Non-Abelian Braiding of Chiral Majorana Fermions

Non-Abelian Braiding of Chiral Majorana Fermions

Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi,
Xiao-Liang Qi, Shou-Cheng Zhang

Princeton Center for Theoretical Science, Princeton University,
Princeton, New Jersey 08544-0001, USA
Stanford Center for Topological Quantum Physics, Stanford University,
Stanford, California 94305-4045, USA
Department of Physics, McCullough Building, Stanford University,
Stanford, California 94305-4045, USA
School of Natural Sciences, Institute for Advanced Study,
Princeton, New Jersey 08540, USA

Correspondence author: B.L. email: biao@princeton.edu
S.-C.Z. email: sczhang@stanford.edu
These authors contribute to the work equally.

Chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain two-dimensonal topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, Majorana zero mode in the bulk of the corresponding topological matter, is known to be a non-Abelian anyon useful in topological quantum computations. Here we show that the propagation of chiral Majorana fermions can naturally lead to non-Abelian braiding just like the Majorana zero modes, and propose a new platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can braid quantum coherent chiral Majorana fermions, implementing the Hadamard gate and the phase gate, where the junction conductance yields a natural readout for the qubit state.

Chiral Majorana fermion, also known as 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 1 dimensional (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 (?). A celebrated example is the chiral topological superconductor (TSC), which carries a Bogoliubov-de Gennes (BdG) Chern number , and can be realized from a quantum anomalous Hall insulator (QAHI) with Chern number in proximity with an -wave superconductor (?, ?, ?). A QAHI-TSC-QAHI junction implemented this way is predicted to exhibit a half quantized conductance plateau induced by chiral Majorana fermions (?, ?), which has been recently observed in the Cr doped (Bi,Sb)Te thin film QAHI system in proximity with Nb superconductor (?). Chiral Majorana fermion could also arise in the Moore-Read state of fractional quantum Hall effect (?) and topologically ordered states of spin systems (?).

A closely related concept, Majorana zero modes (MZMs) which emerge in the bulk vortices of a TSC (?) or at the endpoints of a 1D -wave TSC (?, ?), are known to obey non-Abelian braiding statistics and can be utilized in fault-tolerant topological quantum computations (?, ?, ?, ?, ?, ?). Despite the theoretical progress made during the past decade on employing MZMs in universal quantum computation (?, ?, ?, ?), due to the localized and point-like nature of MZMs, all existing proposed architectures inevitably require nano-scale design and control of the coupling among MZMs. Non-abelian braiding is an essential step towards topological quantum computing, however, it has not yet been experimentally demonstrated with MZMs. In this paper, we propose a novel platform to implement topologically protected quantum gates at mesoscopic scales, which utilizes non-Abelian braiding of chiral Majorana fermions with purely electrical manipulations instead of MZMs.

To begin with, we first show the half quantized two-terminal conductance plateau of a 2D QAHI-TSC-QAHI junction predicted in Refs. (?, ?) can be interpreted as non-abelian braiding of chiral Majorana fermions. As shown in Fig. 1A, the junction consists of two QAHI (?, ?, ?) of Chern number and a chiral TSC of BdG Chern number . The conductance is measured between metallic leads and by driving a current , where no current flows through lead which grounds the TSC. Each edge between the chiral TSC and the vacuum or a QAHI hosts a chiral Majorana fermion edge mode governed by a Hamiltonian , where is the Majorana operator satisfying and the anti-commutation relation , is the Fermi velocity, and is the coordinate of the 1D edge. In contrast, each edge between a QAHI and the vacuum hosts a charged chiral fermion (electron) edge mode with a Hamiltonian , where and are the annihilation and creation operators of the edge fermion, and we have assumed chemical potential for the moment. By defining two Majorana operators and , one can rewrite as , which implies a charged chiral fermion mode is equivalent to two chiral Majorana fermion modes. As a result, the edge states of the junction consist of four chiral Majorana fermion modes () as shown in Fig. 1A, which are related to the charged chiral fermion modes on the QAHI edges as , , and  (?).

