Downloaded by guest on September 27, 2021 h ucincnutneyed aua edu o the for readout and computing quantum gate, natural phase a the yields and Majorana conductance state. gate qubit hybrid chiral junction Hadamard the coherent the the of quantum junction implement and ring use modes to Corbino chi- zero can with A Majorana device computation fermions. of transfor- quantum Majorana perform braiding unitary ral to the same platform in a the propose that to of as leads propagation mation fermions the topological that Majorana in show correspond- applicable chiral we the Hall be Here of to anomalous computations. bulk known quantum the is quantum , in related topological mode a ing closely zero topological of Its Majorana superconductor. 2D the device conventional cousin, certain a hybrid experimentally of and and a insulator state predicted in edge theoretically been observed the has Zhang) as It Fuchun and . arise Nagaosa, self-conjugate Naoto Fradkin, massless can Eduardo a by is which reviewed fermion 2018; 11, Majorana June chiral review The for (sent 2018 5, September Zhang, Shou-Cheng by Contributed 08540 NJ Princeton, Study, Advanced 94305-4045; CA Stanford, University, a www.pnas.org/cgi/doi/10.1073/pnas.1810003115 number Chern num- with Chern (QAHI) (BdG) celebrated insulator Gennes A Hall (1). Bogoliubov–de matter a ber of carries state which topological the edge 2D is example a as realized of be states where can 1D space, fermions physics, 1D Majorana condensed-matter chiral in In predicted sim- unidirectionally. is The propagates it fermion physics. Majorana theoretical chiral in plest ago long proposed T Lian Biao fermions chiral Majorana on based computation quantum Topological rpgto fcia aoaafrin ihprl electrical purely MZMs. with of uses fermions instead which manipulations Majorana scales, chiral mesoscopic of at propagation gates quantum protected cally been yet not topologi- has toward MZMs demonstrated. of step experimentally braiding essential the archi- the computing, an of quantum proposed As cal control MZMs. existing and among design all coupling nanoscale MZMs, require of inevitably MZMs localized tectures the using to point-like on due decade (14–17), and computation past quantum the theo- universal during in the made Despite progress (12–17). fault-tolerant retical in computations used quantum be can topological a and statistics of braiding 1D vortices non-Abelian a bulk of endpoints the the in of emerge quan- states which fractional ordered the topologically of and state (7) (8). systems Moore–Read effect the Hall could Cr- tum in fermion Majorana the arise chiral in The also (6). observed superconductor recently Nb Majorana been the chiral half- has (Bi,Sb) a which by doped exhibit 4), induced to (3, plateau predicted fermions is conductance way quantized this implemented junction an with ity rneo etrfrTertclSine rneo nvriy rneo,N 08544-0001; NJ Princeton, University, Princeton Science, Theoretical for Center Princeton nti ae,w rps ltomt mlmn topologi- implement to platform a propose we paper, this In (MZMs), modes zero Majorana concept, related closely A elfrin samsls emoi atcebigisown its being Majorana– the fermionic massless as a known is also fermion, fermion, Weyl Majorana chiral he N 1 = a,b,1 s n a eraie rmaqatmanomalous quantum a from realized be can and wv uecnutr(–) QAHI–TSC–QAHI A (2–5). superconductor -wave ioQ Sun Xiao-Qi , 2 p Te + | 3 topological ip hnfimQH ytmi rxmt with proximity in system QAHI thin-film hrltplgclsprodco (TSC), superconductor topological chiral p wv S 1,1) r nw oobey to known are 11), (10, TSC -wave b,c,1 | Majorana c blasnVaezi Abolhassan , eateto hsc,Safr nvriy tnod A93544;and 94305-4045; CA Stanford, University, Stanford Physics, of Department p + ip C 1 = S 9 rat or (9) TSC b,c nproxim- in ioLagQi Xiao-Liang , 