. [9] note tht the vliity of the stuk-t fult moel for quntum iruits is limite. They propose n investigte the missinggte fult moel, whih is lrgely tehnology-inepenent n omputtionlly trtle. This pper introues n investigtes hierrhy of logil fult moels of the missing-gte type for reversile k-cnot iruits, se on similr ssumptions to those in [9]. Although our pproh is tehnology-inepenent, we fous on the trppe-ion tehnology [] whih uses the ertin spin n virtionl moes of eletrilly hrge toms (ions) s the quit representtion. Our fult moels re lso pproprite for mny tehnologies tht represent quits s spin, inluing the nuler mgneti resonne tehnology (whih uses the spins of tomi nulei) [5]. The gte opertions re implemente y mens of lser pulses ontrolle y lssil omputer. We ssume tht the intertion etween gte pulses n quits to e the min soure of mlfuntions. This is onfirme y reports on prototype quntum proessor in NMR tehnology []. The reminer of the pper is orgnize s follows. Setion 2 ontins rief review of k-cnot reversile iruits n the trppe-ion tehnology. The fult moels re propose n their properties stuie in Setion 3. Experimentl results re reporte in Setion 4. Setion 5 onlues the pper. Figure 2: Reversile iruit ompose of 2-CNOT, - CNOT n 3-CNOT gte 2 Reversile Ciruits In this setion we summrize k-cnot-se reversile iruits, n isuss representtive quntum tehnology: ion trps. Note tht the moels re lso vli for other tehnologies suh s NMR. 2. k-cnot Ciruits A reversile iruit hs n inputs n n outputs, n mps eh input pttern to unique output pttern, i.e., it omputes ijetive funtion. A k-cnot gte hs k ontrol inputs,... k n one trget input t. It mps the vetor (,..., k,t) to the vetor (,..., k,t ( 2 k )). The vlues t the ontrol inputs re left unhnge, n the vlue on the trget input is inverte if n only if ll the vlues t the ontrol inputs re. A -CNOT gte is often just lle CNOT gte, n 2-CNOT gte is lle Toffoli gte. Figure 2 shows smll reversile iruit ompose of three k-cnot gtes. It hs four wires representing the signls (quits) eing proesse. The three CNOT gtes proess the four signls from left to right. Following stnr onvention, ontrol inputs re shown s lk ots, n the trget input is enote y ringsum. The figure shows how input vetor () is mppe to (). Sine ll its gtes re reversile, eh group of gtes in reversile iruit is lso reversile. Hene, ny ritrry stte n e justifie on eh gte. For exmple, if the vlues () must e pplie to the rightmost 3-CNOT gte of Figure 2 for test purposes, there is unique input vetor tht is esily otine y kwr simultion ( in this se). Furthermore, propgtion of fult effet is trivil: if logi vlue is reple y logi (or vie vers) ue to fult, this will result in ifferent vlue t the output of the iruit. 2.2 Trppe-Ion Tehnology Nielsen n Chung [3] ite four ilities of tehnology s neessry for quntum omputtion. The tehnology must: () roustly represent quntum informtion; (2) perform universl set of unitry trnsformtions; (3) prepre urte initil sttes; n (4) mesure the output results. In the following, we will riefly review how these
/ / / / / Figure 4: Single missing-gte fult (SMGF) n its physil justifition is implemente y lser pulse (or sequene thereof) pplie to quits, n. Their intertion results in the stte of quit eing hnge from to. This is shown in the upper right of Figure 3. Similrly, the seon (-CNOT) gte orrespons to pulse tht hnges the stte of quit from to. The thir gte oes not result in stte hnge on the trget quit ue to logi- vlue t one of its ontrol inputs (); onsequently, the thir pulse oes not hnge the stte of quit. Figure 3: Physil implementtion of the iruit from Figure 2 in trppe-ion tehnology four