It is known that a quantum computer operating on electron-spin qubits with
single-electron Hamiltonians and assisted by single-spin measurements can be
simulated efficiently on a classical computer. We show that the exponential
speed-up of quantum algorithms is restored if single-charge measurements are
added. These enable the construction of a CNOT (controlled NOT) gate for free
fermions, using only beam splitters and spin rotations. The gate is nearly
deterministic if the charge detector counts the number of electrons in a mode,
and fully deterministic if it only measures the parity of that number.Comment: 5 pages including 3 figure

C. Emary

C. W. J. Beenakker

D. P. DiVincenzo

J. M. Elzerman

M. Kindermann

M. A. Nielsen

English

arXiv

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Beenakker, C.W.J.

Vincenzo, D.P. di

Emary, C.

Kindermann, M.

Leiden University Scholary Publications

VOLUME 93, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R Sweek ending9 JULY 2004Charge Detection Enables Free-Electron Quantum ComputationC W J Beenakkei,1 D P DiVincenzo,2 3 C Emaiy,' and M Kindeimann4llnslituut Loientz, Umveisiteit Leiden PO Box 9506 2300 RA Leiden The Netheilands~Depai tinenl of Nanoscience Delft Univeifil) of Technolog}, Lo>ent:\ieg I 2628 CJ Delft The NetheilandsInstitute foi Theoietical Physics Valckenieistiaat 65 1018 XE Amsteidam The NetheilandsDepai tment oj P h y s i c s Massachusetts Institute of Technology Cambndqe Massachusetts 02139 USA(Received 19 Febiuaiy 2004, published 6 July 2004)It is known that a quanlum computei opeiating on elecüon spin qubits with single electionHamillonians and assisted by smgle-spm measuiemenls can be simulated efficiently on a classicalcomputei We show lhal the cxponential speedup of quantum algonthms is lestoied if single-chaigemeasuiements aic added These enable the constuiction of a CNOT (conliollecl NOT) gale foi fieefeimions, usmg only beam sphtleis and spin lotations The gale is neaily deleimmistic if the chaigedetectoi counls the numbei of election s in a mode, and fully deteimmistic if it only measuies the pant}of that numbeiDOI 10 1103/PhysRevLett 93 020501 PACS numbei s 03 67 Lx 03 67 Pp 05 30 Fk 7110-wFlytng qubits tianspoit quantum mfoimation betweendistant memoiy nodes and fotm an essential mgiedient ofa scalable quantum computei [1] Flymg qubits could bephotons [2], but usmg conduction elections in the solidstate foi this puipose lemoves the need to conveit mate-nal qubits to mdiation Since the Coulomb inteiactionbetween fiee elections is stiongly scieened, an mteiac-tion-fiee mechanism foi logical opeiations on elecüomcflymg qubits could be desuable The seaich foi such amechanism is stiongly constiamed by a no-go theoiem[3,4], which states that the exponential speedup of quan-tum ovei classical algoiithms cannot be leached withsingle-election Hamiltomans assisted by single-spinmeasuiements Heie we show that the fü l l powei oi quan-tum computation is lestoied if single-chaige measuie-ments aie added These enable the constiuction of aCNOT (contiolled NOT) gate foi fiee feimions, usmgonly beam sphtteis and spin lotationsThe no-go theoiem [3,4] apphes only to feimions, notto bosons Indeed, in an influential papei [2], Knill,Laflamme, and Milbuin showed that the exponentialspeedup ovei a classical algonthm affoided by quantummechanics can be leached usmg only lineai optics withsingle-photon detectois The detectois inteiact with thequbits, pioviding the nonhneanty needed foi the compu-tation, but qubil-qubil inteiactions (eg , nonlineai opticalelements) aie not jequiied in the bosonic case Thisdiffeience between bosons and feimions explams whythe topic ot "fiee-election quantum computation"(FEQC) is absent in the liteiatuie, in contiast to the activetopic oi "lineai optics quantum computation" (LOQC)[5-12] Heie