1073/pnas.1810003115/-/DCSupplemental at online information supporting contains article This 2 1 hc r nage ttswe rte nteoiia ai of basis original the in written when i states entangled num- are which fermion the preserve have of to we necessary 1A, is ber Fig. sign exchanged. minus in are relative also process MZMs the corresponding In the exchanged, are by to labeled corresponds are states two the number, fermion ooti elzto ftesm aeb aigueo chi- of use chiral QAHI. making complex the and by of TSC states gate the edge of same fermion how states the edge show fermion of we Majorana realization ral following, a the obtain In to involved. the is of braiding pro- no physical if non-Abelian even another fermions, by Majorana exchanges realized that be cess can gate non-Abelian same rela- of anticommutation degree the fermion tion satisfy a operators of MZM states The quantum freedom. two define MZM together single tices a supports vortex Each 1A. Fig. ulse ne the under Published interest. of conflict Theoretical no for declare Institute authors The Kavli F.Z., and Science; Matter China.y A.V. Emergent TokyoPhysics of for and University Center The X.-Q.S., Riken N.N., Urbana–Champaign; and at paper. B.L., y Illinois the of wrote research; University S.-C.Z. E.F., and Reviewers: designed X.-L.Q., A.V., X.-Q.S., S.-C.Z. B.L., and and research; performed X.-L.Q. contributions: Author because Abelian consequence MZMs direct the a that exchanging is robust see of braiding can is MZM one during operation above, gate presented non-Abelian be unitary reasoning not non-Abelian the may From the (12). braiding but the defined, during well phase Abelian the interaction, the of vortices in MZMs of braiding the to no p proposal related our is Since there closely edge. if is the even along states, traveling anyon non-Abelian non-Abelian on realize to operations used gate be Majorana can quantum chiral TSC the the that of show state to edge is fermion proposal our of goal main The Qubits Fermion Majorana Chiral √ owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom work.y To this to equally contributed X.-Q.S. and B.L. γ 1 unu optto schemes. computation 10 com- quantum be its the can realized, by If speed accomplished fermions. putation Majorana naturally mate- chiral are topological of gates propagation of the quantum edges using The the computation rials. on quantum fermions of Majorana platform chiral a propose We Significance + 2 1 i b,c,d γ γ ip 2 |0i {γ 1 γ and n huCegZhang Shou-Cheng and , S,w ei yrveigti rcs,a lutae in illustrated as process, this reviewing by begin we TSC, i 2 12 , b γ and tnodCne o oooia unu hsc,Stanford Physics, Quantum Topological for Center Stanford | 1i j i i } γ γ 34 2 = 3 i γ γ + γ NSlicense.y PNAS 3 δ 4 i 3 ij γ o xml,testate the example, For . i fti ar sacneune h eigenstates the consequence, a As pair. this of |1i γ 1 fw define we If . γ 4 γ 1 12 2 vlet iesae of eigenstates to evolve γ 2 = |0i γ niomtswith anticommutes ,+1 −1, 2 34 , γ  3 ic h otcshv long-range have vortices the Since . d 3 colo aua cecs nttt for Institute Sciences, Natural of School h eutn aems enon- be must gate resulting The . atrta h urnl existing currently the than faster . y epciey hntovortices two When respectively. , y f 12 b,c,2 = 2 1 γ www.pnas.org/lookup/suppl/doi:10. 2 (γ NSLts Articles Latest PNAS → |1i 1 i + γ γ γ 12 1 i i 3 f γ n hstovor- two thus and , i γ 12 , |0i † γ 1 3 2 f γ hrfr,the Therefore, . γ 12 34 ) 3 3 sacomplex a as −γ → which 1, 0, = and vle into evolves −i 2 | The . γ f5 of 1 2 γ 4 ,

PHYSICS that the outgoing states in the two leads are entangled, because ( A 0 12 1 34 + 1 12 0 34) the number operators in these leads do not commute with those of incoming . To be more specific and to make a connection with quantum computation, consider the low-current limit I → 0 where elec- trons are injected from lead 1 one by one, each of which occupies a traveling-wave packet state of ψA. The occupation number 0 or