issues re resse in the trppe-ion tehnology; see [3, 5] for further etils. Quit representtion: The internl stte of n ion serves s the quit representtion; the groun stte ( g ) represents, while the exite stte ( e ) represents. In trppe-ion tehnology, ions re onfine in n ion trp, i.e. etween eletroes, some of whih re groune (hve stti potentil) while others re riven y fst osillting voltge. The Los Almos group use the C + ions with 4 2 S /2 s the groun stte n 3 2 D 5/2 s the exite stte [2]. Unitry trnsformtions: These opertions rotte stte vetors without hnging their length, whih implies reversiility. In the trppe-ion tehnology, ions intert with lser pulses of ertin urtion n frequeny. Quits intert vi shre phonon (quntum of virtionl energy) stte. CNOT funtionlity hs een experimentlly emonstrte for the trppe-ion tehnology (s well s for NMR tehnology) [3]. Initiliztion: Trppe ions re rought into their motionl groun stte... using Doppler n sien ooling. Mesurement: The stte of single ion is etermine y exiting n ion vi lser pulse n mesuring the resulting fluoresene. Figure 3 illustrtes the gte implementtion in trppeion tehnology require for the iruit in Figure 2. The iruit hs four wires,, n, so four ions (quits) re use. Sine the input vetor is (), the quits n re set to the stte n the quits n re set to in the eginning. The leftmost (2-CNOT) gte 3 Fult Moels Next we introue severl fult moels tht re minly motivte y the ion-trp quntum omputing tehnologies isusse in the preeing setion. The si ssumptions re tht quits re represente y the ion stte, gtes orrespon to externl pulses whih ontrol the intertions of the quits, n the gte opertions re errorprone. The fult moels propose re the single missinggte fult (SMGF), the repete-gte fult (RGF), the multiple missing gte fult (MMGF) n the prtil missing-gte fult (PMGF) moels. 3. Single Missing-Gte Fult Moel A single missing-gte fult (SMGF) orrespons to the missing-gte fult isusse in [9]. It is efine s omplete ispperne of one CNOT gte from the iruit. The physil justifition for SMGF is tht the pulse(s) implementing the gte opertion is (re) short, missing, misligne or mistune. Figure 4 shows the iruit from Figure 2 with n SMGF: the first (2-CNOT) gte is missing. The resulting hnges in logil vlues re shown in the formt fultfree vlue/fulty vlue. It n e seen tht the fult effet is oservle on wires n. The right prt of Figure 4 suggests how the pulse orresponing to the first gte is too wek to hnge the vlue on quit from to. The etetion onition for n MGF is tht logi vlue e pplie to ll the ontrol inputs of the gte in question; the vlues on the trget input s well s the vlues on the wires not onnete to the gte re ritrry. The numer of possile SMGFs is equl to the numer of gtes in the iruit. The following hrteriztion of SMGFs in proven in [9].
/ / / / / / / / / X Figure 5: Repete-gte fult (RGF) n its physil justifition Theorem (Properties of SMGFs) Consier reversile iruit onsisting of N CNOT gtes.. There is lwys omplete SMGF test set of N/2 or fewer vetors. 2. There re iruits for whih the miniml omplete SMGF test set hs extly N/2 vetors. 3. By ing one extr wire n severl -CNOT gtes, every iruit n e trnsforme suh tht the resulting iruit retins its originl funtionlity ut hs omplete SMGF test set onsisting of one test vetor. The trnsformtion n e one for ny test vetor, ut there is unique test vetor leing to miniml overhe (numer of require extr -CNOT gtes). SMGFs orresponing to the e gtes re lso overe y tht test vetor. 