The experimental realization of a quantum computer ranks

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1 PUBLISHED ONLINE: 13 FEBRUARY 011 DOI: /NPHYS1915 Non-Aelin sttistics nd topologicl quntum informtion processing in 1D wire networks Json Alice 1 *, Yuvl Oreg, Gil Refel 3, Felix von Oppen 4 nd Mtthew P. A. Fisher 3,5 The synthesis of quntum computer remins n ongoing chllenge in modern physics. Wheres decoherence stymies most pproches, topologicl quntum computtion schemes evde decoherence t the hrdwre level y storing quntum informtion non-loclly. Here we estlish tht key opertion riding of non-aelin nyons cn e implemented using one-dimensionl semiconducting wires. Such wires cn e driven into topologicl phse supporting long-sought prticles known s Mjorn fermions tht cn encode topologicl quits. We show tht in wire networks, Mjorn fermions cn e meningfully rided y simply djusting gte voltges, nd tht they exhiit non-aelin sttistics like vortices in p + ip superconductor. We propose experimentl set-ups tht enle proing of the Mjorn fusion rules nd the efficient exchnge of ritrry numers of Mjorn fermions. This work should open new direction in topologicl quntum computtion tht enefits from physicl trnsprency nd experimentl fesiility. The experimentl reliztion of quntum computer rnks mong the foremost outstnding gols in physics nd hs trditionlly een hmpered y decoherence. In this regrd topologicl quntum computing holds considerle promise, s here one emeds quntum informtion in non-locl, intrinsiclly decoherence-free fshion 1 6. A toy model of spinless, twodimensionl (D) p + ip superconductor nicely illustrtes the key ides. Vortices in such stte ind exotic prticles known s Mjorn fermions, which cost no energy nd therefore generte ground stte degenercy. Becuse of the Mjorns, vortices exhiit non-aelin riding sttistics 7 11 : diticlly exchnging vortices noncommuttively trnsforms the system from one ground stte to nother. Quntum informtion encoded in this ground stte spce cn e controllly mnipulted y riding opertions something the environment finds difficult to chieve. Despite this scheme s elegnce, finding suitle hrdwre poses serious chllenge. Although most effort hs focused on the quntum Hll stte t filling frction 10,1 ν = 5/, numerous relistic lterntive routes to generting non-aelin topologicl phses hve recently ppered Among these, two groups 1, recognized tht one-dimensionl (1D) semiconducting wires cn e engineered, reltively esily, into Kitev s 3 topologicl superconducting stte supporting Mjorn fermions. Motivted y this exciting possiility, we exmine the prospect of exploiting 1D wires for topologicl quntum computtion. The suitility of 1D wires for this purpose is fr from ovious. Mnipulting, riding, nd relizing non-aelin sttistics of Mjorn fermions re ll centrl to topologicl quntum computtion (lthough mesurement-only pproches sidestep the riding requirement 5 ). Wheres Mjorn fermions cn e trnsported, creted, nd fused y gting wire, riding nd non-aelin sttistics pose serious puzzles. Indeed, riding sttistics is ill-defined in 1D ecuse prticles inevitly collide during n exchnge. This prolem cn e surmounted in wire networks, the simplest eing T-junction formed y two perpendiculr wires. Even in such networks, however, non-aelin sttistics does not immeditely follow, s recognized y Wimmer nd collegues 4. For exmple, non-aelin sttistics in D p + ip superconductor is intimtely linked to vortices inding the Mjorns 10,11. We demonstrte tht, despite the sence of vortices, Mjorn fermions in semiconducting wires exhiit non-aelin sttistics nd trnsform under exchnge exctly like vortices in p+ip superconductor. We further propose experimentl setups rnging from miniml circuits (involving one