time 1 of such a fermion wave packet state then defines a qubit A with basis |0Ai and |1Ai. Similarly, we can define the qubits B, γ1 γ2 γ3 γ4 C , and D for ψB , ψC , and ψD , respectively. At each moment 1 0 of time, the real and imaginary parts of the fermionic annihila- 12 34 tion operator of each wave packet state define two self-conjugate ψ Majorana operators localized at the wave packet. When the wave B C f packets move out the superconducting region, they merge with ψC ψB γ3 a different partner and form states of the outgoing qubits. In the evolution of the incident electrons, qubits A and B span the γ4 Z Hilbert space of the initial state |ψi i, while qubits C and D form the Hilbert space of the final state |ψ i. In the same way as the lead 1 QAHITSC QAHI lead 2 f MZM braiding case, the exchange of γ2 with γ3 then leads to a γ 1 H unitary evolution γ      ψ 2 ψ |0C 0D i 1 0 0 1 |0A0B i A D 1 lead 3 ψ  |0C 1D i   0 1 1 0  |0A1B i  i = √ . [1]  |1C 0D i  2  0 −11 0  |1A0B i  |1C 1D i −10 01 |1A1B i Fig. 1. (A) The braiding of vortices in the p + ip TSC. Each pair of vortices supports two states of a single fermion, and the braiding leads to a non- This transformation should be viewed as an S matrix between Abelian operation and maps a product state of vortices 12 and 34 into an incoming and outgoing electron states. Note that the fermion entangled state, as a consequence of exchanging MZMs γ2, γ3.(B) Our pro- parity is conserved in the unitary evolution. If we define a posed device of QAHI–TSC–QAHI junction. The same partner switch as in A new qubit (|0i, |1i) in the odd fermion parity subspace as occurs between incoming electrons from leads A and B and outgoing elec- (|0A1B i , |1A0B i) initially and (|0C 1D i, |1C 0D i) at the final time, trons in leads C and D. (C) Such an exchange leads to a non-Abelian gate the above unitary evolution is exactly a topologically protected that is equivalent to a Hadamard gate H followed by a Pauli-Z gate Z. Hadamard gate H followed by a Pauli-Z gate Z as shown in Fig. 1C; namely, |ψf i = ZH |ψi i, where

The device we propose to study is a 2D QAHI–TSC–QAHI 1 1 1 1 0 junction predicted in refs. 3 and 4. As shown in Fig. 1B, the H = √ , Z = . [2] 2 1 −1 0 −1 junction consists of two QAHIs (18–20) of Chern number C = 1 and a chiral TSC of BdG Chern number N = 1. The con- The same conclusion holds for the even fermion parity subspace. ductance σ12 is measured between metallic leads 1 and 2 by Therefore, the two qubits A and B (C and D) behave effectively driving a current I , where no current flows through lead 3 which as a single qubit, and we can regard qubit A (C) as the data qubit, grounds the TSC. Each edge between the chiral TSC and the while qubit B (D) is a correlated ancilla qubit. vacuum or a QAHI hosts a chiral Majorana fermion edge mode For an electron incident from lead 1 represented by ini- governed by a Hamiltonian HM (x) = −i vF γ(x)∂x γ(x), where ~ tial state |ψi i = |1A0B i, the junction turns it into a final state γ(x) is the Majorana operator satisfying γ(x) = γ†(x) and the √ 0 0 |ψf i = (|0C 1D i + |1C 0D i)/ 2. This implies (SI Appendix) that anticommutation relation {γ(x), γ(x )} = δ(x − x )/2, vF is the the entanglement entropy between left and right halves of the Fermi velocity, and x is the coordinate of the 1D edge. In junction divided by the dashed line in Fig. 2A increases by contrast, each edge between a QAHI and the vacuum hosts a log 2. Indeed, this is verified by our numerical calculation in charged chiral fermion (electron) edge mode with a Hamilto- † † nian HF (x) = −i~vF ψ (x)∂x ψ(x), where ψ(x) and ψ (x) are the annihilation and creation operators of the edge fermion, and we have