Charge detection enables free-electron quantum computation

Beenakker, C.W.J.; Vincenzo, D.P. di; Emary, C.; Kindermann, M.

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Beenakker, C. W. J., Vincenzo, D. P. di, Emary, C., & Kindermann, M. (2004). Charge

detection enables free-electron quantum computation. Retrieved from

https://hdl.handle.net/1887/1294

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VOLUME 93, NUMBER 2

P H Y S I C A L R E V I E W L E T T E R S

9 JULY 2004week ending

Charge Detection Enables Free-Electron Quantum Computation

C W J Beenakkei,

1

D P DiVincenzo,

2 3

C Emaiy,' and M Kindeimann

4

llnslituut 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 Netheilands Institute foi Theoietical Physics Valckenieistiaat 65 1018 XE Amsteidam The Netheilands Depai 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 election Hamillonians and assisted by smgle-spm measuiemenls can be simulated efficiently on a classical computei We show lhal the cxponential speedup of quantum algonthms is lestoied if single-chaige measuiements aic added These enable the constuiction of a CNOT (conliollecl NOT) gale foi fiee feimions, usmg only beam sphtleis and spin lotations The gale is neaily deleimmistic if the chaige detectoi counls the numbei of election s in a mode, and fully deteimmistic if it only measuies the pant} of that numbei

DOI 10 1103/PhysRevLett 93 020501 PACS numbei s 03 67 Lx 03 67 Pp 05 30 Fk 7110-w

Flytng qubits tianspoit quantum mfoimation between

distant memoiy nodes and fotm an essential mgiedient of

a scalable quantum computei [1] Flymg qubits could be

photons [2], but usmg conduction elections in the solid

state foi this puipose lemoves the need to conveit

mate-nal qubits to mdiation Since the Coulomb inteiaction

between fiee elections is stiongly scieened, an

mteiac-tion-fiee mechanism foi logical opeiations on elecüomc

flymg qubits could be desuable The seaich foi such a

mechanism 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 with

single-election Hamiltomans assisted by single-spin

measuiements Heie we show that the füll powei oi

quan-tum computation is lestoied if single-chaige

measuie-ments aie added These enable the constiuction of a

CNOT (contiolled NOT) gate foi fiee feimions, usmg

only beam sphtteis and spin lotations

The no-go theoiem [3,4] apphes only to feimions, not

to bosons Indeed, in an influential papei [2], Knill,

Laflamme, and Milbuin showed that the exponential

speedup ovei a classical algonthm affoided by quantum

mechanics can be leached usmg only lineai optics with

single-photon detectois The detectois inteiact with the

qubits, pioviding the nonhneanty needed foi the

compu-tation, but qubil-qubil inteiactions (eg , nonlineai optical

elements) aie not jequiied in the bosonic case This

diffeience between bosons and feimions explams why

the topic ot "fiee-election quantum computation"