wire nd few gtes) for proing the Mjorn fusion rules, to sclle networks tht permit efficient exchnge of mny Mjorns. The frctionl Josephson effect 13,1 3,5, long with Hssler et l. s recent proposl 6 enle quit redout in this setting. The reltive ese with which Mjorn fermions cn e stilized in 1D wires, comined with the physicl trnsprency of their mnipultion, render these set-ups extremely promising topologicl quntum informtion processing pltforms. Although riding of Mjorns lone does not permit universl quntum computtion 6,7 30, implementtion of these ides would constitute criticl step towrds this ultimte gol. Mjorn fermions in 1D wires We egin y discussing the physics of single wire. Vlule intuition cn e grnered from Kitev s toy model for spinless, p-wve superconducting N -site chin 3 : N H = µ x=1 N 1 c x c x x=1 (tc x c x+1 + e iφ c x c x+1 +h.c.) (1) where c x is spinless fermion opertor nd µ, t > 0, nd e iφ respectively denote the chemicl potentil, tunnelling strength, nd piring potentil. The ulk- nd end-stte structure ecomes prticulrly trnsprent in the specil cse 3 µ = 0, t =. Here it is useful to express c x = 1 e i(φ/) (γ B,x +iγ A,x ) () with γ α,x = γ α,x Mjorn fermion opertors stisfying {γ α,x,γ α,x }=δ αα δ xx. These expressions expose the defining 1 Deprtment of Physics nd Astronomy, University of Cliforni, Irvine, Cliforni 9697, USA, Deprtment of Condensed Mtter Physics, Weizmnn Institute of Science, Rehovot, 76100, Isrel, 3 Deprtment of Physics, Cliforni Institute of Technology, Psden, Cliforni 9115, USA, 4 Dhlem Center for Complex Quntum Systems nd Fchereich Physik, Freie Universität Berlin, Berlin, Germny, 5 Deprtment of Physics, University of Cliforni, Snt Brr, Cliforni 93106, USA. *e-mil: licej@uci.edu. 41 NATURE PHYSICS VOL 7 MAY 011

2 NATURE PHYSICS DOI: /NPHYS1915 z y x A, 1 B, 1 A, B, A, 3 B, 3 A, N B, N B Semiconducting wire s-wve superconductor Figure 1 Mjorn fermions pper t the ends of 1D spinless p-wve superconductor, which cn e experimentlly relized in semiconducting wires 1,., Pictoril representtion of the ground stte of eqution (1) in the limit µ = 0, t =. Ech spinless fermion in the chin is decomposed in terms of two Mjorn fermions γ A,x nd γ B,x. Mjorns γ B,x nd γ A,x+1 comine to form n ordinry, finite-energy fermion, leving two zero-energy end Mjorns γ A,1 nd γ B,N s shown 3., A spin orit-coupled semiconducting wire deposited on n s-wve superconductor cn e driven into topologicl superconducting stte exhiiting such end Mjorn modes y pplying n externl mgnetic field 1,. c, Bnd structure of the semiconducting wire when B = 0 (dshed lines) nd B = 0 (solid lines). When µ lies in the nd gp generted y the field, piring inherited from the proximte superconductor drives the wire into the topologicl stte. chrcteristics of Mjorn fermions they re their own ntiprticle nd constitute hlf of n ordinry fermion. In this limit the Hmiltonin ecomes N 1 H = it γ B,x γ A,x+1 x=1 Consequently, γ B,x nd γ A,x+1 comine to form n ordinry fermion d x = (γ A,x+1 + iγ B,x )/, which costs energy t, reflecting the wire s ulk gp. Conspicuously sent from H, however, re γ A,1 nd γ B,N, which represent end-mjorn modes. These cn e comined into n ordinry (lthough highly non-locl) zero-energy fermion d end = (γ A,1 +iγ B,N )/. Thus there re two degenerte ground sttes which serve s topologiclly protected quit sttes: 0 nd 1=d end 0, where d end 0=0. Figure 1 illustrtes this physics pictorilly. Awy from this limit the Mjorn end sttes no longer retin this simple form, ut survive provided the ulk gp remins finite 3. This occurs when µ < t, where prtilly filled nd pirs. The ulk gp closes when µ =t. For lrger µ, piring occurs in fully occupied or vcnt nd, nd trivil superconducting stte without