ML-aided SOP compensation to increase key exchange rate in QKD systems

ML-Aided SOP Compensa ion o Inc ease Key Exchange Ra e in QKD Sys ems Mo eza Ahmadian*, Ma c Ruiz, Jaume Comellas, and Luis Velasco Op ical Communica ions G oup (GCO), Uni e si a Poli ècnica de Ca alunya (UPC), Ba celona, Spain *e-mail: seyed.mo [email protected] ABSTRACT Secu e communica ions ha e become a equi emen o i ually all kind o applica ions. Cu en ly, wo dis an pa ies can gene a e sha ed andom sec e keys by using public key c yp og aphy. Howe e , quan um compu ing ep esen s one o he g ea es h ea s o he ini e complexi y o he ma hema ics behind public key c yp og aphy. In con as , Quan um Key Dis ibu ion (QKD) elies on p ope ies o quan um mechanics, which enables ea esd opping de ec ion and gua an ees he secu i y o he key. Among QKD sys ems, pola iza ion encoded QKD has been success ully es ed in labo a o y expe imen s and ecen ly demons a ed in closed en i onmen s. In his pape , we p opose a Machine Lea ning (ML) -based pola iza ion acking and compensa ion ha is able o keep sha ed sec e key exchange o high a es e en unde la ge ibe s essing e en s. Exhaus i e esul s using bo h syn he ic and expe imen al da a show ema kable pe o mance, which can simpli y he design o bo h quan um ansmi e and ecei e , as well as enable he use o ae ial op ical cables, hus educing o al QKD sys em cos . Keywo ds: Pola iza ion-encoded Quan um Key Dis ibu ion; Machine Lea ning. 1. INTRODUCTION Quan um Key Dis ibu ion (QKD) has become ma u e in closed, con olled scena ios in iew o he plen y o wo ks a ailable in he li e a u e epo ing ela ed expe imen s. In pola iza ion encoded QKD sys ems, a Quan um T ansmi e (QTx) sends pola ized pho ons, i.e., quan um bi s (qubi ), o a Quan um Recei e (QRx), which decodes hem and gene a es a aw key o a de ined leng h. The aw key is hen dis illed, using a pa allel public channel es ablished be ween ansmi e and ecei e , o co ec possible de ec ion e o s due o op ical ansmission and gene a e a sha ed sec e key. E.g., he au ho s in [1] showed a pola iza ion-based QKD sys em using he BB84 p o ocol ha eaches sha ed sec e Key Exchange Ra es (KER) > 1 Mb/s o dis ances >100 km. Cu en ly, esea ch e o s a e also ocused on demons a ing such pe o mance in eal (mo e challenging) scena ios, including ae ial cables, whe e QKD ansmission migh be se e ely a ec ed by wea he condi ions (e.g., high wind) ha s esses op ical ibe s [2]. Such mechanical s ess changes ibe bi e ingence, which in oduces luc ua ions on he S a e o Pola iza ion (SOP) o he ansmi ed qubi s and, as a esul , Quan um Bi E o Ra e (QBER) inc eases. No e ha QBER is causally ela ed o he e ec i e KER, which educes when QBER inc eases, e.g., om Mb/s o Kb/s o e en b/s as shown in [3]. Since op ical ea esd opping gene a es high QBER, a pos p ocessing phase named key dis illa ion enables i s de ec ion. Howe e , excessi e QBER coming om SOP luc ua ions migh de i e in o alse ea esd opping de ec ion ( h eshold is ypically se wi hin he ange 5%-10%); in such case, sa e y mechanisms agains a acks a e ac i a ed, hus in e up ing (i.e., KER becomes empo a ily 0), o e en blocking ha quan um channel o key exchange. In his wo k, we summa ize he wo k in [4] and p opose a ligh weigh ML-based SOP acking and pola iza ion compensa ion ha uses Deep Neu al Ne wo k (DNN) models o pola iza ion encoded QKD sys ems. Such models accu a ely an icipa e SOP luc ua ions, so adap i e ac ions can be