(FEQC) is absent in the liteiatuie, in contiast to the active

topic 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 oi

quantum a l g o i i t h m s foi fiee ieimions can be lemoved

We accomphsh this by usmg the iacl that the election

c a i i y m g the qubit in its spin degieeol iieedom has also a

chaige degiee of fieedom Spin and chaige commute, so a

measuiement of the chaige leaves the spin qubit

unaf-fected To measuie the chaige the qubit should inteiact

with a detectoi, but no qubit-qubit inteiactions aie

needed

Chaige detectois play a piormnent lole in a vanety of

contexts äs which-path detectois they contiol the

visibil-ity of Ahaionov-Bohm oscillations [13], in combination

with a beam sphttei they piovide a way to entangle two

noninteiacting paiticles [14], in combination with

spin-dependent tunnelmg they enable the leadout oi a spin

qubit [15,16] The expenmental leahzation uses the effect

of the electuc held of the chaige on the conductance of a

neaiby point contact [17] The effect is weak, because of

scieening, but measuiable if the point contact is neai

enough Such a device functions äs an electwmetei It

can count the occupation numbei of a spatial mode (0, l,

01 2 elections with opposite spin) If the point contact is

leplaced by a quantum dot with a lesonant conductance,

then it is possible to opeiate the device äs a pai /f) metei

It can distinguish occupation numbei one (when it is on

lesonance) fiom occupation numbei 0 01 two (when it is

off lesonance)—but it cannot distinguish between 0

and 2 We will considei both types of chaige detectois

in what follows

The geneial foimulation of feimionic quantum

compu-tation [18] is m teims of local modes which can be eithei

empty 01 occupied The annihilation opeiatoi of a local

mode is a

n

, with spatial mode index / = l, 2 3, and

spin index s =T l Foi noninteiacting feimions the

Hamiltoman is bihneai in the cieation and annihilation

opeiatoi s A local measuiement m the computational

basis has piojection opeiatois n,^ = a„«„ and l — n,

s

=

i/

(i

iz

(i

Teihal and one of the authois [3] showed that the

detenmnant of oidei N can be evaluated in a time which scales polynomially with N, the quantum algoiithm can be simulated efficiently on a classical computei This is the no-go theoiem mentioned in the intioduction

We now add measuiements of the local chaige Q, = «,l + 7i,j to the algoiithm The eigenvalues of Ql aie

0, l, 2 The piobability that chaige one is measuied is given by the expectation value of the piojection opeiatoi

P, = l - (l - Q,)2 = ₄

+ α\αάαι\α\ Ο)

The opeiatoi Pt is the sum of two local opeiatois in the

computational basis The piobability that M spatial modes aie smgly occupied theiefoie consists of a sum of an exponentially large numbei (2M) of deteimmants, so

now 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 than sepaiate measuiements of n,j and n^ The fact that paiüal measuiements can add computational powei is a basic pimciple of quantum algonthms [1]

Lei us now see how these formal consideiations could be implemented, by constiuctmg a CNOTgate usmg only beam splitteis, spin lotations, and chaige detectois To constiuct the gate we need one of two new buildmg blocks that aie enabled by chaige detectois The fiist buildmg block is the Bell-state analyzei shown m Fig l Foi this device it does not mattei whethei the chaige detectoi opeiates äs an electiometei 01 äs a panty metei The second buildmg block, shown in Fig 2, conveits a chaige panty measuiement to a spinpaiity measuiement We piesent each device in tuin and then show how to constiuct the CNOTgate

The Bell-state analyzei makes it possible to telepoit [19] the spin state a\ T) + ß\ 1) of election A to anothei

FIG 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 äs panty meteis (measming p, = q, mod2) The fiisl chaige de-tectoi can identify the spin smglet slale hlO), which is the only one oi the ioui Bell states (2)-(4) lo show (p, = 0) Smcc (l ® er )ΙλΙΛι) = — l^o) tne second chaige dctecloi can identify

|Ψι) wheu p2 = 0 Finally smce (l <8> σ,σ )|Ψ2) = |Φ()) the

thnd chaige detectoi can identify the two lemaming states hP2) (whcn /?·, = 0) and |Ψ·,} (when p, = 1)

election A', usmg a thnd election B that is entangled with

A' The telepoitation is peifoimed by measunng thejomt

state of A and B in the Bell basis

(2)

= (i m

|Ψ

3

> = (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 with 100% success piobability) without usmg inteiactions be-tween the qubits Howevei, it has been noted that this theoiem does not apply to qubits that possess an addi-tional degiee of fieedom [22], and that is how we will woik aiound it

In Fig l we show how a deteimimstic Bell measuie-ment foi feimions can be peifoimed usmg thiee 50/50 beam sphtteis, thiee chaige detectois, and two local spin lotations (lepiesented by Pauh matuces σ, and σ ) The beam sphttei scatteis two elections into the same aim (bunching) if they aie in the smglet state (2), and into two diffeient aims (antibunching) if they aie in one of the tnplet states (3)-(5) (This can be easily undeistood [23] fiom the antisymmetiy of the wave function undei pai-ticle exchange, demanded by the Pauh pimciple The smglet state is antisymmetnc in the spin degiee of fiee-dom, so the spatial pait of the wave function should be