Mjorns emerges. Relizing Kitev s topologicl superconducting stte experimentlly requires spinless system (tht is, with one pir of Fermi points) tht p-wve pirs t the Fermi energy. Both criteri cn e stisfied in spin orit-coupled semiconducting wire deposited on n s-wve superconductor y pplying mgnetic field 1, (see Fig. 1). The simplest Hmiltonin descriing such wire reds H = dx ψ x h x m µ i huê σ x gµ BB z σ ψ z x +( e iϕ ψ x ψ x +h.c.) (3) The opertor ψ αx corresponds to electrons with spin α, effective mss m, nd chemicl potentil µ. (We suppress the spin indices except in the piring term.) In the third term, u denotes the spin orit 31,3 strength, nd σ = (σ x,σ y,σ z ) is vector of Puli c E k mtrices. This coupling fvours ligning spins long or ginst the unit vector ê, which we ssume lies in the (x,y) plne. The fourth term represents the Zeemn coupling due to the mgnetic field B z < 0. Note tht spin orit enhncement cn led 33 to g. Finlly, the lst term reflects the spin-singlet piring inherited from the superconductor y mens of the proximity effect. To understnd the physics of eqution (3), consider first B z = = 0. The dshed lines in Fig. 1c illustrte the nd structure here clerly no spinless regime is possile. Introducing mgnetic field genertes nd gp B z t zero momentum, s the solid line in Fig. 1c depicts. When µ lies in this gp the system exhiits single pir of Fermi points s desired. Turning on wekly compred to the gp then effectively p-wve pirs fermions in the lower nd with momentum k nd k, driving the wire into Kitev s topologicl phse 1,. (Singlet piring in eqution (3) genertes p-wve piring ecuse spin orit coupling fvours opposite spins for k nd k sttes.) Quntittively, relizing the topologicl phse requires 1, < gµ B B z /, which we herefter ssume holds. The opposite limit > gµ B B z / effectively violtes the spinless criterion ecuse piring strongly intermixes sttes from the upper nd, producing n ordinry superconductor without Mjorn modes. In the topologicl phse, the connection to eqution (1) ecomes more explicit when gµ B B z mu, where the spins nerly polrize. One cn then project eqution (3) onto simpler onend prolem y writing ψ x (u(e y +ie x )/gµ B B z ) x x nd ψ x x, with x the lower-nd fermion opertor. To leding order, one otins H eff dx x h x m µ eff x + eff e iϕeff x x x +h.c. (4) where µ eff =µ+gµ B B z / nd the effective p-wve pir field reds eff e iϕeff u gµ B B z eiϕ (e y +ie x ) (5) The dependence of ϕ eff on ê will e importnt elow when we consider networks of wires. Eqution (4) constitutes n effective low-energy Hmiltonin for Kitev s model in eqution (1) in the low-density limit. From this perspective, the existence of end- Mjorns in the wire ecomes mnifest. We exploit this correspondence elow when ddressing universl properties such s riding sttistics, which must e shred y the topologicl phses descried y eqution (3) nd the simpler lttice model, eqution (1). We now seek prcticl method to mnipulte Mjorn fermions in the wire. As motivtion, consider pplying gte voltge to djust µ uniformly cross the wire. The excittion gp otined from eqution (3) t k = 0 vries with µ s E gp (k = 0) = gµ B B z +µ For µ <µ c = (gµ B B z /) the topologicl phse with end Mjorns emerges, wheres for µ >µ c topologiclly trivil phse ppers. A uniform gte voltge thus llows the cretion or removl of the Mjorn fermions. However, when µ =µ c the ulk gp closes, nd the excittion spectrum t smll momentum ehves s E gp (k) hv k, with velocity v = u /(gµ B B z ). The gp closure is clerly undesirle, s we would like to mnipulte Mjorn fermions without generting further qusiprticles. This prolem cn e circumvented y employing keyord of loclly tunle gtes s in Fig., ech impcting µ over finite NATURE PHYSICS VOL 7 MAY