aken a he QRx o e e se hem be o e hey p oduce nega i e impac . The p oposed sys em is speci ically designed o maximize pe o mance, i.e., o educe alse ea esd opping de ec ion and inc ease e ec i e KER, in scena ios exposed o en i onmen al e en s. The p oposed app oach will enable cos educ ion o QKD sys ems as: i) QTx speci ica ions can be elaxed since SOP impe ec ions can be co ec ed by he QRx; and ii) he ha dwa e design o he QRx can be simpli ied and ely on so wa e. 2. ML-BASED FAST QUANTUM KEY DISTRIBUTION In his sec ion, we i s b ie ly p esen he main concep s and used no a ion. Ra he han an exhaus i e desc ip ion o QKD sys ems, we i s p esen he essen ial concep s ega ding ansmission, p opaga ion, and pho ons measu emen o aw keys exchange unde he BB84 p o ocol [5]. Nex , we iden i y oppo uni ies and p opose solu ions o accele a e he dis ibu ion o keys o e a quan um channel in he p esence o SOP luc ua ions. P elimina y concep s In BB84, he QTx con inuously gene a es aw keys con aining sequences o pai s o Boolean alues, each pai con aining a basis (B) and bi (b). The pai <B( ), b( )> gene a ed a ime is de ined by he quan um s a e |q( )〉, which can be de ined as a posi ion on he Bloch sphe e. The e o e, |q( )〉 can be al e na i e exp essed: i) in Euclidean coo dina es <x( ), y( ), z( )>, wi h one componen o axis X, Y, and Z, espec i ely; o ii) in pola coo dina es <θ( ), φ( )>, ep esen ed by azimu h and ellip ici y angles, espec i ely. In p ac ice, |q( )〉 is encoded as a single pho on, which ansla es in o a single poin on he uni a y Poinca é sphe e; Bo h Bloch and Poinca é sphe es a e exchangeable i axes X, Y, and Z o he o me ma ch S okes S2, S3, and S1, espec i ely, in he la e . © 2023 IEEE. Pe sonal use o his ma e ial is pe mi ed. Pe mission om IEEE mus be ob ained o all o he uses, in any cu en o u u e media, including ep in ing/ epublishing his ma e ial o ad e ising o p omo ional pu poses,c ea ing new collec i e wo ks, o esale o edis ibu ion o se e s o lis s, o euse o any copy igh ed componen o his wo k in o he wo ks. DOI 10.1109/ICTON59386.2023.10207413 E ec s ela ed o ibe p opaga ion and ea esd opping al e |q( )〉. Le us deno e |p( )〉 = <θp( ), φp( )> as he eal pola iza ion o he ecei ed pho on. We adop he QRx ha dwa e a chi ec u e p oposed in [6], whe e he QRx is equipped wi h an Elec onic Pola iza ion Con olle (EPC) ollowed by a Pola iza ion Beam Spli e (PBS). The pho on i s eaches he EPC, which is in cha ge o pola iza ion alignmen . Speci ically, gi en a e e ence pola iza ion s a e ( ) (he ea e deno ed as o a ion) de ined by he uple <θ ( ), φ ( )>, he EPC pe o ms a e e sal ope a ion o align he pho on de ec o wi h he con igu ed pola iza ion s a e. Hence, i is wo h no ing ha he o a ion wi h con igu a ion θ ( )=θp( ) and φ ( )=φp( ) is he one pe ec ly aligned wi h he s a e |p( )〉 o ecei ed pho on. Be o e he pho on passes h ough he PBS, a basis is selec ed, which en ails selec ing a speci ic axis in he sphe e o de ec he pho on and ex ac i s bi [5]. Two main condi ions lead o e oneous bi ex ac ion: i) i he sphe e is pe ec ly aligned wi h |p( )〉, he bi is w ongly decoded i QRx selec s he w ong basis; and ii) e en i QRx selec ed he co ec basis, bi e o can be p oduced i he e is misalignmen be ween ( ) and |p( )〉. Besides he quan um channel, a pa allel secu e public channel is used o key dis illa ion pu poses. QRx s a s sending a subse o decoded bi s and basis o QTx in o de o quan i y bi e o s, i.e., QBER. In