Abstract
The realization of Majorana zero modes is in the centre of intense theoretical and experimental investigations. Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan–Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a threesite chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.
Introduction
Majorana zero modes (MZMs) give rise to a nontrivial Hilbert space encoded in their degenerate states. When two MZMs are exchanged, their degenerate subspace can be transformed by a unitary operator that signals their nonAbelian statistics^{1}. In addition, quantum information encoded in the degenerate subspace of MZMs is naturally immune to local errors. The degeneracy of these states is protected by the fermion parity symmetry; thus, it cannot be lifted by any local symmetrypreserving perturbation. These unique characteristics make MZMs of interest to fundamental physics, whereas they can potentially provide novel and powerful methods for quantum information storing and processing^{2}.
The investigation of MZMs has attracted great attention. Rapid theoretical developments^{3,4,5} have greatly simplified the technological requirements and made it possible to experimentally observe MZMs. A few positive signatures of the formation of MZMs have been reported recently in solidstate systems^{6,7,8,9,10,11}. Nevertheless, the central characteristic of exotic statistics requires the trapping and transport of MZMs, to perform their exchange. This level of manipulation of MZMs is beyond our current control of solidstate systems.
In the following, we overcome this hurdle by taking advantage of the quantum simulation approach^{12}: although the simulated system is not experimentally accessible with the current technology, the quantum simulator and its measurement results provide information about the simulated system. We use a photonic quantum simulator to investigate the exchange of MZMs supported in the onedimensional Kitaev Chain Model (KCM). In principle, we can directly map the Fock space of the Majorana system to the space of the quantum simulator. Nevertheless, to make our approach easily accessible to other scalable systems, such as trapped ions^{13} or Josephson junctions^{14}, where spin modes are directly available, we divide the map into two consecutive steps. First, we perform the mapping of the Majorana system to the spin1/2 system via the Jordan–Wigner (JW) transformation^{15,16,17,18}. Next, we perform the mapping of the spin system to the spatial modes of single photons. The JW transformation^{19} is a nonlocal transformation that relates the states and evolutions of the topological superconducting Kitaev chain to the states and evolutions of a spin1/2 chain. The corresponding spin system can then be simulated by our optical simulator. The simulated system has only two Majoranas (separated by a gap from all other excitations), any adiabatic manipulation can only result in a U(1) × U(1) transformation that carries the hallmark of their anyonic statistics. In this way, we are able to demonstrate the exchange of two MZMs in a threesite Kitaev chain encoded in the spatial modes of photons. We further demonstrate that quantum information encoded in the degenerate ground state is immune to local phase and flip noise errors.
Owing to the JW transformation, some of the nonlocal properties of the MZMs are lost in the simulated spin system. Indeed, local singleparticle measurements can reveal the fusion space of the corresponding MZMs, something that is not possible in the fermionic picture. At the same time, some of the local properties of the MZMs are also lost. For example, two isolated Majorana fermions, located at the end points of the chain, correspond to excitons in the spin system that are spread along the whole chain. Nevertheless, as the spectra of the two systems are the same, their corresponding quantum evolutions are equivalent. This gives us the means to investigate the statistical behaviour of the MZMs as well as their resilience to errors. Our current simulation only probes the Abelian part of the exchange, as it employs two isolated MZMs. However, our technology and methodology can be used to demonstrate the nonAbelian statistics of Majorana fermions in which four MZMs are needed. This establishes our photonic quantum simulator as a novel platform to study the exotic physics of Majorana fermions and their possible applications to quantum information^{20}. Thus, our results extend the capabilities of optical simulators.