FIG 2 Gate that conveits a chaige panty measuiement to a spin panty measuiement The shaded box at the nght lepiesents the cncint shown at the lefl A pan of elections is incident in aims a and b A polaiizmg beam sphttei (double dashed l ine) liansmils spin up and leflccts spin down A chaige detectoi lecoids bunching (p = 0) 01 antibunching (p = 1) and passes the elections on to a second polaiizmg beam sphttei If each election at the mput is in a spin eigenslate l T) οι | |), then Output equals mput and p measiues the spin panty (p = l if the two spms aie aligned, and p = 0 if they aie opposite) The gate can be used to encode a qubit [ T) us the two pailicle state l T)l T) and l I) äs | |)| |) Foi that puipose the mput consists of the qubit lo bc encoded in a i m a plus an ancilla in aim b in the state (| f) + | |))/\/2 The Output is the lequnecl two-paiticle state in a i m s t and cl Ιοί p = 1 Foi p = 0 it becomes the icquiicd state aftei a spin-flip (fr,) opeiation on the election in a i m d

VOLUME 93, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 9 JULY 2004week ending

symmetiic, and vice veisa foi the tnplet state) Lei p, be the chaige ql measuied by detectoi i, mod2 So p, = 0

means bunchmg and /?, = l means antibunching aftei beam sphttei ι The quantity

control m

control out

= P\ + P\PI

(6)

takes on the value 0, l, 2, 01 3 dependmg on whethei the incident state is |Ψ0), |Ψ,), |Ψ2). O1 1^3 X lespectively

The measuiement of 2> is theiefoie the lequned piojective measuiement 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 a

panty metei (measuimg p,)

In Fig 2 we show how a chaige detectoi opeiating äs a panty metei can be used to measuie in a nondestiuc-tive way whethei two spms aie the same 01 opposite

"Nondestiuctive" means without measunng whethei the spm is up 01 down The device consists of two polai-izmg beam splitteis in seiies, with the chaige detectoi in between (A polaiizmg beam sphttei fully tiansmits T and fully leflects | ) At the mput two elections aie incident m diffeient aims Input equals Output if each election is m a spm eigenstate The measuied chaige panty then lecoids whethei the two spms aie the same 01 opposite We will lefei to this device äs an encodei, because it can deteimimstically entangle a qubit in the a i b i t i a i y state a\ T) + ß\ I) and an ancilla in the fixed state (| T) + | Ι))/λ/2 into the two-paiticle entangled state

a\ 1)1 ί) + ß\ 1)1 I)

To constiuct a CNOTgate usmg the Bell-state analyzei we follow Ref [2], wheie it was shown that telepoitation can be used to conveit a piobabilistic logical gate mto a neaily deteiministic one It is well known that a pioba-bihstic CNOTgate can be constiucted fiom beam sphtteis and single-qubit opeiations The design of Pittman et al [7] has success piobability | and woiks foi feimions äs well äs bosons It consumes an entangled pan of ancillas, which can be cieated piobabihstically usmg a beam sphttei and chaige detectoi [14] Because the gate is not deteiministic, it cannot be used in a scalable way mside the computation Howevei, the CNOT gate can be lepeat-edly executed off-hne, independent of the piogiess of the quantum algoiithm, until it has succeeded Two Bell measuiements telepoit the CNOTopeiation into the com-putation [24], when needed In this way a quantum algo-i algo-i t h m can be executed usmg only salgo-ingle-paalgo-italgo-icle Hamiltonians and single-paiticle measuiements

In Fig 3 we show how to constiuct a CNOT gate usmg the encodei Oui design was inspned by that of Pittman et al [7], but lathei than being piobabilistic it is exactly deteiministic We take two encodei s in senes, wilh a change of basis on gomg tiom the fnst to the second encodei The change of basis is the Hadamaid tiansioimation

ancilla out

(measured)

target m

target out

FIG 3 Deteiministic CNOTgate foi noninteiacting elections Each shaded box conlams a pan of polaiizmg beam sphtteis and a chaige detecloi, äs descnbed in Fig 2 The foui Hadamaid gates H = (<rv + σ )/V2 lotate the spms enteiing

and leaving the second box The mput of the CNOTgate consists of the contiol and taiget qubits plus an ancilla m the state (l T) + l i))/V2 The spm of the ancilla is measuied at the outpuL The outcome of that measuiement togethei with the two paiities pt p-, measuied by the chaige detectois deteimme