case ha QBER exceeds a gi en h eshold, e.g., 10%, ea esd opping in he quan um channel is assumed, which igge s a sa e y mechanism, such as QKD in e up ion. O he wise, QKD is assumed o be secu e enough. Nex , bases need o be e i ied, since hey we e andomly selec ed a he QRx side. To ha end, key si ing is pe o med, whe e QTx sends o QRx he sequence o used bases h ough he public channel, so ha QRx can check hem and disca d he w ong ones. A e he bases a e synch onized, e o cascading is conduc ed o co ec he e oneous bi s, which esul s in o a co ec ed si ed key. In he end, a po ion o he si ed key is selec ed as he inal sha ed sec e key o ampli y p i acy. This p ocess esul s in o a maximum achie able KER when QBER is low, and i will be no iceably educed when QBER inc eases. ime SOP Measu emen (o( )) (a) Reac i e SOP o a ion +m Ro a ion ( ( )) TOTRKey Dis ibu ion QBER ime Key Dis ibu ion ime SOP Measu emen +m In e media e Ro a ions TOTRTR (b) ML-based adap i e SOP o a ions QBER ime Excessi e QBER Excessi e QBER Fig. 1. Reac i e (a) and ML-based adap i e (b) SOP o a ion. s1 s2 s3 QTx HQV R Si e A |q( )〉 Si e B Key Dis illa ion Engine Key Dis illa ion Engine Quan um channel Public channel QRx Elec onic Pola iza ion Con olle (EPC) Pola iza ion Beam Spli e (PBS) ML-based SOP T acking and Ro a ion Manage QBER h ( )dis ance( ,|o〉) |o〉 |o( )〉 Fig. 2. Sys em a chi ec u e. ime θo( ) θp( )(b) # o a ions ime # o a ions ime (a) (b) (c) unnecessa y o a ion excessi e QBER high QBER e y high QBER 28 o a ions 28 o a ions Fig. 3. Example o ope a ion (a) and pe o mance o he eac i e (b) and ML-based adap i e (c) SOP o a ion. Oppo uni ies and p oposed solu ions Fo illus a i e pu poses, Fig. 1a shows he ope a ion o he quan um channel wi h ime based on he app oach p oposed in [6]. A egula ime in e als o size m, he QTx sends a numbe o qubi s wi h a p ede ined pola iza ion ha a e used o moni o he cu en SOP, deno ed |o( )〉, a he QRx. Based on he measu ed SOP, he QRx compu es he needed o a ion (deno ed ( )) o compensa e he pola iza ion d i . Once he o a ion is pe o med, he quan um communica ion sys em exchanges pola iza ion-encoded keys. I he alue o m is la ge enough compa ed o he ime o moni o ing (TO) and o a ion (TR), his scheme in oduces a small o e head, while allows o eac quickly o changes in he SOP. Fig. 1a also includes a possible e olu ion o he QBER om one o a ion o he nex . In he p esence o SOP luc ua ions, i migh happen ha he o a ion pe o med a he s a ing o a pe iod does no allow o keep he QBER unde a desi ed h eshold (deno ed QBER h), e.g., 1%, un il he nex pola iza ion s a e is measu ed, and a new o a ion is pe o med. A possible solu ion o deal wi h scena ios wi h la ge SOP luc ua ions would be o educe m, which would esul in a highe sys em o e head, especially du ing he ime when luc ua ions a e small o negligible. Fo ha , m can be de ined dynamically, which would en ail a way o synch onize QTx and QRx eal- ime. In iew o his, we p opose an app oach o ack SOP luc ua ions and apply ML o p edic he nex pola iza ion s a es based on such acking. Then, o a ions can be planned o be pe o med a any in e media e ime om one SOP measu emen o he nex ; he numbe o o a ions would a y om none o se e al, so he ob ained QBER is always unde QBER h (Fig. 1b). Because o a ions can be planned o be pe o med a in e media e imes, accu a e es ima ion o u u e s a es is o pa amoun impo ance o he p oposed sys em. A med wi h such p edic i e