3 NATURE PHYSICS DOI: /NPHYS1915 e f c d length Lgte of the wire. When given gte loclly tunes the chemicl potentil cross µ = µc, finite excittion gp Egp h vπ/lgte remins. (Roughly, the gte cretes potentil well tht supports only k lrger thn π/lgte.) Assuming g µb Bz / nd h u 0.1 ev Å yields velocity v 104 m s 1 ; the gp for 0.1 µm wide gte is then of order 1 K. We consider this conservtive estimte hevy-element wires such s InS nd/or nrrower gtes could generte sustntilly lrger gps. Locl gtes llow Mjorn fermions to e trnsported, creted, nd fused, s outlined in Fig.. As one germintes pirs of Mjorn fermions, the ground stte degenercy increses, s does our cpcity to topologiclly store quntum informtion. Specificlly, n Mjorns generte n ordinry zero-energy fermions, with occuption numers tht specify topologicl quit sttes. Aditiclly riding the Mjorn fermions to mnipulte these quits, however, is impossile in single wire. Thus we now turn to the simplest rrngement permitting exchnge the T-junction of Fig. 3. Mjorn riding nd non-aelin sttistics First, we explore the properties of the junction where the wires in Fig. 3 meet (see the Supplementry Informtion for more detils). It is instructive to view the T-junction s three segments meeting t point. When only one segment relizes topologicl phse, single zero-energy Mjorn fermion exists t the junction. When two topologicl segments meet t the junction, s in Fig. 3 nd, genericlly no Mjorn modes exist there. To see this, imgine 414 g d Figure Applying keyord of individully tunle gtes to the wire llows locl control of which regions re topologicl (drk lue) nd non-topologicl (light lue), nd hence mnipulte Mjorn fermions while mintining the ulk gp. As nd illustrte, sequentilly pplying the leftmost gtes drives the left end of the wire non-topologicl, therey trnsporting γ1 rightwrd. Nucleting topologicl section of the wire from n ordinry region or vice vers cretes pirs of Mjorn fermions out of the vcuum s in c. Similrly, removing topologicl region entirely or connecting two topologicl regions s sketched in d fuses Mjorn fermions into either the vcuum or finite-energy qusiprticle. c h Figure 3 A T-junction provides the simplest wire network tht enles meningful ditic exchnge of Mjorn fermions. Using the methods of Fig., one cn rid Mjorns ridged y either topologicl region (drk lue lines) s in d, or non-topologicl region (light lue lines) s in e h. The rrows long the topologicl regions in d re useful for understnding the non-aelin sttistics, s outlined in the min text. decoupling the topologicl segments so tht two nery Mjorn modes exist t the junction; restoring the coupling genericlly comines these Mjorns into n ordinry, finite-energy fermion. As n illustrtive exmple, consider the setup of Fig. 3 nd model the left nd right topologicl segments y Kitev s model with µ = 0 nd t = in eqution (1). (For simplicity we exclude the non-topologicl verticl wire in Fig. 3.) Suppose furthermore tht φ = φl/r in the left/right chins nd tht the fermion cl,n t site N of the left chin couples wekly to the fermion cr,1 t site 1 of the right chin vi H = (cl,n cr,1 + h.c.). Using eqution (), the Mjorns t the junction couple s follows, i φl φr L R H cos γb,n γa,1 (6) nd therefore generlly comine into n ordinry fermion3. An exception occurs when the regions form π-junction tht is, when φl φr = π which fine-tunes their coupling to zero. Importntly, coupling etween end Mjorns in the semiconductor context is governed y the sme φl φr dependence s in eqution (6) (refs 1,). Finlly, when three topologicl segments meet, gin only single Mjorn mode exists t the junction without finetuning. Three Mjorn modes pper only when ll pirs of wires simultneously form mutul π junctions (which is possile ecuse the superconducting phses re defined with respect to direction in ech wire; see the Supplementry Informtion). NATURE PHYSICS VOL 7 MAY 011