assumed chemical potential µ = 0 for the moment. A B 1.0 † By defining two Majorana operators γ1 = (ψ + ψ )/2 and γ2 = 0.8 † (ψ − ψ )/2i [hereafter γi = γi (x) is short for chiral Majorana S EE0 -S 0.6 H (x) H (x) = −i v (γ ∂ γ + log 2 fermion], one can rewrite F as F ~ F 1 x 1 QAHIQA TSCT C Q QAHIH 0.4 γ2∂x γ2), which implies a charged chiral fermion mode is equiv- alent to two chiral Majorana fermion modes. As a result, the 0.2 edge states of the junction consist of four chiral Majorana 0 0204060 fermion modes γi (1 ≤ i ≤ 4) as shown in Fig. 1B, which are t (a.u.) related to the charged chiral fermion modes on the QAHI edges as ψA = γ1 + iγ2, ψB = γ4 + iγ3, ψC = γ1 − iγ3, and Fig. 2. (A) The setup for numerical computation of entanglement entropy. We use a lattice model of QAHI–TSC–QAHI junction, add an initial edge ψD = γ4 + iγ2 (3). Our key observation is that the same kind of partner switch of wave packet on a QAHI edge, and then examine the time evolution of the state and the entanglement entropy between the left and the right part Majorana fermions as that of the vortex braiding occurs in this of the lattice separated by the dashed line. (B) Evolution of entanglement device between incoming and outgoing electrons. An incoming entropy SE between left and right halves of the junction (divided by dashed electron from lead A becomes a nonlocal fermion simultaneously line in A) with time t (arbitrary unit) after an electron above the Fermi on the two edges of the TSC described by γ1 and γ2. If we mea- sea is injected from lead 1, where SE0 is the entanglement entropy of the sure the number of outgoing electrons in leads C and D, we find Fermi sea.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1810003115 Lian et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 AITCjnto ssoni i.3A Fig. QAHI–TSC– in geometry shown Corbino conductance as a junction implement QAHI–TSC to propose we remains conductance iymatrix sity of factor a propagation phase is random the state process a in the if introduced instance, whether is For thus gate. and quantum propagation coherent the fermions during Majorana not chiral whether confirm conductance The Gate Quantum Testable A fermion Majorana case chiral topologically this of are processes in equivalent. process two the state above so Appendix), final the (SI the propagation as and same TSC state the the initial are on propagating the fermion However, Majorana edge. no is there process, of edge QAH the and fuse vortices, the σ braid can vortices one more two create σ can one Meanwhile, channel. ine al. et Lian with device Corbino the as gates same φ the gives that gates a process quantum single-qubit with of series ring a where to QAHI equivalent is Corbino gate junction voltage a a a and TSC, of the into consists IV junction Corbino fan-shaped The (A) tion. 3. Fig. which after bulk, TSC entangle- the where 2A), (Fig. junction the entropy of ment model lattice a noteHletsaeo w eryvortices nearby two of space (fermion) electron Hilbert the the dragging of into operation an imagine can one t has in sea. Fermi provided electron the are the above calculation 1 this lead Since of from details injected is More electron an after 2,2) oecnrtl,we h electron state the TSC when incident the concretely, an in More of operators 22). vortex (21, four Appendix ) of (SI bulk braiding and fusion to alent qubit final of basis charge the identifying fact in connects 2) are (lead 1 lead conductance two-terminal quantized (1 between qubit initial σ of that with A G 4 4 12 1 = ocnr hte h ytma unu aei coherent, is gate quantum a as system the whether confirm To sw aedsusd h bv rcs stplgclyequiv- topologically is process above the discussed, have we As ial,oecnda h tt nteHletsaeof space Hilbert the in state the drag can one Finally, . |h − = ntebl fteTCi h aumfso hne.Next, channel. fusion vacuum the in TSC the of bulk the in rvdsantrlmaueeto h vra probability overlap the of measurement natural a provides TSC σ ,respectively. 