Topological quantum computation based on chiral Majorana fermions Biao Liana,b,1, Xiao-Qi Sunb,c,1, Abolhassan Vaezib,c, Xiao-Liang Qib,c,d, and Shou-Cheng Zhangb,c,2 aPrinceton Center for Theoretical Science, Princeton University, Princeton, NJ 08544-0001; bStanford Center for Topological Quantum Physics, Stanford University, Stanford, CA 94305-4045; cDepartment of Physics, Stanford University, Stanford, CA 94305-4045; and dSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 Contributed by Shou-Cheng Zhang, September 5, 2018 (sent for review June 11, 2018; reviewed by Eduardo Fradkin, Naoto Nagaosa, and Fuchun Zhang) The chiral Majorana fermion is a massless self-conjugate fermion Chiral Majorana Fermion Qubits which can arise as the edge state of certain 2D topological The main goal of our proposal is to show that the chiral Majorana matters. It has been theoretically predicted and experimentally fermion edge state of the TSC can be used to realize non-Abelian observed in a hybrid device of a quantum anomalous Hall quantum gate operations on electron states, even if there is no insulator and a conventional superconductor. Its closely related non-Abelian anyon traveling along the edge. Since our proposal cousin, the Majorana zero mode in the bulk of the correspond- is closely related to the braiding of MZMs in vortices of the ing topological matter, is known to be applicable in topological p + ip TSC, we begin by reviewing this process, as illustrated in quantum computations. Here we show that the propagation of Fig. 1A. Each vortex supports a single MZM γi , and thus two vor- chiral Majorana fermions leads to the same unitary transfor- tices together define two quantum states of a fermion degree of mation as that in the braiding of Majorana zero modes and freedom. The MZM operators satisfy the anticommutation rela- propose a platform to perform quantum computation with chi- 1 tion fγi , γj g = δij . If we define f12 = 2 (γ1 + iγ2) as a complex ral Majorana fermions. A Corbino ring junction of the hybrid y fermion number, the two states are labeled by f f12 = 0, 1, which device can use quantum coherent chiral Majorana fermions 12 corresponds to iγ1γ2 = −1, +1, respectively. When two vortices to implement the Hadamard gate and the phase gate, and are exchanged, the corresponding MZMs also are exchanged. the junction conductance yields a natural readout for the PHYSICS In the process in Fig. 1A, we have γ2 ! γ3, γ3 ! −γ2. The qubit state. relative minus sign is necessary to preserve the fermion num- ber parity iγ2γ3 of this pair. As a consequence, the eigenstates quantum computing j topological j Majorana of iγ1γ2 and iγ3γ4 evolve to eigenstates of iγ1γ3 and −iγ2γ4, which are entangled states when written in the original basis of iγ1γ2 and iγ3γ4. For example, the state j1i12j0i34 evolves into he chiral Majorana fermion, also known as the Majorana– 1 p j0i j1i + j1i j0i . Since the vortices have long-range TWeyl fermion, is a massless fermionic particle being its own 2 12 34 12 34 antiparticle proposed long ago in theoretical physics. The sim- interaction, the Abelian phase during the braiding may not be plest chiral Majorana fermion is predicted in 1D space, where well defined, but the non-Abelian unitary operation is robust it propagates unidirectionally. In condensed-matter physics, 1D (12). From the reasoning presented above, one can see that the chiral Majorana fermions can be realized as quasiparticle edge non-Abelian gate during MZM braiding is a direct consequence states of a 2D topological state of matter (1). A celebrated of exchanging MZMs γ2, γ3. The resulting gate must be non- example is the p + ip chiral topological superconductor (TSC), Abelian because iγ1γ2 anticommutes with iγ1γ3. Therefore, the which carries a Bogoliubov–de Gennes (BdG) Chern num- same non-Abelian gate can be realized by another physical pro- ber N = 1 and can be realized from a quantum anomalous cess that exchanges Majorana fermions, even if no braiding of the Hall insulator (QAHI) with Chern number C = 1 in proxim- non-Abelian anyon is involved. In the following, we show how ity with an s-wave superconductor (2–5). A QAHI–TSC–QAHI to obtain a realization of the same gate by making use of chi- junction implemented this way is predicted to exhibit a half- ral Majorana fermion edge states of the TSC and complex chiral quantized conductance plateau induced by chiral Majorana fermion edge states of the QAHI. fermions (3, 4), which has been recently observed in the Cr- doped (Bi,Sb)2Te3 thin-film QAHI system in proximity with Significance the Nb superconductor (6). The chiral Majorana fermion could also arise in the Moore–Read state of the fractional quan- We propose a platform of quantum computation using the tum Hall effect (7) and topologically ordered states of spin chiral Majorana fermions on the edges of topological mate- systems (8). rials. The quantum gates are naturally accomplished by the A closely related concept, Majorana zero modes (MZMs), propagation of chiral Majorana fermions. If realized, its com- which emerge in the bulk vortices of a p + ip TSC (9) or at putation speed can be 103 faster than the currently existing the endpoints of a 1D p-wave TSC (10, 11), are known to obey quantum computation schemes. non-Abelian braiding statistics and can be used in fault-tolerant topological quantum computations (12–17). Despite the theo- Author contributions: X.-L.Q. and S.-C.Z. designed research; B.L., X.-Q.S., and A.V. retical progress made during the past decade on using MZMs performed research; and B.L., X.-Q.S., A.V., X.-L.Q., and S.-C.Z. wrote the paper.y in universal quantum computation (14–17), due to the localized Reviewers: E.F., University of Illinois at Urbana–Champaign; N.N., The University of Tokyo and point-like nature of MZMs, all existing proposed archi- and Riken Center for Emergent Matter Science; and F.Z., Kavli Institute for Theoretical tectures inevitably require nanoscale design and control of the Physics China.y coupling among MZMs. As an essential step toward topologi- The authors declare no conflict of interest. y cal quantum computing, the braiding of MZMs has not yet been Published under the PNAS license.y experimentally demonstrated. 1 B.L. and X.-Q.S. contributed equally to this work.y In this paper, we propose a platform to implement topologi- 2 To whom correspondence should be addressed. Email: [email protected] cally protected quantum gates at mesoscopic scales, which uses This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. propagation of chiral Majorana fermions with purely electrical 1073/pnas.1810003115/-/DCSupplemental.y manipulations instead of MZMs. www.pnas.org/cgi/doi/10.1073/pnas.1810003115 PNAS Latest Articles j 1 of 5 Downloaded by guest on September 27, 2021 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 electrons. 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 A. The occupation number 0 or time 1 of such a fermion wave packet state then defines a qubit A with basis j0Ai and j1Ai. Similarly, we can define the qubits B, γ1 γ2 γ3 γ4 C , and D for B , C , and D , 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 j i i, while qubits C and D form the Hilbert space of the final state j 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 0 1 0 10 1 ψ 2 ψ j0C 0D i 1 0 0 1 j0A0B i A D 1 lead 3 ψ B j0C 1D i C B 0 1 1 0 CB j0A1B i C i = p : [1] @ j1C 0D i A 2 @ 0 −11 0 A@ j1A0B i A j1C 1D i −10 01 j1A1B 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 (j0i, j1i) in the odd fermion parity subspace as occurs between incoming electrons from leads A and B and outgoing elec- (j0A1B i , j1A0B i) initially and (j0C 1D i, j1C 0D i) at the final time, trons in leads C and D.