4 NATURE PHYSICS DOI: /NPHYS1915 Recll from eqution (5) tht the spin orienttion fvoured y spin orit coupling determines the effective superconducting phse of the semiconducting wires. Two wires t right ngles to one nother therefore exhiit π/ phse difference, well wy from the pthologicl limits mentioned ove. One cn thus lwys trnsport Mjorn fermions cross the junction without generting spurious zero-modes. T-junctions llow exchnge of Mjorns residing on either the sme or different topologicl regions. Figure 3 d illustrtes counterclockwise rid for the former cse, wheres Fig. 3e h illustrtes the ltter. Although the Mjorns cn now e exchnged, their non-aelin sttistics remins to e proven. Let us first recll how non-aelin sttistics of vortices rises in spinless D p+ip superconductor 10,11. Ultimtely, this cn e deduced y considering two vortices which ind Mjorn fermions γ 1 nd γ. As the spinless fermion opertors effectively chnge sign on dvncing the superconducting phse y π, one introduces rnch cuts emnting from the vortices; crucilly, Mjorn fermion chnges sign whenever crossing such cut. On exchnging the vortices, γ (sy) crosses the rnch cut emnting from the other vortex, leding to the trnsformtion rule γ 1 γ nd γ γ 1, which is generted y the unitry opertor U 1 = exp(πγ γ 1 /4). With mny vortices, the nlogous unitry opertors U ij implementing exchnge of γ i nd γ j do not generlly commute, implying non-aelin sttistics. Following n pproch similr to Stern nd collegues 34, we now rgue tht Mjorn fermions in wires trnsform exctly like those ound to vortices under exchnge, nd hence lso exhiit non-aelin sttistics. This cn e estlished most simply y considering the exchnge of two Mjorn fermions γ 1 nd γ, s illustrted in Fig. 3 d. At ech step of the exchnge, there re two degenerte ground sttes 0 nd 1=f 0, where f = (γ 1 +iγ )/ nnihiltes 0. In principle, one cn deduce the trnsformtion rule from the Berry phses χ n Im dtn t n cquired y the mny-ody ground sttes n = 0 nd 1, lthough in prctice these re hrd to evlute. As exchnge sttistics is universl property, however, we re free to deform the prolem to our convenience provided the energy gp remins finite. As first simplifiction, ecuse the semiconductor Hmiltonin nd Kitev s model in eqution (1) cn e smoothly connected, let us consider the cse where ech wire in the T-junction is descried y the ltter. More importntly, we further deform Kitev s Hmiltonin to e purely rel s we exchnge γ 1,. The sttes 0 nd 1 cn then lso e chosen rel, leding to n enormous simplifiction: lthough these sttes still evolve nontrivilly the Berry phse ccumulted during this evolution vnishes. For concreteness, we deform the Hmiltonin such tht µ<0 nd t = = 0 in the non-topologicl regions of Fig. 3. For the topologicl segments, relity implies tht the superconducting phses must e either 0 or π. It is useful to visulize the sign choice for the piring with rrows s in Fig. 3. (To e concrete, we tke the piring e iφ c j c j+1 such tht the site indices increse moving rightwrd/upwrd in the horizontl/verticl wires; the cse φ = 0 then corresponds to rightwrd/upwrd rrows, wheres leftwrd/downwrd rrows indicte φ = π.) To void generting π junctions, when two topologicl segments meet t the junction, one rrow must point into the junction while the other must point out. With this simple rule in mind, we see in Fig. 3 tht lthough we cn successfully swp the Mjorns while keeping the Hmiltonin rel, we inevitly end up reversing the rrows long the topologicl region. In other words, the sign of the piring hs flipped reltive to our initil Hmiltonin. To