2, π/ /2 | IV ψ 3 R ψ ψ φ unu nefrnei h AITCQH–S obn junc- Corbino QAHI–TSC–QAHI–TSC the in interference Quantum i noteQHeg of edge QAH the onto f C rbblt otne nola .Ti ils()ahalf- a (3) yields This 2. lead into tunnel to probability G i |ψ |ψ = sapaegt otoldby controlled gate phase a is s-wave SC wv uecnutro o fi hc rvsrgosI and II regions drives which it of top on superconductor s-wave Voltage Gate and lead 2 QAHI QAHI i i ρ |1 i| i f lead 1 σ ψ 2 A and V (|0 = )e 12 S C G 0 ψ III ψ I E B D 2 B ewe ed1adla .Tejnto can junction The 2. lead and 1 lead between i /h r |ψ γ ψ γ σ γ ψ nrae ihtime with increases C 3 1 ileov noamxdfia tt ihaden- a with state final mixed a into evolve will 4 rpgt nolas1ad2 respectively, 2, and 1 leads into propagate 1 = ψ D 12 A f . 1 |1 D TSC i uigsc otxbadn n fusing and braiding vortex a such During . σ D II A 12 fteaoejnto,hwvr cannot however, junction, above the of ne hscmo ai;namely, basis; common this under /2 h0 i σ 0 γ 2 1 B [1 = ψ A rbblt ortr ola and 1 lead to return to probability i C A and (B 1 ece h onayo h TSC, the of boundary the reaches V (ψ D − 3 G .Acrigy h conductance the Accordingly, ). | ψ B BC tr(ρ σ Such (B) edge. bottom the at added is + C with ) 2 σ |1 1 V ψ r ntefrincfusion fermionic the in are φ H H ψ Z Z n htof that and f σ C G G f ψ |ψ i with (C . 12 0 C i D t hψ i ψ = and h1 i and C ssoni i.2B, Fig. in shown as σ γ te 1 3 i φ γ (ψ C γ 2 | n fuse and , 2 h Z braiding MZM The D) )]e G ψ /h i n esr the measure and 0 = 0 σ D D D r oeetor coherent are γ 1 σ 2 3 ,we 1B), (Fig. ) = ueinitial pure a , hl the while |)/2, /h 2 γ and 4 IAppendix. SI e and 2 = D /2h φ σ e γ ZHR G 1 2 σ 2 σ σ φ C = γ /2h 4 Since . 2 2 nthe in σ 3 G φ and 0 = onto 12 with G (D and

γ π . ZH 3 σ /2 = γ 1 ) 4 , h nlsaewl etemxmlymxdsaedsrbdby described state mixed maximally the be matrix will density state final the of tude function a as oscillates which junc- Corbino this of is tion conductance two-terminal the Therefore, (ψ I ψ eV qubit corresponding the on acting gate phase a induces length a in term potential chemical a as junction Corbino states the edge of 3A. states Majorana Fig. in edge chiral the four voltage, are gate through. passing zero current no At has and superconductor length the a gate grounds covering III voltage the region into A QAHI IV of 6). edge and (4, II regions phase two TSC the driving field magnetic a of fan-shaped top a attaching by realized be by given roughly is which ∆pL/~, amplitude peak-to-valley The σ 4. Fig. odcac ilcntnl be constantly will conductance (ψ modes when inlpaegt,wihlast hs shift phase a to leads which gate, phase there of tional when landscape edges nonzero general, QAHI a Such the In disorders. along are gate. nonuniform voltage is and by nonzero covered is interval the except edges junction. the in fermions Majorana of amplitude lation unitary total a with 3B, Fig. in shown evolution as gates quantum of series riigpoesta eut ntesm o-bla aea the as gate non-Abelian same the φ in results that process braiding etfo ed1rpeetdb h nta state initial the is by state finial represented the 1 lead from dent AB 12 G D fw eadtecagdcia demdso AIregion QAHI of modes edge chiral charged the regard we If ofrw aeasmdceia potential chemical assumed have we far So h aevoltage gate The B G acltdfor calculated 0 = nege ntr evolution unitary a undergoes A |ψ l and e G φ and 2 f /~v ueial calculated Numerically γ /h G aeadthe and case i 2 ψ = C = |ψ and ncnrs,i h ytmlsschrnecompletely, coherence loses system the if contrast, In . F D 1 = σ ψ e hsi qiaett necag fMajorana of exchange an to equivalent is this 2, π/ steaclaqbt h ucincnb iwda a as viewed be can junction the qubit, ancilla