3.2 Repete-Gte Fult Moel A repete-gte fult (RGF) is n unwnte replement of CNOT gte y severl instnes of the sme gte. The physil justifition for n RGF is the ourrene of long or uplite pulses. Figure 5 (left) shows the iruit from Figure 2 with uplite first gte. It n e seen tht the fult effet is ientil to tht of the SMGF (Figure 4). Figure 5 (right) illustrtes the oule trnsition on quit first from to, n then k to ue to long or uplite pulse. As generliztion, the following theorem hols: Theorem 2 (Properties of RGFs) Consier n RGF tht reples gte y k instnes of the sme gte.. If k is even, the effet of the RGF is ientil to the effet of the SMGF with respet to the sme gte. 2. If k is o, the fult is reunnt, i.e., it oes not hnge the funtion of the iruit. Proof: It suffies to see tht two ientil gtes in series ompute the ientity funtion. Sine the ontrol n the trget inputs of oth gtes re lote on the sme wire, the only wire whose vlue n e hnge is the wire t on whih the trget inputs of the gtes re ple. In Figure Figure 6: Multiple missing-gte fult (MMGF), n iruit frgment for whih the optiml MMGF test set is lrger thn ny SMGF test set 5, t orrespons to wire. For test vetor tht pplies logi- vlue to t lest one ontrol input of the first gte, the sme hols for the seon gte, n the vlue on wire t is unhnge y either gte. Otherwise, the vlue on wire t is inverte y oth gtes, n so ens up unhnge. Theorem 2 is lso n immeite onsequene of k- CNOT gte s unitrity property. From this theorem we onlue tht n SMGF test set lso etets ll irreunnt RGFs. 3.3 Multiple Missing-Gte Fult Moel This moel ssumes tht gte opertions re isture for severl onseutive yles, so tht severl onseutive gtes re missing from iruit. An exmple involving two missing gtes is shown in Figure 6 (left). Note tht the MMGF efinition oes not mth our usul unerstning of multiple fult, whih implies tht severl istint single fults re present in the sme time. We lso restrit multiple fults to one or more onseutive gtes. Hene for the iruit from Figure 6 (left), removing the mile n the rightmost gte yiels vli MMGF, ut removing the leftmost n the rightmost gtes oes not. This fult moel is justifie y the ssumption tht the lser implementing gte opertions is more likely to e isture for perio of time exeeing one gte opertions thn to e isture for short time, then perform error-free, n then e isture gin. Clerly, SMGFs re suset of the MMGFs. In n N-gte iruit, the numer of possile MMGFs is N(N + )/2, qurti funtion of N, wheres the orresponing numer of multiple SMGFs is exponentil in N. It hs een proven for stuk-t fults in reversile iruits tht omplete single fult test set overs ll multiple fults [8]. This is not true for SMGFs n MMGFs, however, espite the restrition tht the missing gtes must e onseutive. This is emonstrte y the two-gte iruit frgment shown in Figure 6 (right). The SMGF orresponing to the left (3-CNOT) gte requires the test vetor (X) for etetion, where X stns for on t re. The SMGF for the seon (2-CNOT) gte requires (XX), so the optiml SMGF test set onsists of one test vetor, e.g., (). However, this vetor oes not etet the MMGF efine y removl of oth gtes, lthough it
/ Figure 7: Prtil missing-gte fult (PMGF) n its physil justifition is omplete SMGF test set. The MMGF is not reunnt, s vetor (X) etets it. Furthermore, s every SMGF is lso n MMGF, the vetor (X) lso must e inlue in ny omplete MMGF test set. Hene, the optiml size of omplete MMGF test set is two. We hve seen ove tht the size of the optiml test set for SMGFs is one. Hene, omplete SMGF test set oes not over ll MMGFs. 