Figure 1: Braiding of chiral Majorana fermions in a QAHI-TSC-QAHI junction. (A) Paths of the four chiral Majorana edge modes in the junction. (B) Topology of the world lines of wave packets of chiral Majorana fermions , whose spatial projections are exactly the paths of in the junction. (C) In the braiding of , one can regard the left two Majoranas as the data qubit, and the right two as an ancilla qubit. (D) Such a four-Majorana braiding is equivalent to a Hadamard gate followed by a Pauli-Z gate . (E) Evolution of entanglement entropy between left and right halves of the junction (divided by dashed line in Fig. 1A) with time (arbitrary unit) after an electron above the fermi sea is injected from lead , where is the entanglement entropy of the fermi sea.

Our key observation is the paths of four chiral Majorana modes in Fig. 1A are topologically equivalent to an exchange braiding of and as shown in Fig. 1B and Fig. 1C, which takes and  (?). This braiding turns the two incident charged chiral fermion modes and into the two outgoing modes and , and is non-Abelian. To be more specific and to make a connection with quantum computation, consider the low current limit where electrons are injected from lead one by one, each of which occupies a travelling wave packet state of . The occupation number or of such a wave packet state then defines a qubit with basis and . Similarly, we can define the qubits , and for , and , respectively. At each moment of time, the real and imaginary parts of the fermionic annihilation operator of each wave packet state define two self-conjugate Majorana operators localized at the wave packet. As time increases, these local Majorana operators move along the paths of , and yield four braiding world lines as shown in Fig. 1B, thus naturally obey the non-Abelian braiding statistics of MZMs. In the evolution of the incident electrons, qubits and span the Hilbert space of the initial state , while qubits and form the Hilbert space of the final state . The braiding of with then leads to a unitary evolution

(1)

Note that the fermion parity is conserved in the unitary evolution. If we define a new qubit in the odd fermion parity subspace as initially and at the final time, the above unitary evolution is exactly a topologically protected Hadamard gate followed by a Pauli-Z gate as shown in Fig. 1D, namely, , where

(2)

The same conclusion holds for the even fermion parity subspace. Therefore, the two qubits A and B (C and D) behaves 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 represented by initial state , the junction turns it into a final state . This implies (?) that the entanglement entropy between left and right halves of the junction divided by the dashed line in Fig. 1A increases by . Indeed, this is verified by our numerical calculation in a lattice model of the junction (?), where the entanglement entropy increases with time as shown in Fig. 1E after an electron is injected from lead above the fermi sea. Since and propagate into leads and , respectively, the electron has probability to return to lead , and probability to tunnel into lead . This yields (?) a half-quantized two-terminal conductance . Since lead (lead ) connects () with () (Fig. 1C), we are in fact identifying the charge basis of final qubit () with that of initial qubit (). Accordingly, the conductance provides a natural measurement of the overlap probability between and under this common basis, namely, .

However, the conductance of such a junction cannot tell whether chiral Majorana fermions are coherent or not during the propagation. For instance, if a random phase factor is introduced in the propagation of and , a pure initial state will evolve into a mixed final state with a density matrix , while the conductance remains .

To tell whether the system is coherent, we propose to implement a Corbino geometry QAHI-TSC-QAHI-TSC junction as shown in Fig. 2A, and measure the conductance between lead and lead . The junction can be realized by attaching a fan-shaped -wave superconductor on top of a QAHI Corbino ring, with a proper out-of-plane magnetic field driving the two regions II and IV into the TSC phase (?, ?). A voltage gate is added on the bottom edge of QAHI region III covering a length of the edge. Lead grounds the superconductor and has no current passing through. At zero gate voltage, the edge states of the Corbino junction are four chiral Majorana edge states () as shown in Fig. 2A.

Figure 2: Braiding of chiral Majorana fermions in the QAHI-TSC-QAHI-TSC Corbino junction. (A) The Corbino junction consists of a Corbino QAHI ring with a fan-shaped -wave superconductor on top of it which drives regions II and IV into TSC, and a voltage gate is added at the bottom edge. (B) Such a junction is equivalent to a series of single-qubit quantum gates , where is a phase gate controlled by . (C) The braiding of chiral Majorana fermions in the junction at . (D) The braiding of chiral Majorana fermions at .

The gate voltage on the bottom edge of region III behaves as a chemical potential term for in a length . In the language of quantum computation, this induces a phase gate

(3)

acting on the corresponding qubit , where the phase shift is tunable via . Accordingly, the fermion operator undergoes a unitary evolution , which is equivalent to an exchange braiding of Majorana modes and when .