complete the exchnge we must then perform guge trnsformtion which restores the Hmiltonin to its originl form. This cn e ccomplished y multiplying ll fermion cretion opertors y i; in prticulr, f = (γ 1 iγ )/ if = (γ +iγ 1 )/. It follows tht γ 1 γ nd γ γ 1, which the unitry trnsformtion L i R i Figure 4 Experimentl set-ups tht llow the proing of non-aelin sttistics nd Mjorn-fermion fusion rules., Exmple of semiconductor wire network which llows for efficient exchnge of mny Mjorn fermions. Adjcent Mjorns cn e exchnged s in Fig. 3, wheres non-djcent Mjorns cn e trnsported to the lower wire to e similrly exchnged., Miniml set-up designed to detect the non-trivil Mjorn fusion rules. Mjorns γ 1, re first creted out of the vcuum. In the left pth, γ is shuttled rightwrd, nd Mjorns γ 3,4 lwys comine to form finite-energy stte which is unoccupied. In the right pth, γ 3,4 re lso creted out of the vcuum, nd then γ nd γ 3 fuse with 50% proility into either the vcuum or finite-energy qusiprticle. The Josephson current flowing cross the junction llows the deduction of the presence or sence of this extr qusiprticle. U 1 = exp(πγ γ 1 /4) genertes s in the D p + ip cse. (Note tht one could lterntively multiply ll fermion cretion opertors y i insted of i to chnge the sign of the piring, which would led to the slightly different trnsformtion γ 1 γ nd γ 1 γ. The miguity disppers if one exchnges the Mjorns while keeping the superconducting phses fixed s one would in prctice; see the Supplementry Informtion for detiled discussion.) We stress tht this result pplies lso in the physiclly relevnt cse where gtes trnsport the Mjorns while the superconducting phses remin fixed; we hve merely used our freedom to deform the Hmiltonin to expose the nswer with miniml formlism. Furthermore, ecuse Fig. 3e h lso represents counterclockwise exchnge, it is nturl to expect the sme result for this cse. The Supplementry Informtion nlyses oth types of exchnges from complementry perspective (nd when the superconducting phses re held fixed), confirming their equivlence. There we lso estlish rigorously tht in networks supporting ritrrily mny Mjorns exchnge is implemented y set of unitry opertors U ij nlogous to those in D p+ip superconductor. (The Methods section outlines the nlysis.) Thus the sttistics is non-aelin s dvertised. Discussion The keyord of gtes shown in Fig. nd the T-junction of Fig. 3 provide the sic elements llowing mnipultion of topologicl quits in semiconducting wires. In principle, single T-junction cn 4 NATURE PHYSICS VOL 7 MAY

5 support numerous well-seprted Mjorn modes, ech of which cn e exchnged with ny other. (First, crete mny Mjorns in the horizontl wire of the T-junction. To exchnge given pir, shuttle ll intervening Mjorns down to the end of the verticl wire nd then crry out the exchnge using the methods of Fig. 3.) However, networks consisting of severl T-junctions such s the set-up of Fig. 4 enle more efficient Mjorn exchnge. In the figure, ll djcent Mjorn fermions cn e immeditely swpped using Fig. 3, wheres non-djcent Mjorns cn e shuttled down to the lower wire to e exchnged. This ldder configurtion strightforwrdly scles up y introducing extr rungs nd/or legs. As Fu nd Kne suggested in the topologicl insultor context 13, fusing Mjorn fermions cross Josephson junction provides redout method for the topologicl quit sttes. We illustrte the physics with the schemtic set-up of Fig. 4, which extends the experiments proposed in refs 1, to llow the Mjorn fusion rules to e directly proed. Here semiconducting wire ridges two s-wve superconductors with initil phses ϕ L/R i ; we ssume ϕ i ϕ L i ϕ R i = π. Three gtes drive the wire from n initilly non-topologicl ground stte into topologicl