3.4 Prtil Missing-Gte Fult Moel A prtil missing-gte fult is result of prtilly misligne or mistune gte pulses. It turns k-cnot gte into k -CNOT gte, with k < k. We ll k k the orer of PMGF. Figure 7 shows first-orer PMGF ffeting the thir ontrol input of the rightmost gte, n the wek pulse tht fils to mke intert with, n. An SMGF n e seen s -orer PMGF. Theorem 3 (Properties of PMGFs). A k-cnot requires k test vetors to etet ll firstorer PMGFs n k + vetors if the SMGFs must lso e etete. 2. An m-orer PMGF omintes m first-orer PMGFs, i.e. it is etete y ny test vetor tht etets one of the first-orer PMGFs. Proof:. The etetion onition for first-orer PMGF ffeting the ith ontrol input is to ple on the ith ontrol input n on ll other ontrol inputs, sine this is the only se for whih the fult-free n the fulty gte proue ifferent vlues on the trget noe. Hene, ifferent first-orer PMGFs nnot e etete y the sme test vetor. Sine there re k first-orer PMGFs, k test vetors re require. To etet the SMGF ffeting the sme gte, we must ple on ll ontrol inputs. As this is inomptile with the etetion onitions of ny of the k first-orer PMGFs, n itionl vetor is neee for SMGF etetion. 2. A higher-orer PMGF is etete when is pplie to t lest one of the ffete ontrol inputs n is pplie to ll other ontrol inputs. Thus, ny of the test vetors eteting first-orer PMGF ffeting one of the Ciruit N n S M P S+M+P 2of5 8 6 4 5 8 2of52 2 7 2 2 3 4 3 7t 6 3 2 2 2 3 4 49t 6 4 3 3 5 5 5mo5t 7 6 6 5 7 6sym2 2 2 2 3 4 9sym2 28 2 3 3 hm3t 5 3 2 2 3 3 hm7t 24 7 4 4 4 4 hw4t 7 4 2 4 5 6 hw5t 56 5 5 5 9 9 hw6t 26 6 8 8 5 6 hw7t 29 7 3 4 24 25 mo5ers 2 6 3 4 6 8 mo5 8 5 4 2 4 mo52 9 5 2 2 3 r32 4 4 2 2 3 3 r53 2 7 2 3 8 8 r532 2 8 2 2 3 4 r53rmg 3 7 3 4 8 r732 2 3 3 4 4 r84 28 5 3 3 4 xor5 4 5 2 Averge 3.3 3.83 5.77 6.8 Tle : Ext ATPG results (S = Single MGF, M = Multiple MGF, P = Prtil MGF, S+M+P = ll moels omine) ontrol inputs ffete y the higher-orer PMGF lso etets the higher-orer PMGF. Note tht test vetor with on more thn one ontrol input my etet higher-orer PMGFs ut no first-orer PMGFs. The first result implies lower oun on the numer of test vetors for iruit. It lso implies tht no iruit hving k-cnot with k > is testle for ll PMGFs with just one test vetor. This is ifferent thn the SMGFs se. It is possile to efine PMGFs with respet to trget noe. However, the resulting fult orrespons to SMGF on the sme gte. 4 Experimentl Results We implemente n ext utomti test pttern genertion (ATPG) metho for the vrious types of MGFs se on integer liner progrmming. The numer of test vetors is the ojetive funtion to e minimize, n the etetion onitions re enoe s onstrints. We pplie this ATPG tehnique to the enhmrk iruits in [4]. The first three olumns of Tle give the iruit s nme, the numer of gtes N, n the numer of wires n. The
ompute numer of test vetors for the single missinggte, multiple missing-gte, n prtil missing-gte fult moels re reporte in the next three olumns of the tle. Dshes inite iruits for whih the ATPG proeure i not terminte within resonle time. The finl olumn ontins the numer of test vetors in miniml set tht etets ll these fults. The ottom line of the tle ontins verge vlues. Sine repete-gte fults (RGFs) re either ientil to SMGFs or reunnt, the vlues for the SMGF moel re lso vli for the RGF se. As mentione ove, the MMGF moel pplies to fults ffeting onseutive gtes n not to ritrry multiple SMGFs. Sine first-orer PMGFs re ominte y higher-orer PMGFs, our experiments only trgete first-orer PMGFs, so the resulting test sets re omplete with respet to ritrry PMGFs. It n e seen tht SMGF n MMGF test sets re of ientil size in 4 out of 23 ses. However, there re iruits for whih the test set sizes re onsierly lrger for MMGFs thn for SMGFs. This onfirms our sttement tht SMGFs re not ominte y MMGFs. Note tht ll of the results in the tle re provly optiml. Aoring to Theorem 3, lower oun on the size