If we regard the charged chiral edge modes of QAHI region I ( and ) as the data qubit, and those of QAHI region III ( and ) as the ancilla qubit, the junction can be viewed as a series of quantum gates as shown in Fig. 2B, with a total unitary evolution . Fig. 2C and 2D show the Majorana braiding pictures corresponding to and , respectively. For an electron incident from lead represented by the initial state , the finial state is

(4)

Therefore, the two-terminal conductance of this Corbino junction is

(5)

which oscillates as a function of with a peak-to-valley amplitude . In contrast, if the system loses coherence completely, the final state will be the maximally mixed state described by density matrix , and the conductance will constantly be . Therefore, the oscillation amplitude of measures the coherence of the chiral Majorana fermions in the junction.

So far we have assumed chemical potential on all QAHI edges except the interval covered by voltage gate. In general, is nonzero, and is nonuniform along the QAHI edges when there are disorders. Such a nonzero landscape of contributes an additional phase gate, which leads to a phase shift , with being a fixed phase (?).

Figure 3: Numerically calculated oscillation for the Corbino junction. (A) calculated for and as a function of , respectively. (B) The peak-to-valley amplitude of in units of with respect to , which is roughly given by .

There are mainly two effects contributing to the decoherence of chiral Majorana fermions. The first is the non-monochromaticity of the incident electron wave packet, which is characterized by a momentum uncertainty for a wave packet of width . In general, the (effective) path lengths of the four chiral Majorana modes in Fig. 2A may differ by a length scale , and the oscillation is sharp only if . As a demonstration, we numerically examine the time evolution of an electron wave packet from lead within an energy window on a lattice model of the Corbino junction and calculate  (?). Fig. 3A shows as a function of for and , respectively, where is the QAHI bulk gap. The modulation of the amplitude by is due to the effective change of as a result of the change in on the edge covered by voltage gate . Fig. 3B shows the peak-to-valley amplitude as a function of , where we find the amplitude roughly decays as . In the experiments, the temperature yields a momentum uncertainty , where is the Boltzmann constant. For the Cr-doped (Bi,Sb)Te thin film QAHI with superconducting proximity studied in Ref. (?), the Fermi velocity is of order eV (?), and the temperature reaches as low as mK. This requires a path length difference m or smaller, which is experimentally feasible (?, ?).

The second effect causing decoherence is the inelastic scattering. The inelastic scattering of charged chiral fermions mainly originates from the electron-phonon coupling, which yields an inelastic scattering length at temperature  (?, ?, ?). For integer quantum Hall systems, exceeds m at mK (?), while is expected to be smaller for QAHI (?). In contrast, since the electron-phonon coupling is odd under charge conjugation, the neutral chiral Majorana fermions are immune to phonon coupling. Instead, their lowest order local interaction is of the form  (?), which is highly irrelevant. Therefore, of in TSCs should be much longer than that of in QAHIs. If the interference is to be observed, the sizes of the QAHI and TSC regions in the junction have to be within their inelastic scattering lengths , respectively.

In summary, we have introduced the appealing possibility of performing topological quantum computations using propagations of 1D chiral Majorana fermions, which are intrinsically equivalent to the non-Abelian 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 conductance oscillation in the Corbino junction, if observed, will also unambiguously prove the existence of quantum coherent chiral Majorana fermions in the experiment (?, ?, ?). Finally, if error corrections of the phase gate  (?, ?) and nondemolitional four-Majorana charge measurement for implementing the controlled NOT gate (?, ?, ?) can be performed in this scheme, it will be highly promising to achieve universal quantum computation in devices hosting chiral Majorana fermions.

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Acknowledgments

B.L. acknowledges the support of Princeton Center for Theoretical Science at Princeton University. X.-Q.S. and S.-C.Z. acknowledges 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 EPiQS Initiative through Grant GBMF4302. X.-L.Q. acknowledges support from David and Lucile Packard Foundation.

Supplementary Material

The supplementary material is organized as follows. In Sec. 1 we show the 2D lattice Hamiltonians of QAHI and TSC we use for calculations of entanglement entropy change in the QAHI-TSC-QAHI junction and conductance in the Corbino junction. Sec. 2 gives the details of entanglement entropy numerical calculation for a QAHI-TSC-QAHI junction lattice model during the evolution of an incident electron above the fermi sea. Sec. 3 reviews the generalized Landauer-Büttiker formula for two-terminal conductance of a superconducting junction, while Sec. 4 shows the numerical calculation for oscillation of a Corbino junction in a 2D lattice as a function of the gate voltage . Sec. 5 shows that the nonzero chemical potential on QAHI edges induces a phase shift to in the formula of in the Corbino junction. Finally, in Sec. 6, we provide a Bloch sphere illustration of the single qubit quantum gate that we propose to implement by the Corbino junction.