phse. Importntly, the order in which one pplies these gtes qulittively ffects the physics. As we now discuss, only in the left pth of Fig. 4 cn the quit stte t the junction e determined in single mesurement. Consider first germinting Mjorn fermions γ 1 nd γ y pplying the left gte. Assuming f A = (γ 1 + iγ )/ initilly costs finite energy s γ 1 nd γ seprte, the system initilizes into ground stte with f A unoccupied. Applying the centrl nd then right gtes shuttles γ to the other end (see the left pth of Fig. 4). As nrrow insulting rrier seprtes the superconductors, n ordinry fermion f B = (γ 3 + iγ 4 )/ rises from two coupled Mjorns γ 3,4 t the junction. Similr to eqution (6), the energy of this mode is well-cptured y 1 3 H J i i γ 3 γ 4 = i (f B f B 1), where i = δcos(ϕ i /) with non-universl δ. The system hs een prepred in ground stte, so the f B fermion will e sent if i > 0 ut occupied otherwise. Suppose we now vry the phse difference cross the junction wy from its initil vlue to ϕ. The mesured Josephson current (see Supplementry Informtion for pedgogicl derivtion) will then e I = e h de dϕ = eδ h sgn(i )sin(ϕ/)+i e (7) where E is the ground-stte energy nd I e denotes the usul Cooper-pir-tunnelling contriution. The first term on the right reflects single-electron tunnelling originting from the Mjorns γ 3,4. This frctionl Josephson current exhiits 4π periodicity in ϕ, ut π periodicity in the initil phse difference ϕ i. The right pth in Fig. 4 yields very different results, reflecting the nontrivil Mjorn fusion rules. Here, fter creting γ 1,, one pplies the rightmost gte to nuclete nother pir γ 3,4. Assuming f A nd f B defined s ove initilly cost finite energy, the system initilizes into the ground stte 0, 0 stisfying f A/B 0, 0 =0. Applying the centrl gte then fuses γ nd γ 3 t the junction. To understnd the outcome, it is useful to re-express the ground stte in terms of f A = (γ 1 + iγ 4 )/ nd f B = (γ + iγ 3 )/. In this sis 0,0=( 0,0 i 1,1 )/, where fa,b nnihilte 0,0 nd 1,1 =fa f B 0,0. Following our previous discussion, f B cquires finite energy t the junction, lifting the degenercy etween 0,0 nd 1,1. Mesuring the Josephson current then collpses the wvefunction with 50% proility onto either the ground stte, or n excited stte with n extr qusiprticle loclized t the junction. In the former cse eqution (7) gin descries the current, wheres in the ltter cse the frctionl contriution simply chnges sign. NATURE PHYSICS DOI: /NPHYS1915 In more complex networks, such s tht of Fig. 4, fusing the Mjorns cross Josephson junction nd in prticulr mesuring the sign of the frctionl Josephson current similrly llows quit redout. Alterntively, the interesting recent proposl of Hssler et l. 6 for reding quit sttes vi ncillry non-topologicl flux quits cn e dpted to these setups (nd indeed ws originlly discussed in terms of n isolted semiconducting wire 6 ). To conclude, we hve introduced surprising new venue for riding, non-aelin sttistics, nd topologicl quntum informtion processing networks of one-dimensionl semiconducting wires. From fundmentl stndpoint, the ility to relize non- Aelin sttistics in this setting is remrkle. Perhps even more ppeling, however, re the physicl trnsprency nd experimentl promise of our proposl, prticulrly given the fets lredy chieved in ref. 35. Although topologicl quntum informtion processing in wire networks requires much experimentl progress, oserving the distinct fusion chnnels chrcteristic of the two pths of Fig. 4 would provide remrkle step en route to this gol. And ultimtely, if riding in this setting cn e supplemented y π/8 phse gte nd topologicl chrge mesurement of four Mjorns, wire networks my provide fesile pth to universl quntum computtion 6,7 30. Methods In the