the as ) f 12 l stnbevia tunable is C −i G i stedt ui n hs fQH einIII region QAHI of those and qubit data the as ) γ = ntelnug fqatmcmuain this computation, quantum of language the In . ρ (1 = φ AICriorn,wt rprout-of-plane proper a with ring, Corbino QAHI 4 ∆p f G namely, ; ZHR (|0 = /2 V |h −  L/~ 2 π/ σ G cos φ A 12 R G ψ ntebto deo einIIbehaves III region of edge bottom the on ≈ 1 φ ZH f B ae epciey o neeto inci- electron an For respectively. case, G esrstechrneo h chiral the of coherence the measures φ n 8a ucinof function a as 18 and 0 V y |ψ γ 2 h0 i σ G = of G 4 12 | i H ψ → i| codnl,tefrinoperator fermion the Accordingly, . |  σ 0 A silto o h obn ucin (A) junction. Corbino the for oscillation i G σ 2 y 12 A i.3 Fig. i. 1 0 e 1 ) γ 12 = V = −i 1 B e 2 nuisof units in h B G D sin and , 2 | = eV φ ψ V + i G hr h hs shift phase the where , = ihapa-ovle ampli- peak-to-valley a with e D 2|. 2|/η/ |η/ + G |1 G s 2 cos + 1 → wv uecnutron superconductor -wave /2h 0 i C ψ A γ saddo h bottom the on added is γ  NSLts Articles Latest PNAS sin 2 D † 0 i and e B −γ → l µ ψ e i hrfr,teoscil- the Therefore, . (1 G 2 φ 2 h1 i φ D otiue naddi- an contributes /h G 2 fteeg.La 3 Lead edge. the of ≤ µ G D φ ψ for A ihrsetto respect with 0 = |1 V G i D hw h MZM the shows 4 0 G φ ≤ . A nparticular, In . |ψ epciey (B) respectively. , ψ e B G h 0 2 sshown as 4) nalQAHI all on D n the and |)/2, i B → , i = i =  φ γ . | G 4 1 | N A + + φ f5 of 3 0 G 1 = i [3] [5] [4] φ η B γ = 0 = i, µ 2 ,

PHYSICS with φ0 being a fixed phase (SI Appendix). Experimentally, the highly irrelevant. Therefore, lin of γi in TSCs should be much gate voltage VG and thus φG can be well controlled by current longer than that of ψi in QAHIs. If the σ12 interference is to techniques at a high precision level (23). be observed, the sizes of the QAHI and TSC regions in the junction have to be within their inelastic scattering lengths lin , Decoherence respectively. There are mainly two effects contributing to the decoherence of chiral Majorana fermions. The first one is the nonmonochro- Conclusion maticity of the incident electron wave packet, which is charac- In summary, we have introduced the appealing possibility of terized by a momentum uncertainty ∆p ≈ 2π~/lW for a wave performing topological quantum computations via propaga- packet of width lW . In general, the (effective) path lengths of tions of 1D chiral Majorana fermion wave packets, which are the four chiral Majorana modes γi (1 ≤ i ≤ 4) in Fig. 3A may dif- physically equivalent to the braiding of MZMs. The Corbino fer by a length scale ∆L, and the σ12 oscillation is sharp only if junction above gives a minimal demonstration of single-qubit ∆p∆L < 2π~. As a demonstration, we numerically examine the quantum-gate operations with chiral Majorana fermions, and time evolution of an electron wave packet from lead 1 within the conductance of the junction provides a natural readout for an energy window vF [−∆p/2, ∆p/2] on a lattice model of the the final qubit states. Most importantly, this circumvents two Corbino junction and calculate σ12 (SI Appendix). Fig. 4A shows main experimental difficulties in quantum computations with σ12 as a function of VG /Eg for ∆p∆L/~ ≈ 0 and 18, respectively, MZMs: the braiding operation of MZMs and the readout of the where Eg is the QAHI bulk gap. The modulation of the σ12 qubit states. The high velocity of chiral Majorana edge modes 3 amplitude by VG is due to the effective change of ∆L as a result also makes the quantum gates 10 times faster than those of of the change in vF on the edge covered by voltage gate VG . other quantum computation schemes (31, 32). Furthermore, the 2 Fig. 4B shows the peak-to-valley amplitude y = ∆σ12/(e /h) as development of a single-electron source (33) makes the injec- a function of η = ∆p∆L/~, where we find the amplitude roughly tion and detection of a single-electron wave packet qubit on decays as y = | sin(η/2)/(η/2)|. In the experiments, the tempera- edges possible. Yet in the current stage we still face difficulties ture T yields a momentum uncertainty ∆p ≈ kB T /vF , where kB which are also encountered by the MZM quantum computa-