we would hke to open up the ioimei topic,by demonstiatmg how the constiamt on the efficiency oiquantum a lgo i i thms foi fiee ieimions can be lemovedWe accomphsh this by usmg the iacl that the electionc a i i y m g the qubit in its spin degieeol iieedom has also achaige degiee of fieedom Spin and chaige commute, so ameasuiement of the chaige leaves the spin qubit unaf-fected To measuie the chaige the qubit should inteiactwith a detectoi, but no qubit-qubit inteiactions aieneededChaige detectois play a piormnent lole in a vanety ofcontexts äs which-path detectois they contiol the visibil-ity of Ahaionov-Bohm oscillations [13], in combinationwith a beam sphttei they piovide a way to entangle twononinteiacting paiticles [14], in combination with spin-dependent tunnelmg they enable the leadout oi a spinqubit [15,16] The expenmental leahzation uses the effectof the electuc held of the chaige on the conductance of aneaiby point contact [17] The effect is weak, because ofscieening, but measuiable if the point contact is neaienough Such a device functions äs an electwmetei Itcan count the occupation numbei of a spatial mode (0, l,01 2 elections with opposite spin) If the point contact isleplaced by a quantum dot with a lesonant conductance,then it is possible to opeiate the device äs a pai /f) meteiIt can distinguish occupation numbei one (when it is onlesonance) fiom occupation numbei 0 01 two (when it isoff lesonance)—but it cannot distinguish between 0and 2 We will considei both types of chaige detectoisin what followsThe geneial foimulation of feimionic quantum compu-tation [18] is m teims of local modes which can be eitheiempty 01 occupied The annihilat ion opeiatoi of a localmode is an , with spatial mode index / = l, 2 3, andspin index s =T l Foi noninteiacting feimions theHamiltoman is bihneai in the cieation and annihi lat ionopeiatoi s A local measuiement m the computationalbasis has piojection opeiatois n,^ = a„«„ and l — n,s =i/ ( iiz ( i Teihal and one of the authois [3] showed that thepiobability of the outcome oi any sei of such local mea-suiements is the squaie loot ot a deteimmant Smce a020501-l 0031-9007/04/93(2)/020501(4)$2250 © 2004 The Amencan Physical Society 020501-1VOLUME 93, NUMBER 2 P H Y S I C A L R E V I E W L E I T E R Sweek ending9 JULY 2004detenmnant of oidei N can be evaluated in a time whichscales polynomially with N, the quantum algoi i thm canbe simulated efficiently on a classical computei This isthe no-go theoiem mentioned in the intioductionWe now add measuiements of the local chaige Q, =«,l + 7i,j to the algoiithm The eigenvalues of Ql aie0, l, 2 The piobability that chaige one is measuied isgiven by the expectation value of the piojection opeiatoiP, = l - (l - Q,)2 = 4 + α\αάαι\α\ Ο)The opeiatoi Pt is the sum of two local opeiatois in thecomputational basis The piobability that M spatialmodes aie smgly occupied theiefoie consists of a sumof an exponentially large numbei (2M) of deteimmants, sonow a classical Simulation need no longei scale polyno-mially with the numbei of modes Notice that a measuie-ment of Q, contams less infoimation about the state thansepaiate measuiements of n,j and n^ The fact that paiüalmeasuiements can add computational powei is a basicpimciple of quantum algonthms [1]Lei us now see how these formal consideiations couldbe implemented, by constiuctmg a CNOTgate usmg onlybeam splitteis, spin lotations, and chaige detectois Toconstiuct the gate we need one of two new buildmgblocks that aie enabled by chaige detectois The fiistbuildmg block is the Bell-state analyzei shown m Fig lFoi this device it does not mattei whethei the chaigedetectoi opeiates äs an electiometei 01 äs a panty meteiThe second buildmg block, shown in Fig 2, conveits achaige panty measuiement to a spinpaii ty measuiementWe piesent each device