Appendix A Model Hamiltonian for simulation

In this section, we present the 2D lattice model Hamiltonian that we will use for later numerical calculations. The structures that we study in the main text consists of a quantum anomalous Hall insulator (QAHI), where we add s-wave superconductivity pairing to induce chiral topological superconductor (TSC) or add voltage gate to change the chemical potential of edge states. The lattice model Hamiltonian for QAHI we adopt is as follows:

(6)

where are fermion operators in momentum space and , and are Pauli matrices. We work in the dimensionless unit with lattice constant and set , , and . The band parameters are chosen such that the the valence band has a non-trivial Chern number and therefore describe a QAHI. In the calculation for the QAHI-TSC-QAHI junction or the Corbino junction, we write the above Hamiltonian in the real space with an open boundary condition at the edges between the junction and the vacuum.

The TSC is realized by adding an -wave superconductivity pairing into the Hamiltonian Eq.(6), where are fermion operators in the real space. We choose to set in the superconducting regions, which drives the regions into a TSC. We model the static electrical potential induced by voltage gate with a chemical potential term , where inside the gated region outside. The full model Hamiltonian can be summarized as

(7)

In all simulations, the model Hamiltonian will be kept at the fixed parameters where , , , , and . Several useful quantities are the Fermi velocity of the edge modes, which is equal to at zero chemical potential. The energy gap is for the QAHI regions, and is in the TSC regions.

Appendix B Entanglement entropy during the braiding of

In this section, we discuss the entanglement entropy change of the QAHI-TSC-QAHI junction during the propagation of an incident electron from lead . In the case of the Majorana zero mode(MZM), if one splits a system into two subsystems and its complement , the braiding of one MZM in subsystem with another MZM in subsystem creates an entanglement entropy for the subsystem . This is also expected to be true in our case, where MZMs are replaced by wave packets of chiral Majorana fermion. Indeed, a nonvanishing increment in the value of entanglement entropy is a generic signature of non-Abelian statistics as the initial state will mix with other degenerate states upon braiding of non-Abelian anyons, and as a result, the entanglement entropy associated with an entanglement cut that segregates the pair of anyons undergoing braiding will increase afterwards. On the contrary, for Abelian states, the entanglement entropy remains unchanged upon braiding of anyons. Thus, the evolution of entanglement entropy throughout the braiding operation is a powerful tool that can easily distinguish non-Abelian from Abelian statistics.

We design the Hamiltonian defined in Eq.(7) for a QAHI-TSC-QAHI junction on a lattice as shown in Fig. S1. The length of each QAHI region in direction is while the length of TSC region in x direction is . A cut along direction in the TSC region is made at a distance to the boundary of TSC and the left QAHI. We define subsystem as the subsystem to the left of the cut and we denote its compliment in Fig. S1. The entanglement entropy of subsystem is given by

(8)

where is the reduced density matrix of the quantum state of subsystem A. With the BdG Hamiltonian adopted, the system consists of non-interacting fermionic quasiparticles. We denote the annihilation operators of the BdG quasiparticle eigenstates as , . The many-particle state for the fermi sea of the system is then satisfying .

Figure S1: The geometry of a QAHI/TSC/QAHI junction. We align the QAHI regime, TSC regime and the other QAHI regime in the x direction. The length of each QAHI regime in x direction is while the length of TSC regime in x direction is . A cut along y direction at the TSC regime is made at a distance to the boundary of TSC and the left QAHI. We define subsystem as the subsystem to the left of the cut and we denote its compliment . For illustration purpose, we only label the compliment subsystem . The position of the initial wave packet is centered at a distance to the boundary of vacuum and the left QAHI. The simulation of Fig. 1E in the main text is run at the geometry parameters , , and .

We then consider the evolution of an electron wave packet state injected from lead , given by , where is a chosen creation operator of an electron wave packet at time located near lead on the QAHI edge, and is its time evolution. The wave packet is restricted within an energy window , which is smaller than the minimal bulk gap of the system.