ool, an op imiza ion 3 p oblem can be sol ed o decide no only when o pe o m he o a ions, bu also he alue o each o a ion o minimize he numbe o o al o a ions ha a e pe o med; his would esul in o a educed o e head, while assu ing a con ained QBER. In he example o QBER e olu ion in Fig. 1b, no ini ial o a ion is needed, as QBER was ini ially low, whe eas wo o a ions a e pe o med a in e media e imes. In pa icula , he i s o a ion is pe o med o compensa e SOP a a u u e s a e, as e ealed by he e olu ion o he QBER ha p og essi ely educes un il a minimum and inc eases again eaching a alue close o QBER h be o e he second o a ion is pe o med. Fig. 2 shows a schema ic iew o a quan um communica ion channel es ablished be ween emo e si es A and B. Wi hou assuming any speci ic pola iza ion based QTx implemen a ion, le us conside ha a qubi is gene a ed by andomly selec ing one linea pola iza ion (poin s H, V, R, and Q on he sphe e a si e A in Fig. 2). Then, he pe ec ly pola ized pho on is sen o he QRx. When he pho ons a e ecei ed and measu ed a he QRx side, he SOP posi ion migh ha e d i ed. Fig. 2 ep oduces he EPC and PBS modules in he QRx based on he a chi ec u e p oposed in [6]. The ob ained QBER will be below QBER h i he s a e o he ecei ed pho ons is wi hin an a ea cen e ed in he cu en e e ence pola iza ion s a e wi h adius d h. When he e e ence pola iza ion s a e o he QRx is o a ed, he a ea o ole able QBER h also mo es co e ing a di e en egion. In he p oposed sys em, a ML- based module is in cha ge o acking SOP and deciding he o a ions o be pe o med, as illus a ed in Fig. 2. An illus a i e example o he ope a ion is p esen ed in Fig. 3. Fig. 3a shows he e olu ion pola iza ion angle θ o he eal pho ons s a e |p( )〉 and measu ed s a e |o( )〉, bo h a he QRx. In addi ion, linea (polynomial o deg ee 1) in e pola ion connec ing wo measu ed pola iza ion s a es is ep esen ed. No e ha al hough linea in e pola ion is used o he sake o simplici y in he d awing, highe deg ees can be used. In Fig. 3b-c, he o a ions ha a e pe o med unde he eac i e and adap i e app oaches a e shown. We assume he e he same pe iod m o bo h app oaches. In he eac i e app oach (Fig. 3b), one single o a ion is pe o med once he cu en s a e |o〉 is measu ed a e TO, which esul s in o 28 o a ions o he sample in Fig. 3a. Howe e , as many as 15 o he o a ions a e unnecessa y, because a he ime hey a e pe o med, he measu ed pola iza ion s a e is wi hin he a ea o low QBER. On he con a y, he e a e 4 pe iods wi h high and e y high QBER, due o la ge SOP luc ua ions in hose pe iods. In con as , he p oposed ML-based SOP acking and o a ion planning app oach, is able o achie e low QBER e en du ing la ge SOP luc ua ions (Fig. 3c), due o i s abili y o p edic u u e pola iza ion s a es and plan he needed o a ions. No e ha he o al numbe o o a ions unde he ML-based app oach is equi alen (i can be e en lowe ) o he eac i e app oach, which ensu es high e iciency. Tha ac , combined o he educed QBER, esul s in as e KER. 3. ML-based SOP T acking and Ro a ion Manage In his sec ion, we i s p esen he p ocedu e used o measu e and p edic he e olu ion o pho ons’ pola iza ion s a e based on he combina ion o he quan um s a e omog aphy heo y and DNN models. Nex , he p ocedu e o plan he sequence o Poinca é sphe e o a ions ha needs o be ca ied ou o achie e accu a e pola iza ion alignmen based on he SOP p edic ion is desc ibed. SOP moni o ing and p edic ion As in oduced in he p e ious sec