of omplete PMGF test set is the mximum numer of ontrol inputs of ny gte in the iruit. For the iruits onsiere, this oun is tight for iruits 3 7t, mo5 n mo52, whih onsist of -CNOTs n 2- CNOTs, n iruit xor5 whih onsists of -CNOTs only. Overll, the PMGF test sets re lrger thn SMGF test sets for ll ut three iruits, n they re lmost twie s lrge on verge. The test sets eteting ll the fults (SMGF, MMGF n PMGF) re 8% lrger thn PMGF test sets, on verge. This is onsistent with the sttement tht PMGF test is not SMGF test for the sme gte. Nevertheless, the test sets for PMGFs n the omintion of ll moels re of ientil size for seven iruits. This must e ue to fortuitous etetion of non-pmgfs y PMGF test vetors otine for ifferent gte. 5 Conlusions Effiient testing n fult-tolernt esign methos re require for reliztion n opertion of quntum iruits. In this pper, we hve onsiere the test genertion prolem for reversile k-cnot iruits, whih form restrite lss of quntum iruits, ut pture some of their key hrteristis suh s lossless informtion proessing n no loning. Although onventionl fult moels for CMOS ICs hve een pplie to reversile iruits, their vliity for quntum tehnologies ppers to e quite limite. We hve propose fmily of fult moels for reversile iruits, se on the onepts of missing n repete gte opertions. These fult moels re ll efine t the logil level, whih implies high egree of tehnology inepenene n low omputtionl omplexity. While pplile, in priniple, to ll known quntum tehnologies, they seem to e espeilly well-suite to trppe-ion tehnology (n spin tehnologies suh s NMR). Issues for future reserh inlue moeling other types of gte efets, n eling with the stte superposition n mesurement properties of generl quntum iruits. Aknowlegment John Hyes ontriute to this work while visiting the University of Freiurg uner n wr from the Alexner von Humolt Fountion. 6 Referenes [] C.H. Bennett. Notes on the history of reversile omputtion. IBM J. Res. n Develop., 32:6 23, 988. [2] A. Peres. Reversile logi n quntum omputers. Phys. Rev. A, 32(6):3266 3276, 2 985. [3] M.A. Nielsen n I.L. Chung. Quntum Computtion n Quntum Informtion. Cmrige Univ. Press, 2. [4] E. Rieffel. An introution to quntum omputing for non-physiists. ACM Computer Surveys, 32(3):3 335, 9 2. [5] Quntum Computtion (QC) Romp. http://qist.lnl.gov/, 24. [6] E. Knill, R. Lflmme, A. Ashikhmin, H. Brnum, L. Viol, n W. H. Zurek. Introution to quntum error orretion. Los Almos Siene, 27:88 22, 22. [7] K.M. Oenln n A.M. Despin. Impt of errors on quntum omputer rhiteture. Tehnil report, Univ. of Southern Cliforni, 996. [8] K.N. Ptel, J.P. Hyes, n I.L. Mrkov. Fult testing for reversile iruits. In VLSI Test Symp., pges 4 46, 23. [9] J.P. Hyes, I. Polin, n B. Beker. Testing for missinggte fults in reversile iruits. In Asin Test Symp., pges 5, 24. [] J.I. Cir n P. Zoller. Quntum omputtions with ol trppe ions. Phys. Rev. Lett., 74(2):49 494, 995. [] L. Steffen, L.M.K. Vnersypen, n I.L. Chung. Towr quntum omputtion: five-quit quntum proessor. IEEE Miro, 2(2):24 34, 3 2. [2] R.J. Hughes, D.F.V. Jmes, J.J. Gomez, M.S. Gulley, M.H. Holzsheiter, P.G. Kwit, S.K. Lmoreux, C.G. Peterson, V.D. Snerg, M.M. Shuer, C.M. Simmons, C.E. Thorurn, D. Tup, P.Z. Wng, n A.G. White. The Los Almos trppe ion quntum omputer experiment. Tehnil report, Los Almos Ntionl Lortory, 997. LA- UR-97-33. Aville from http://li-www.lnl.gov/giin/getfile?4296.pf. [3] C. Monroe, D.M. Meekhof, B.E. King, W.M. Itno, n D.J. Wineln. Demonstrtion of funmentl quntum logi gte. Phys. Rev. Lett., 75(25):474 477, 995. [4] D. Mslov, G. Duek, n N. Sott. Reversile Logi Synthesis Benhmrks Pge. http://www.s.uvi./ mslov/, 24.