Supplementry Informtion we provide rigorous, systemtic derivtion of non-aelin sttistics of Mjorn fermions in wire networks, thus estlishing solid mthemticl foundtion for the results otined in the min text. As the nlysis is rther lengthy, here we riefly outline the pproch. We first define the mny-ody ground sttes in the presence of ritrrily mny Mjorn fermions in n ritrry wire network. We then estlish three importnt generl results tht gretly fcilitte the derivtion of non-aelin sttistics. (1) If two Mjorns re exchnged without disturing ny other Mjorns in the network, ll of these other Mjorns simply fctor out in the sense tht their presence in no wy ffects how the degenerte ground sttes trnsform. () If we know how given pir of Mjorns trnsforms under exchnge in some miniml setting, then the sme trnsformtion holds when ritrrily mny extr Mjorns re introduced, provided they re fr from those eing exchnged. These first two properties re rther nturl nd follow from the loclity of the Mjorn wvefunctions. (3) The trnsformtion of the degenerte ground sttes under exchnge (up to n overll non-universl phse) cn e deduced solely y understnding how the Mjorn opertors trnsform. This provides n enormous simplifiction, s it distills the prolem down to understnding the ehviour of the single-ody Mjorn opertors eing rided. It follows from these results tht to understnd non-aelin sttistics in wire networks composed of trijunctions, it suffices to deduce how the Mjorn opertors trnsform under the two types of rids shown in Fig. 3. We susequently nlyse these exchnges (when the superconducting phses re held fixed, s would e the cse in prctice) nd show tht the opertors trnsform similrly to vortices in D p+ip superconductor, therey estlishing non-aelin sttistics. Interestingly, the picture we develop in the Supplementry Informtion closely resemles Ivnov s construction for non-aelin sttistics of vortices, despite their sence in wire networks. Very crudely, s the Mjorn fermions move long the network to e exchnged, the effective p-wve superconducting phses they feel vry, in loose nlogy to wht hppens when Mjorn fermions ound to vortices rid one nother. It is lso interesting to note tht it is not only the clockwise versus counterclockwise nture of the rid tht determines how the Mjorn opertors trnsform, unlike in D p+ip superconductor. In ddition to the hndedness, the superconducting phses of the wires forming the junction lso ply criticl role in governing the outcome of n exchnge. For exmple, counterclockwise exchnge with given set of superconducting phses cn hve the sme effect s clockwise exchnge when the superconducting phses re modified. Thus, wire networks feture more ville knos tht one cn tune to control how n exchnge impcts quit sttes, which my hve useful pplictions. Received 9 June 010; ccepted Decemer 010; pulished online 13 Ferury 011 References 1. Kitev, A. Fult-tolernt quntum computtion y nyons. Ann. Phys. 303, 30 (003).. Freedmn, M. H. P/NP, nd the quntum field computer. Proc. Ntl Acd. Sci. USA 95, (1998). 416 NATURE PHYSICS VOL 7 MAY 011

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DuBridge Foundtion, ISF, BSF, DIP nd SPP 185 grnts, Pckrd nd Slon fellowships, the Institute for Quntum Informtion under NSF grnts PHY nd PHY , nd the Ntionl Science Foundtion through grnt DMR Author contriutions All uthors contriuted to the inception of the ides in the mnuscript, design of networks nd proposed experimentl setups, nd proof of non-aelin sttistics. Additionl informtion The uthors declre no competing finncil interests. Supplementry informtion ccompnies this pper on Reprints nd permissions informtion is ville online t Correspondence nd requests for mterils should e ddressed to J.A. NATURE PHYSICS VOL 7 MAY