in tuin and then show how toconstiuct the CNOTgateThe Bell-state analyzei makes it possible to telepoit[19] the spin state a\ T) + ß\ 1) of election A to anotheiFIG l Bell state analyzei foi noninleiactmg elections, con-sistmg öl thiee 50/50 bcam sphlleis (dashed honzonlal hnes),loui mmois (solid hoiizona] lines), two local spin lotations(Pauh malnces <r, and σ,), and thiee chaige detectois(squaies) The chaige detectois may opciate eilhei äs electio-metei s (counting the occupation q, — 0 l 2 in an aim) 01 äspanty meteis (measming p, = q, mod2) The fiisl chaige de-tectoi can identify the spin smglet slale hlO), which is the onlyone oi the ioui Bell states (2)-(4) lo show (p, = 0) Smcc (l ®er )Ι λΙΛι) = — l^o) tne second chaige dctecloi can identify|Ψι) w h e u p2 = 0 F i n a l l y smce (l <8> σ,σ )|Ψ2) = |Φ()) thet h n d chaige detectoi can i d e n t i f y the two l e m a m i n g stateshP2) (whcn /?·, = 0) and |Ψ·,} (when p, = 1)election A', usmg a thnd election B that is entangled withA' The telepoitation is peifoimed by measunng thejomtstate of A and B in the Bell basis(2)= (i m|Ψ3> = (l ίί> - l ü»M(3)(4)(5)A no-go theoiem [20,21] says that such a Bell measuie-ment cannot be done deteimimstically (meanmg with100% success piobability) without usmg inteiactions be-tween the qubits Howevei, it has been noted that thistheoiem does not apply to qubits that possess an addi-tional degiee of fieedom [22], and that is how we willwoik aiound itIn Fig l we show how a deteimimstic Bell measuie-ment foi feimions can be peifoimed usmg thiee 50/50beam sphtteis, thiee chaige detectois, and two local spinlotations (lepiesented by Pauh matuces σ, and σ ) Thebeam sphttei scatteis two elections into the same aim(bunching) if they aie in the smglet state (2), and into twodiffeient a ims (antibunching) if they aie in one of thetnplet states (3)-(5) (This can be easily undeistood [23]fiom the ant isymmetiy of the wave function undei pai-ticle exchange, demanded by the Pauh pimciple Thesmglet state is antisymmetnc in the spin degiee of fiee-dom, so the spatial pa i t of the wave function should beFIG 2 Gate that conveits a chaige panty measuiement to aspin panty measuiement The shaded box at the nght lepiesentsthe cncint shown at the lefl A pan of elections is incident inaims a and b A po la i izmg beam sphttei (double dashed l ine)liansmils spin up and leflccts spin down A chaige detectoilecoids bunching (p = 0) 01 antibunching (p = 1) and passesthe elections on to a second pola i izmg beam sphttei If eachelection at the mput is in a spin eigenslate l T) οι | |), thenOutput equals mput and p measiues the spin panty (p = l ifthe two spms aie a l igned, and p = 0 i f they aie opposite) Thegate can be used to encode a qubit [ T) us the two pailicle statel T)l T) and l I) äs | |)| |) Foi that puipose the mput consists ofthe qubi t lo bc encoded in a im a plus an anci l la in aim b in thestate (| f) + | |))/\/2 The Output is the lequnecl two-paiticlestate in a ims t and cl Ιοί p = 1 Foi p = 0 i t becomes thei c q u i i c d state aftei a s p i n - f l i p (fr,) opeiation on the electionin a i m d020501-2 020501-2VOLUME 93, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R Sweek ending9 JULY 2004symmetiic, and vice veisa foi the tnplet state) Lei p, bethe chaige ql measuied by detectoi i, mod2 So p, = 0means bunchmg and /?, = l means antibunching afteibeam sphttei ι The quantitycontrol m control out= P\ + P\PI (6)takes on the value 0, l, 2, 01 3 dependmg on whethei theincident state is |Ψ0), |Ψ,), |Ψ2). O1 1^3 X lespectivelyThe measuiement of 2> is theiefoie the lequned piojectivemeasuiement in the Bell basis It is a destiuctive mea-suiement, so it does not mattei whethei the chaige de-tectoi opeiates äs an electiometei (measunng qt) 01 äs apanty metei (measuimg p,)In Fig 2 we show how a chaige