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# Topological quantum computation based on chiral Majorana fermions

Contributed by Shou-Cheng Zhang, September 5, 2018 (sent for review June 11, 2018; reviewed by Eduardo Fradkin, Naoto Nagaosa, and Fuchun Zhang)

## Significance

We propose a platform of quantum computation using the chiral Majorana fermions on the edges of topological materials. The quantum gates are naturally accomplished by the propagation of chiral Majorana fermions. If realized, its computation speed can be

## Abstract

The chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain 2D topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of a quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, the Majorana zero mode in the bulk of the corresponding topological matter, is known to be applicable in topological quantum computations. Here we show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes and propose a platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can use quantum coherent chiral Majorana fermions to implement the Hadamard gate and the phase gate, and the junction conductance yields a natural readout for the qubit state.

The chiral Majorana fermion, also known as the Majorana–Weyl fermion, is a massless fermionic particle being its own antiparticle proposed long ago in theoretical physics. The simplest chiral Majorana fermion is predicted in 1D space, where it propagates unidirectionally. In condensed-matter physics, 1D chiral Majorana fermions can be realized as quasiparticle edge states of a 2D topological state of matter (1). A celebrated example is the _{2}Te_{3} thin-film QAHI system in proximity with the Nb superconductor (6). The chiral Majorana fermion could also arise in the Moore–Read state of the fractional quantum Hall effect (7) and topologically ordered states of spin systems (8).

A closely related concept, Majorana zero modes (MZMs), which emerge in the bulk vortices of a

In this paper, we propose a platform to implement topologically protected quantum gates at mesoscopic scales, which uses propagation of chiral Majorana fermions with purely electrical manipulations instead of MZMs.

## Chiral Majorana Fermion Qubits

The main goal of our proposal is to show that the chiral Majorana fermion edge state of the TSC can be used to realize non-Abelian quantum gate operations on electron states, even if there is no non-Abelian anyon traveling along the edge. Since our proposal is closely related to the braiding of MZMs in vortices of the *A*. Each vortex supports a single MZM *A*, we have

The device we propose to study is a 2D QAHI–TSC–QAHI junction predicted in refs. 3 and 4. As shown in Fig. 1*B*, the junction consists of two QAHIs (18⇓–20) of Chern number *B*, which are related to the charged chiral fermion modes on the QAHI edges as

Our key observation is that the same kind of partner switch of Majorana fermions as that of the vortex braiding occurs in this device between incoming and outgoing electrons. An incoming electron from lead A becomes a nonlocal fermion simultaneously on the two edges of the TSC described by

To be more specific and to make a connection with quantum computation, consider the low-current limit *Z* gate Z as shown in Fig. 1*C*; namely, *A* and *B* (*C* and *D*) behave effectively as a single qubit, and we can regard qubit *A* (*C*) as the data qubit, while qubit *B* (*D*) is a correlated ancilla qubit.

For an electron incident from lead 1 represented by initial state *SI Appendix*) that the entanglement entropy between left and right halves of the junction divided by the dashed line in Fig. 2*A* increases by *A*), where the entanglement entropy *B*, after an electron is injected from lead 1 above the Fermi sea. More details of this calculation are provided in *SI Appendix*. Since *B*), we are in fact identifying the charge basis of final qubit C (D) with that of initial qubit A (B). Accordingly, the conductance

As we have discussed, the above process is topologically equivalent to fusion and braiding of four vortex operators in the TSC bulk (*SI Appendix*) (21, 22). More concretely, when the electron of an incident state *SI Appendix*), so the two processes are topologically equivalent.

## A Testable Quantum Gate

The conductance

To confirm whether the system as a quantum gate is coherent, we propose to implement a Corbino geometry QAHI–TSC–QAHI–TSC junction as shown in Fig. 3*A* and measure the conductance *A*.

The gate voltage

If we regard the charged chiral edge modes of QAHI region I (*B*, with a total unitary evolution *C* and *D* shows the MZM braiding process that results in the same non-Abelian gate as the

So far we have assumed chemical potential *SI Appendix*). Experimentally, the gate voltage

## Decoherence

There are mainly two effects contributing to the decoherence of chiral Majorana fermions. The first one is the nonmonochromaticity of the incident electron wave packet, which is characterized by a momentum uncertainty *A* may differ by a length scale *SI Appendix*). Fig. 4*A* shows *B* shows the peak-to-valley amplitude _{2}Te_{3} thin-film QAHI with superconducting proximity studied in ref. 6, the Fermi velocity is of order

The second effect causing decoherence is the inelastic scattering. The inelastic scattering of charged chiral fermions

## Conclusion

In summary, we have introduced the appealing possibility of performing topological quantum computations via propagations of 1D chiral Majorana fermion wave packets, which are physically equivalent to the braiding of MZMs. The Corbino junction above gives a minimal demonstration of single-qubit quantum-gate operations with chiral Majorana fermions, and the conductance of the junction provides a natural readout for the final qubit states. Most importantly, this circumvents two main experimental difficulties in quantum computations with MZMs: the braiding operation of MZMs and the readout of the qubit states. The high velocity of chiral Majorana edge modes also makes the quantum gates

## Acknowledgments

B.L. acknowledges the support of the Princeton Center for Theoretical Science at Princeton University. X.-Q.S. and S.-C.Z. acknowledge support from the US Department of Energy, Office of Basic Energy Sciences under Contract DE-AC02-76SF00515. A.V. acknowledges the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems Initiative through Grant GBMF4302. X.-L.Q. acknowledges support from the David and Lucile Packard Foundation.

## Footnotes

↵

^{1}B.L. and X.-Q.S. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: sczhang{at}stanford.edu.

Author contributions: X.-L.Q. and S.-C.Z. designed research; B.L., X.-Q.S., and A.V. performed research; and B.L., X.-Q.S., A.V., X.-L.Q., and S.-C.Z. wrote the paper.

Reviewers: E.F., University of Illinois at Urbana–Champaign; N.N., The University of Tokyo and Riken Center for Emergent Matter Science; and F.Z., Kavli Institute for Theoretical Physics China.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1810003115/-/DCSupplemental.

Published under the PNAS license.

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