which opeialions ac σ, one has lo apply to contiol and taiget at

the Output in oidei to complele the CNOT opeiation Foi the contiol, ac — er, li p2 = 0 while σι — l if p->_ = l Foi the

taiget, σ, = σν if Ine ancilla is down and pt = l, 01 ii

the ancilla is up and p\ = 0 Otheiwise, σ, = l The calcu lalion 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 if the spm of the contiol qubit is | Contiol and taiget aie mput into sepaiate encodeis The ancilla of the encodei foi the contiol is fed back into the encodei foi the taiget At the Output, the spm of the ancilla is measuied Con-ditioned on the outcome of that measuiement and on the two paiities measuied by the encodeis, a Pauh matiix has to be applied to contiol and taiget to complete the CNOT opeiation

The computational powei of the panty detectois is lemaikable The CNOT gate of Fig 3 lequnes a single ancilla to achieve a 100% success piobability, while the optimal design of LOQC needs n ancillas m a specially piepaied entangled state foi a l — l/«2 success

piobabil-ity [8] In this lespect it would seem that FEQC is computationally moie poweiful than LOQC, but we em-phasize that Fig 3 applies to bosons äs well asieimions If panty detectois could be leahzed foi photons (and theie exist pioposals in the hteiatuie [6]), then the design of Fig 3 would diamatically simplify existing Scheines foi LOQC

a chaige detectoi opeiatmg äs an electiometei) 01 exactly deteimmistically (usmg an encodei with a chaige detec-toi opeiatmg äs a panty metei) Unlikephotons, elections inteiact stiongly if biought close togethei, so theie is no need to iely exclusively on single-paiticle Hamiltonians We expect that FEQC would ultimately be used foi flymg qubits [25], while othei gate designs based on shoit-iange inteiactions [15,26] would be piefened foi stationaiy qubits

The 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 a two-dimensional election gas The time-iesolved detec-tion lequned toi the opeiadetec-tion äs a logical gate has not yet been leahzed The cunently achievable time lesolution foi chaige detection is μ& [16], while the lesolution lequned foi balhstic elections in a semiconductoi is in the ps lange That time scale is not maccessible [29], but it might not be possible to leach the lequned single-election sensitivity due to the unavoidable shot noise in the chaige detectoi In the light of this, is could be moie piactical to stait with isolated elections in an anay of quantum dots, lathei than with flymg qubits, in oidei to investigate the potential and hmitations of oui theoietical concept on a piesently accessible time scale

We have benefitted fiom discussions with B M Teihal This woik was suppoited by the Dutch Science Foundation NWO/FOM, by the US A i m y Re-seaich Office (Giants No DAAD 19-02-0086 and No 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 Computaiion

and Quantum Infoimation (Cambudge Umveisity,

Cambudge, 2000)

E Knill, 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 Knill, 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 Pieskill, Phy« Rev A 64, 012310 (2001)

[7J T B Pittman, B C Jacobs, and J D Fianson, Phys Rev A 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 A 68, 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, and V 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, Appl Phys Lett 84, 4617 (2004)

[17] M Field, C G Smith, M Peppei, D A Ritchie, J E F Fiosl, G A C Jones, and D G Hasko, Phys Rev Lelt 70, 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 - A Suommcn,

Phys Rev A 59, 3295 (1999)

[22] P G Kwiat and H Wemfuitei, Phys Rev A 58, 2623(R) (1998)

|23] G Buikaid, D Loss, and E V Sukhoiukov, Phys Rev B 61, 16303(R) (2000)

[24] D Gottesman and I L Chuang, Natuie (London) 402, 390 (1999)

[25] A E Popescu and R lonicioiu, cond-mat/0306401 [26] B E Kane, Natuie (London) 393, 133 (1998)

[27] M Henny, S Obeiholzei, C Stiunk, T Hemzel, K Ensslin, M Holland, and C Schonenbeigei, Science 284, 296 (1999)

[28] W D Ohvei, J Kim, R C Liu, and Υ Yamamoto, Science 284, 299 (1999)

[29] E A Shanei and S A Lyon, cond-mat/0308460 [30] C W J Beenakkei, D P DiVincenzo, C Emary, and

M Kindeimann, quanl-ph/0401066 (see appcndix)