detectoi opeiating äs apanty metei can be used to measuie in a nondestiuc-tive way whethei two spms aie the same 01 opposite"Nondestiuctive" means without measunng whetheithe spm is up 01 down The device consists of two polai-izmg beam splitteis in seiies, with the chaige detectoiin between (A polaiizmg beam sphttei fully tiansmits Tand fully leflects | ) At the mput two elections aieincident m diffeient aims Input equals Output if eachelection is m a spm eigenstate The measuied chaigepanty then lecoids whethei the two spms aie the same01 opposite We will lefei to this device äs an encodei,because it can deteimimstically entangle a qubit in theaibi t iaiy state a\ T) + ß\ I) and an ancilla in the fixedstate (| T) + | Ι))/λ/2 into the two-paiticle entangled statea\ 1)1 ί) + ß\ 1)1 I)To constiuct a CNOTgate usmg the Bell-state analyzeiwe follow Ref [2], wheie it was shown that telepoitationcan be used to conveit a piobabilistic logical gate mto aneaily deteiministic one It is well known that a pioba-bihstic CNOTgate can be constiucted fiom beam sphtteisand single-qubit opeiations The design of Pittman et al[7] has success piobability | and woiks foi feimions äswell äs bosons It consumes an entangled pan of ancillas,which can be cieated piobabihstically usmg a beamsphttei and chaige detectoi [14] Because the gate is notdeteiministic, it cannot be used in a scalable way msidethe computation Howevei, the CNOT gate can be lepeat-edly executed off-hne, independent of the piogiess of thequantum algoiithm, until it has succeeded Two Bellmeasuiements telepoit the CNOTopeiation into the com-putation [24], when needed In this way a quantum algo-i i thm can be executed usmg only single-paiticleHamiltonians and single-paiticle measuiementsIn Fig 3 we show how to constiuct a CNOT gateusmg the encodei Oui design was inspned by that ofPittman et al [7], but lathei than being piobabilistic it isexactly deteiministic We take two encodei s in senes,wilh a change of basis on gomg tiom the fnst to thesecond encodei The change of basis is the Hadamaidtiansioimationancilla out(measured)target m target outFIG 3 Deteiministic CNOTgate foi noninteiacting electionsEach shaded box conlams a pan of polaiizmg beam sphtteisand a chaige detecloi, äs descnbed in Fig 2 The fouiHadamaid gates H = (<rv + σ )/V2 lotate the spms enteiingand leaving the second box The mput of the CNOTgate consistsof the contiol and taiget qubits plus an ancilla m the state(l T) + l i))/V2 The spm of the ancilla is measuied at theoutpuL The outcome of that measuiement togethei with thetwo paiities pt p-, measuied by the chaige detectois deteimmewhich opeialions ac σ, one has lo apply to contiol and taiget atthe Output in oidei to complele the CNOT opeiation Foi thecontiol, ac — er, li p2 = 0 while σι — l if p->_ = l Foi thetaiget, σ, = σ ν i f Ine anci l la is down and pt = l, 01 iithe ancilla is up and p\ = 0 Otheiwise, σ, = l The calculalion is given m Ref [30]l i) - (l I) + l !»/V2, 11)-(l i)- I))/V2 (7)The CNOTopeiation flips the spm of the taiget qubit i f thespm of the contiol qubit is | Contiol and taiget aie mputinto sepaiate encodeis The ancilla of the encodei foi thecontiol is fed back into the encodei foi the taiget Atthe Output, the spm of the ancilla is measuied Con-ditioned on the outcome of that measuiement and on thetwo paiities measuied by the encodeis, a Pauh mati ix hasto be applied to contiol and taiget to complete the CNOTopeiationThe computational powei of the panty detectois islemaikable The CNOT gate of Fig 3 lequnes a singleancilla to achieve a 100% success piobability, while theoptimal design of LOQC needs n ancillas m a speciallypiepaied entangled state foi a l — l/«2 success piobabil-ity [8] In this lespect it would seem that FEQC iscomputationally moie poweiful than LOQC, but we em-phasize that Fig 3 applies