ion, SOP can be a ec ed by pe u ba ions on he ibe , du ing he moni o ing pe iod s a ing a ime , he QTx sends a numbe o pho ons wi h a known pola iza ion and he QRx measu es hem in di e en axes o accu a ely es ima e he cu en s a e |o( )〉, de ined by he uple <θo( ), φo( )>. Speci ically, he QTx gene a es n pho ons wi h H pola iza ion (i.e., <B,b> = <0,0>), which a e p opaga ed h ough he quan um channel. A he QRx side, he ecei ed pho ons a e sepa a ed in h ee di e en chunks o n/3 pho ons, one o each o he h ee axes X, Y, and Z measu emen s. The decoded bi s can con ain some 1’s due o he combina ion o he selec ed axes o measu emen , he luc ua ions o he SOP du ing p opaga ion, and he cu en o a ion con igu a ion in he EPC. Then, we de ine he QBER o a chunk as he sum o he ex ac ed bi s (numbe o e oneous bi s) o e he leng h o he chunk (n/3). A e ansmi ing and decoding all n pho ons, measu emen esul s a e a ailable o each axis, i.e., QBER( ) = {X, Y, Z}. The measu emen along he Z axis is enough o compu e θ( ), whe eas φ( ) equi es om measu emen s along X and Y axes o es ima e sine and cosine o φ( ), espec i ely. Once he cu en pola iza ion s a e |o( )〉 is es ima ed, i is used o p edic he SOP e olu ion un il he nex moni o ing pe iod. Cu en ly es ima ed s a e |o( )〉 and he se o pas pola iza ion s a e es ima ions along wi h he DNN model used o SOP p edic ion. The objec i e is o gene a e sequence O con aining he cu en es ima ed s a e |o( )〉 and he p edic ion o he nex k consecu i e and e enly dis ibu ed pola iza ion s a es connec ing |o( )〉 and he expec ed one o he nex moni o ing pe iod, i.e., |o( +m)〉. Sequence O is de e mined by using DNN-based o ecas ing and polynomial i ing sequen ially. The DNN is used o accu a ely o ecas a disc e e ime-dependen e en ahead in ime, whe eas polynomial is used o in e pola e unknown pola iza ion s a es be ween known s a es. The p ocedu e is as ollows; he las es ima ed pola iza ion s a e is s o ed in he SOP da abase and he las es ima ed pola iza ion s a es wi hin he p e ious ime window w a e e ie ed ha a e used o eed a DNN model ha p edic s |o( +m)〉. The DNN has 2·⎿w/m⏌ inpu s ( o angles θ and φ o hose las SOP alues), se e al hidden laye s using he anh ac i a ion unc ion, and wo ou pu s o angles θ and φ o p edic ed s a e |o( +m)〉. Nex , he las w es ima ed pola iza ion s a es oge he wi h he p edic ed |o( +m)〉 a e used o in e pola e a polynomial-based model g. To inc ease he accu acy o he in e pola ion p ocedu e, g is a compound model wi h ou l-deg ee polynomials used o es ima e sin(θ), cos(θ), sin(φ), and cos(φ) as a unc ion o ime in he ange [ , +m]. Finally, g is used o ob ain k p edic ions be ween |o( )〉 and |o( +m)〉. Ro a ion plan compu a ion based on SOP p edic ion A e he SOP p edic ion phase, he p oblem o inding which o a ions need o be applied wi hin he ime in e al [ , +m] is sol ed. This p oblem can be modeled as an op imiza ion p oblem and s a ed as ollows: Gi en: •The sequence O o p edic ed s a es, each o a ela i e ime i∈[0, m] and de ined as O(i) = <θo(i), φo(i)>. •The se o candida e o a ions R, whe e e e y o a ion is de ined by <θ , φ >. R includes he o a ion 0 cu en ly con igu ed in he EPC. •A ci cula a ea o adius dmax [ ad] de ined o a a ge QBER and hus, de e mining he need o o a ions. A candida e o a ion ∈R ha becomes ac i e a ela i e ime j is alid o s a e p edic ions |o〉∈O | i≥j i and only i dis ance( , |o〉) ≤ dmax. Ou pu : The o a ions plan P = [< , i>], whe e e e y