is the Boltzmann constant. For the Cr-doped (Bi,Sb)2Te3 thin- tion scheme: the error correction of the phase gate RφG (34, film QAHI with superconducting proximity studied in ref. 6, the 35) and nondemolitional four-Majorana implementation of the controlled not gate (14, 35, 36). If one could overcome these Fermi velocity is of order ~vF ∼ 3 eV·A˚ (24), and the temper- ature T reaches as low as 20 mK. This requires a path-length difficulties, one may in principle achieve universal quantum com- difference ∆L ∼ 100 µm or smaller, which is experimentally putation using chiral Majorana fermion devices, which would feasible (6, 25). have a high computation speed. Finally, we remark that the The second effect causing decoherence is the inelastic scatter- conductance oscillation in the Corbino junction, if observed, ing. The inelastic scattering of charged chiral fermions ψi mainly will also unambiguously prove the existence of quantum coher- originates from the electron– coupling, which yields an ent chiral Majorana fermions in the experiment (6, 22, 30, −p/2 37–39). inelastic scattering length lin ∝ T at temperature T (26– 2 28). For integer quantum Hall systems, lin exceeds 10 µm at ACKNOWLEDGMENTS. B.L. acknowledges the support of the Princeton T ∼ 20 mK (29), while lin is expected to be smaller for QAHI Center for Theoretical Science at . X.-Q.S. and S.-C.Z. (20). In contrast, since the electron–phonon coupling is odd acknowledge support from the US Department of Energy, Office of Basic Energy Sciences under Contract DE-AC02-76SF00515. A.V. acknowledges the under charge conjugation, the neutral chiral Majorana fermions Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum γi are immune to phonon coupling. Instead, their lowest-order Systems Initiative through Grant GBMF4302. X.-L.Q. acknowledges support 2 3 local interaction is of the form γi ∂x γi ∂x γi ∂x γi (30), which is from the David and Lucile Packard Foundation.

1. Qi XL, Zhang SC (2011) Topological insulators and superconductors. Rev Mod Phys 15. Alicea J (2012) New directions in the pursuit of Majorana fermions in solid state 83:1057–1110. systems. Rep Prog Phys 75:076501. 2. Qi XL, Hughes TL, Zhang SC (2010) Chiral topological superconductor from the 16. Aasen D, et al. (2016) Milestones toward Majorana-based . Phys quantum Hall state. Phys Rev B 82:184516. Rev X 6:031016. 3. Chung SB, Qi XL, Maciejko J, Zhang SC (2011) Conductance and noise signatures of 17. Karzig T, et al. (2017) Scalable designs for quasiparticle-poisoning-protected Majorana backscattering. Phys Rev B 83:100512. topological quantum computation with Majorana zero modes. Phys Rev B 95:235305. 4. Wang J, Zhou Q, Lian B, Zhang SC (2015) Chiral topological superconductor and half- 18. Liu CX, Qi XL, Dai X, Fang Z, Zhang SC (2008) Quantum anomalous Hall effect in integer conductance plateau from quantum anomalous Hall plateau transition. Phys Hg1−y Mny Te quantum wells. Phys Rev Lett 101:146802. Rev B 92:064520. 19. Yu R, et al. (2010) Quantized anomalous Hall effect in magnetic topological insulators. 5. Strubi¨ G, Belzig W, Choi MS, Bruder C (2011) Interferometric and noise signatures of Science 329:61–64. Majorana fermion edge states in transport experiments. Phys Rev Lett 107:136403. 20. Chang CZ, et al. (2013) Experimental observation of the quantum anomalous Hall 6. He QL, et al. (2017) Chiral Majorana fermion modes in a quantum anomalous Hall effect in a magnetic . Science 340:167–170. insulator–superconductor structure. Science 357:294–299. 