to bosons äs well asieimions Ifpanty detectois could be leahzed foi photons (and theieexist pioposals in the hteiatuie [6]), then the design ofFig 3 would diamatically simplify existing Scheinesfoi LOQCIn conclusion, we have shown that fiee-election quanturn computation (FEQC) is possible in pnnciple eitheineaily deteimimstically (usmg a Bell-state analyzei with020501-3 020501-3VOLUME 93, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R Swcek ending9 JULY 2004a chaige detectoi opeiatmg äs an electiometei) 01 exactlydeteimmistically (usmg an encodei with a chaige detec-toi opeiatmg äs a panty metei) Unlikephotons, electionsinteiact stiongly if biought close togethei, so theie is noneed to iely exclusively on single-paiticle HamiltoniansWe expect that FEQC would ultimately be used foi flymgqubits [25], while othei gate designs based on shoit-iangeinteiactions [15,26] would be piefened foi stationaiyqubitsThe two mgiedients of the cucuits consideied heie,beam sphtteis [27,28] and chaige detectors [13,16,17],have both been leahzed by means of pomt contacts in atwo-dimensional election gas The time-iesolved detec-tion lequned toi the opeiation äs a logical gate has not yetbeen leahzed The cunently achievable time lesolutionfoi chaige detection is μ& [16], while the lesolutionlequned foi balhstic elections in a semiconductoi is inthe ps lange That time scale is not maccessible [29], but itmight not be possible to leach the lequned single-electionsensitivity due to the unavoidable shot noise in the chaigedetectoi In the light of this, is could be moie piactical tostait with isolated elections in an anay of quantum dots,lathei than with flymg qubits, in oidei to investigate thepotential and hmitations of oui theoietical concept on apiesently accessible time scaleWe have benefitted fiom discussions with B M TeihalThis woik was suppoited by the Dutch ScienceFoundation NWO/FOM, by the US A i m y Re-seaich Office (Giants No DAAD 19-02-0086 andNo DAAD 19-01-C-0056), and by the Cambndge-MIT Institute, Ltd'Peimanent addiess IBM, T J Watson Reseaich Centei,PO Box 218, Yoiklown Heights, NY 10598, USA[1] M A Nielsen and I L Chuang, Quantum Computaiionand Quantum Infoimation (Cambudge Umveisity,Cambudge, 2000)E Knil l , R Laflamme, and G J Milburn, Natuic(London) 409, 46 (2001)[3] B M Teihal and D P DiVincenzo, Phys Rev A 65,032325 (2002)[4] E K n i l l , quant-ph/0108033[5] M Koashi, T Yamamolo, and N Imoto, Phys Rev A 63,030301 (2001)[2;[6] D Gottesman, A Kitacv, and J Piesk i l l , Phy« Rev A64, 012310 (2001)[7J T B Pit tman, B C Jacobs, and J D Fianson, Phys RevA 64, 062311 (2001)[8] T D Fianson, M M Donegan, M J Fitch, B C Jacobs,and T B Pittman, Phys Rev Leu 89, 137901 (2002)[9] H F Hol mann and S Takeuchi, Phys Rev A 66, 024308(2002)[10] T C Ralph, N K Langfoid, T B Bell, and A G White ,Phys Rev A 65, 062324 (2002)[11] J L Dodd,TC Ralph, and G J Milbum, Phys Rev A68, 042328 (2003)[12] J D Fianson, B C Jacobs, and T B Pittman, quant-ph/0401133[13] E Buks, R Schuslei, M Heiblum, D Mahalu, andV Umansky, Natuie (London) 391, 871 (1998)[14] S Böse and D Home, Phys Rev Lett 88, 050401 (2002)[15] D Loss and D P DiVincenzo, Phys Rev A 57, 120(1998)[16] J M Elzeiman, R Hanson, L H Willems van Beveien,L M K Vandeisypen, and L P Kouwenhoven, ApplPhys Lett 84, 4617 (2004)[17] M Field, C G Smith, M Peppei, D A Ritchie, J E FFiosl, G A C Jones, and D G Hasko, Phys Rev Lelt70, 1311 (1993)[18] S B Biavyi and A Yu Kilaev, quant-ph/0003137[19] C H Bennctt, G Biassaid, C Ciepeau, R Jozsa,A Pcies, and W K Wootleis, Phys Rev Lett 70, 1895(1993)[20] L Vaidman and N Yoian, Phys Rev A 59, 116 (1999)[21] N Lutkenhaus, J Calsamiglia, and K - 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Charge detection enables free-electron quantum computation

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