elemen de ines he ela i e ime i∈[0, m] when candida e o a ion ∈R needs o be con igu ed in he EPC. Objec i e: minimize he numbe o o a ions o be pe o med. To educe he complexi y o he o a ion plan p oblem, we conside ha se R includes he cu en o a ion 0 and all p edic ed pola iza ion s a es in O. The e o e, a i ial easible solu ion would consis in pe o ming k o a ions, one o each p edic ed s a e. To e icien ly sol e he o a ion plan op imiza ion p oblem, we designed he as de e minis ic g eedy algo i hm (Algo i hm I). A p e-compu a ion phase is un o ind he subse o p edic ed pola iza ion s a es ha can be se ed om each candida e o a ion. Then, an i e a i e p ocedu e is execu ed o build he plan (sequence) o o a ions un il all pola iza ion s a es a e assigned o, a leas , one o he selec ed o a ions. A e e y i e a ion, he g eedy cos o e e y o a ion is compu ed. Such cos is de ined as a weigh ed sum o h ee componen s, wi h weigh s β1 >> β2 >> 1. The h ee componen s accoun : i) whe he he o a ion co e s e e ence pola iza ion s a e |o e 〉, which is ini ialized wi h he measu ed pola iza ion s a e and upda ed wi h he las s a e co e ed by he o a ion when a new o a ion is pe o med. This componen ies o os e selec ing new o a ions ha o e lap wi h he p e ious one, which o ces building he plan as a sequence ha acks he e olu ion o O; ii) whe he he o a ion is he cu en ly ac i e one o no , so as o educe he numbe o o a ions; and iii) he numbe o pola iza ion s a es co e ed by he candida e o a ion. The candida e o a ion wi h he highes g eedy cos is selec ed and added o he incumben solu ion. Then, he ela i e ime o pe o m he nex o a ion is compu ed and he se o co e ed pola iza ion s a es Oin and e e ence s a e |o e 〉 a e upda ed. Finally, he o a ion plan is e u ned. Algo i hm I. Heu is ic o he Ro a ion Plan P oblem INPUT: O, R, dmax OUTPUT: P 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: P  {}; i  0; Oin  {}; |o e 〉  O[0] o ∈ R do o |o〉 ∈ O do i dis ance( , |o〉) > dmax hen con inue .O.append(|o〉) while Oin <> O do o each ∈ R do i |o e 〉 ∈ .O hen x1  1 else x1  0 i = 0 hen x2  1 else x2  0 x3  | .O| .cos  β1·x1 + β2·x2 + x3 ’  a gmax( .cos ∀ ∈R) P  P U < ’, i> Oin  Oin U ’.O |o e 〉  ’.O[-1] i  |o e 〉.i e u n P 4. CONCLUSION A ML-based SOP acking and pola iza ion compensa o has been p esen ed ha migh signi ican ly educe he cos o pola iza ion encoded QKD sys ems by simpli ying he speci ica ions o quan um ansmi e and ecei e and enabling he use o ae ial op ical ibe cables. The p oposed sys em is based on h ee main componen s: i) a SOP moni o ing p ocedu e able o p ecisely es ima e he cu en pola iza ion s a e while minimizing o e head; ii) a ligh weigh ML-based SOP p edic ion ha is able o accu a ely o ecas u u e SOP e olu ion wi h ine g anula i y; iii) a Poinca é sphe e o a ion planne , which decides when o a ions need o be pe o med and he magni ude o such o a ions o compensa e pola iza ion d i and keep QBER unde a gi en h eshold. ACKNOWLEDGEMENTS The esea ch leading o hese esul s has ecei ed unding om he Eu opean Commission HORIZON ALLEGRO (G.A. 101092766) and he AEI IBON (PID2020-114135RB-I00) p ojec s and om he ICREA ins i u ion. REFERENCES [1] M. Khan e al., “Analysis o achie able dis ances o BB84 and KMB09 QKD p o ocols,” Quan um In o, ol. 18, 2020. [2] R. Liu e al., “Analysis o pola iza ion luc ua ion in long-dis ance ae ial ibe o QKD sys em design,” OFT, 2019. [3] B. 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