The entanglement entropy of the noninteracting fermion states (i.e., Slater determinant states) and are given by (?, ?)

(9)

respectively, where and are eigenvalues of the correlation matrices defined as follows:

(10)

Here is the electron annihilation operator on site in the subsystem A, while , are the spin indices. The correlation matrix of the fermi sea can be calculated from the eigenstate operators . Once the commutators of with the , are determined, the correlation matrix of the wave packet state can be calculated based on , and the entanglement entropy can be calculated numerically.

We calculate the time evolution of the entanglement entropy using geometry parameters , , and . We set the wave packet to contain quasiparticle states in an energy window . The wave packet is created by projecting an electron wave packet onto the quasiparticle states in this energy window. Summary of the geometry parameters is given in Fig. S1, and the evolution of the entanglement entropy is plotted in Fig. 1E of the main text. We can clearly that after when the wave packet has left the TSC regime, the entanglement entropy increase of subsystem A is quantized at .

Appendix C Calculation of the two terminal conductance

In this supplementary section, we briefly review the calculation of the two terminal conductance for the Corbino junction. The two terminal conductivity from the lead 1 to the lead 2 can be obtained from the generalized Landauer-Buttiker formula (?):

(11)

where is the current flowing out of the lead , is the voltage of the lead , and , are the normal transmission and Andreev transmission probabilities from leads to (), while and are the normal reflection and Andreev reflection from the lead back to itself, respectively. As a consistency check, the conductance of the Corbino junction calculated this way should agree with our prediction in the main text based on chiral Majorana fermion braiding.

We simulate the time evolution of an electron wave packet initialized inside the lead 1 region using the Hamiltonian from Eq. (7). At the time when the wave packet reflects (transmits) to the lead 1 (lead 2) neighbourhoods, we stop the time evolution and compute the probability of reflection and transmission, namely , , and , from the wave function. Note that if we connect the electron source directly across leads 1 and 2, we also have an additional constrain:

(12)

From Eq. (11) and Eq. (12), we can then solve for two terminal conductivity .

Appendix D Decoherence effect from non-monochromaticity

In the main text, we have discussed the decoherence effect from the non-monochromaticity of the incident electron wave packet. The non-monochromaticity is described by the momentum uncertainty of the electron wave packet together with a length scale characterizes the length difference of the four chiral Majorana modes (). In this section, we shall discuss the precise definition of these parameters in simulation and the method to study the dependence of oscillation amplitude on them.

As shown in Fig. S2, we put the Corbino junction on a cylindrical lattice with left and right vertical dashed lines identified, which is equivalent to the Corbino geometry. We can consider an incident electron wave packet from the lead 1. In simulation, we obtain a wave packet of momentum uncertainty in the following way. We initialize an electron wave packet broader than . Then we project this wave function onto the energy eigenspace of Hamiltonian from Eq.(7) in the energy window and normalize the projected wave function as . We shall define as the initial electron wave packet with momentum uncertainty . Notice that this initial condition is slightly different from the calculation for entanglement entropy in section B because the negative energy state represents a hole of quasiparticle which is impossible to generate from ground state with no quasiparticles at zero temperature. Here we are considering the non-monochromaticity of electron wave packet from the finite temperature effect and this initial condition is physical.

A suitable perspect is to consider the electron wave packet as a superposition of wave packets of two Majorana fermions. Upon time evolution, the fate of the two Majorana fermions is either recombination to a particle/hole at the lead 1 or at the lead 2. For the process that the wave packet ends up back at the lead 1, the probability is contributed by two paths shown as two blue lines in Fig. S2. In a precise fashion, this can be interpretted as a interferometry of chiral Majorana fermions: the electron wave packet passes through a ”beam splitter” , travels through two arms as through the chiral Majorana mode and recombines at the lead 1. The length difference of the two arms of the interferometry is and we can expect the interference effect in the probability of propagating back to be measurable when . For the process that the wave packet transmits to the lead 2, similarly, the probability is contributed by two paths shown as two red lines in Fig. S2. The length difference of the two paths is and the condition for the interference is . For illustration purpose, we study the case when so that so that a unique length scale is defined.