21. Nayak C, Simon SH, Stern A, Freedman M, Das Sarma S (2008) Non-Abelian 7. Moore G, Read N (1991) Nonabelions in the fractional . Nucl Phys and topological quantum computation. Rev Mod Phys 80:1083–1159. B 360:362–396. 22. Bonderson P, Kitaev A, Shtengel K (2006) Detecting non-Abelian statistics in the ν = 8. Kitaev A (2006) Anyons in an exactly solved model and beyond. Ann Phys 321: 5/2 fractional quantum Hall state. Phys Rev Lett 96:016803. 2–111. 23. An S, et al. (2011) Braiding of Abelian and non-Abelian anyons in the fractional 9. Read N, Green D (2000) Paired states of fermions in two dimensions with breaking of quantum Hall effect. arXiv:1112.3400. parity and time-reversal symmetries and the fractional quantum Hall effect. Phys Rev 24. Liu CX, et al. (2010) Model Hamiltonian for topological insulators. Phys Rev B B 61:10267–10297. 82:045122. 10. Kitaev AY (2001) Unpaired Majorana fermions in quantum wires. Phys-Usp 44: 25. Fox EJ, et al. (2017) Part-per-million quantization and current-induced breakdown of 131–136. the quantum anomalous Hall effect. arXiv:1710.01850. 11. Lutchyn RM, Sau JD, Das Sarma S (2010) Majorana fermions and a topological 26. Thouless DJ (1977) Maximum metallic resistance in thin wires. Phys Rev Lett 39: phase transition in semiconductor-superconductor heterostructures. Phys Rev Lett 1167–1169. 105:077001. 27. Pruisken AMM (1988) Universal singularities in the integral quantum Hall effect. Phys 12. Ivanov DA (2001) Non-Abelian statistics of half-quantum vortices in p-wave Rev Lett 61:1297–1300. superconductors. Phys Rev Lett 86:268–271. 28. Huckestein B, Kramer B (1990) One-parameter scaling in the lowest Landau band: 13. Kitaev A (2003) Fault-tolerant quantum computation by anyons. Ann Phys 303: Precise determination of the critical behavior of the localization length. Phys Rev Lett 2–30. 64:1437–1440. 14. Alicea J, Oreg Y, Refael G, von Oppen F, Fisher MPA (2011) Non-Abelian statistics 29. Koch S, Haug RJ, Klitzing Kv, Ploog K (1991) Size-dependent analysis of the metal- and topological processing in 1D wire networks. Nat Phys 7: insulator transition in the integral quantum Hall effect. Phys Rev Lett 67:883– 412–417. 886.

4 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1810003115 Lian et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 4 odro ,Cak J aa ,Stne 21)Ipeetn rirr phase arbitrary Implementing (2010) K Shtengel C, Nayak DJ, Clarke P, Bonderson 34. superconducting a in qubits logical Building (2017) M Steffen JM, Chow JM, Gambetta 32. 3 F 33. charge with modes edge fermion Majorana H neutral Probing 31. (2009) CL Kane L, Fu 30. ine al. et Lian ae ihIiganyons. Ising with gates 316:1169–1172. system. computing quantum 469:155–203. transport. v ,e l 20)A ndmn oeetsnl-lcrnsource. single-electron coherent on-demand An (2007) al. et G, eve ` fnrH osC lt 20)Qatmcmuigwt rpe ions. trapped with computing Quantum (2008) R Blatt C, Roos H, affner ¨ hsRvLett Rev Phys 102:216403. hsRvLett Rev Phys p unu Inf Quantum npj 104:180505. 3:2. hsRep Phys Science 9 iso ,Aheo R(00 hoyo o-bla ar-eo nefrmtyin interferometry Fabry-Perot non-Abelian of Theory (2010) AR Akhmerov J, non-Abelian Nilsson the probe 39. to experiments Proposed (2006) BI Halperin interferometry A, detected Stern Electrically (2009) 38. CWJ Beenakker J, Nilsson AR, Akhmerov 37. the gates with Clifford computation quantum ideal Universal (2006) with S computation Bravyi quantum 36. Universal (2005) A Kitaev S, Bravyi 35. oooia insulators. topological state. Hall quantum insulator. topological a in fermions Majorana of state. Hall ancillas. noisy and hsRvA Rev Phys hsRvA Rev Phys hsRvLett Rev Phys 73:042313. hsRvB Rev Phys 71:022316. 81:205110. 96:016802. hsRvLett Rev Phys NSLts Articles Latest PNAS ν = rcinlquantum fractional 5/2 102:216404. | ν = f5 of 5 5/2

PHYSICS