Figure S2: Illustration of chiral Majorana interferometry: A band of QAHI with two TSC regimes induced by proximity to a s-wave superconductor. The lengths of of QAH regimes, TSC regimes and the voltage gate regime are denoted as , , and , respectively. If one consider an incident electron wave packet from the lead 1, we can decompose it into a superposition of two Majorana fermions. Two red lines are paths for those Majorana fermions to travel from the lead 1 to the lead 2 while two blue lines are paths for those Majorana fermions to travel back to the lead 1. The probability for a charge from the lead 1 to transmit/reflect is contributed by the red/blue paths. The path difference of two transmitted/reflected paths from the lead 1 is .

In simulation, we fix the geometry parameters at , and and vary from 0 to 30. For each , we initialize a wave packet at the lead 1 region with momentum uncertainty . We can simulate the time evolution of the wave packet and obtain as described in the previous section for from 0 to 1. At and , the dependence of on is shown in Fig. 3A in the main text with an oscillation feature. We can also observe similar oscillation for other and the peak-to-valley oscillation amplitude has a dependence on shown in Fig. 3A in the main text.

Appendix E Phase shift of due to nonzero chemical potential

In this section we discuss the phase shift of in the two terminal conductance of the Corbino junction due to chemical potential and static disorders on the QAHI edges. When the chemical potential on a QAHI edge is nonzero, the Hamiltonian of the corresponding charged chiral edge state is

(13)

Solving the Shrödinger equation yields an electron wave function

(14)

where is an arbitrary function of . Therefore, a chiral fermion wave packet accumulates a phase after propagation from to which is fixed by the function of chemical potential . In contrast, a chiral Majorana fermion always has zero chemical potential as ensured by the particle-hole symmetry of TSC.

In the Corbino junction as shown in Fig. 2A of the main text, assume charged chiral state () accumulates an additional chemical potential induced phase during propagation on the corresponding QAHI edge. In the odd fermion parity subspace , the total unitary transformation becomes

(15)

which is equivalent to insertion of two additional phase gates. As a result, an initial state transforms into a final state

(16)

where . Therefore, the conductance becomes

(17)

Appendix F Bloch sphere illustration of the Corbino junction

In this section, we present an illustration for the time evolution of the qubit on its Bloch sphere after injecting an electron wave packet from lead 1. As shown in Fig. 2 in the main text, the charged chiral fermion modes on the QAHI edges are labeled as , , and . If we regard the charged chiral edge modes of QAHI region I ( and ) as the data qubit, and those of QAHI region III ( and ) as the ancilla qubit, the junction can be viewed as a series of quantum gates as shown in Fig. 2B in the main text, with a total unitary evolution . The initial state of the wave packet is occupying a fermion state. The electron wave packet will then approach the TSC II region and leave this region as chiral fermion mode or . If we define the qubit state as before the wave packet approaches the TSC II region and after the wave packet leaves the TSC II region, the time evolution of such a process can be viewed as the operator acting on a qubit which is initialized at state at north pole of its Bloch sphere. The operator is a rotation of along axis and upon the operation, the qubit rotates to direction on the Bloch sphere. After leaving the TSC II region, the wave packet may enter the voltage gate and the effect of voltage gate is to contribute additional phase to state while to state and therefore is a rotation of along z axis in the Bloch sphere of qubit . Before reaching leads, the wave packet must also approach the TSC IV region and leave this region as chiral fermion mode or . The time evolution of such a process can be viewed as the operator rotating the qubit by along y axis on the Bloch sphere if we define the qubit state in as before the wave packet approaches the junction and after the wave packet leaves the junction. From Fig. S3(A-D), we can clearly see the time evolution of the qubit on the Bloch sphere of the process that we have described in this paragraph and the final state at polar angle and azimuthal angle on the Bloch sphere. This is an illustrative derivation of Eq. (4) in the main text.

Figure S3: The time evolution of a qubit. (A). The electron wave packet is ejected from the lead 1 and occupies one state of fermion. The qubit at this time is initialized at . (B). The effect of the QAHI I–TSC II–QAHI III junction is a rotation of along y axis on the Bloch sphere if we define the qubit state as before the wave packet approaches the junction and after the wave packet leaves the junction. (C). The effect of the voltage gate is a rotation of along z axis state on the Bloch sphere of qubit . (D). The effect of the QAHI III–TSC IV–QAHI I junction is a rotation of along y axis on the Bloch sphere if we define the qubit state as before the wave packet approaches the junction and after the wave packet leaves the junction.
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