Results
Majorana braiding
To clearly illustrate the properties of MZMs, we first consider the fermionic KCM^{5}. This is the simplest model that supports two MZMs, γ_{A} and γ_{B} at its boundaries and exhibits twofold degeneracy in its ground state. The braiding properties of Majorana fermions can be investigated by exchanging two MZMs. To perform such an exchange, an adiabatic evolution between carefully constructed Hamiltonians () have been proposed to probe the statistics of the MZMs in the KCM^{21,22}. The exchange of γ_{A} and γ_{B} induces the braiding evolution (ref. 21) acting on the twofolddegenerate groundstate space. As the MZMs belong to the same chain with a fixed fusion channel, the braiding matrix is diagonal. Its elements are given in terms of geometric phases^{23} on the basis of 0_{Lf}〉 (odd fermion parity) and 1_{Lf}〉 (even fermion parity), where the subscript f denotes fermionic states and L is the number of sites of the chain (see Supplementary Note 1). According to the work of Pancharatnam^{24} and the Bargmann invariants^{25}, the geometric phases resulting from the braiding process can be directly determined through projective measurements
where ϕ_{g} is the geometric phase associated with m_{Lf}〉, which represent the basis for the ground states of the Hamiltonian (m=0 or 1). Here, P_{j} projects the system into the ground space of , for j=1,...,n. The geometric phases are gauge invariant and uniquely determined by the Hamiltonians , whereas the details of the adiabatic processes between them are not essential^{26,27}. In general, the projective measurement can be expressed as an imaginarytime evolution (ITE) operator with a sufficiently large evolution time t (ref. 28). The adiabatic requirement preserves the fermion parity of the initial state and guarantees that the offdiagonal elements in U_{ex} remain zero^{26}. Therefore, the whole information of U_{ex} can be read out from ITE operators (ref. 29). By employing the method of dissipative evolution^{30}, each projective measurement can be efficiently completed with some finite probability (a general circuit is given in Supplementary Note 2). We shall employ this method to experimentally probe the geometric phases obtained during the exchange of two MZMs by simulating a spin model that is equivalent to the KCM.
To complete the exchanging of two MZMs, a fourfermion scheme has been proposed in the superconducting wire system^{21}, which is further reduced to a threefermion scheme in our work. The threefermion KCM can be described in terms of six Majorana fermions, denoted by γ_{j} (, for ), as shown in Fig. 1. The exchanging process can be completed by a set of adiabatic processes among three different Hamiltonians, as illustrated in Fig. 1a–d. The initial Hamiltonian is . The MZMs γ_{1} and γ_{6} are isolated at the boundary of the chain to form two MZMs, namely A and B (see Fig. 1a), and the degenerate groundstate space of is spanned by 0_{3f}〉 and 1_{3f}〉 (L=3). The other two Hamiltonians are as follows: , where γ_{3} and γ_{6} are isolated to form MZMs (see Fig. 1b), and , where γ_{1} and γ_{3} are isolated to form MZMs (see Fig. 1c). The MZMs cross during the adiabatic transition from to . To complete the exchange process, we adiabatically transform back into and the Majorana mode A is driven to the location of γ_{6} (see Fig. 1d). The exchange of A and B is thus completed, introducing the unitary operation in the groundstate space on the basis of 0_{3f}〉 and 1_{3f}〉.
Through the JW transformation, a general KCM can be transformed into a onedimensional transversefield Ising model (TFIM)^{15}. However, these two models have some differences in their physics (see Supplementary Note 3). In the fermionic system, the total fermion parity is fixed. The braiding effect in a single KCM is an overall phase. This phase cannot be directly measured as the superposition state with different fermionic parity is impossible. Despite that, the state of the spin model can be prepared in any superposition state. Hence, the relative geometric phase obtained during the exchange can be measured. As the KCM and the TFIM in the ferromagnetic region have the same spectra and their corresponding quantum evolutions are equivalent, the geometric phases obtained from the braiding evolution U_{ex} are invariant under this mapping (see Supplementary Note 4). As a result, the wellcontrolled spin system offers a good platform to determine the exchange matrix and investigate the statistical behaviour of MZMs.
The Hamiltonians involved in the threefermion braiding scheme of the KCM, that is, , and , are transformed through the JW transformation into a spin1/2 system as follows (see Supplementary Note 5 for more details):
where , and are Pauli matrices of the ith spin. The ground states of these Hamiltonians are twofold degenerated. The basis of the ground states of H_{0} are denoted by 0_{3s}〉 and 1_{3s}〉, (s indicates spin states) corresponding to the basis of 0_{3f}〉 and 1_{3f}〉 in KCM through JW transformation, respectively. The detailed form of in the ground space of H_{0}, which is the same as in the ground space of , can be obtained by the tomography process, where the initial ground states of H_{0} are obtained from a randomly chosen input state φ_{r}〉 by implementing with a sufficiently large t, that is, (not normalized). The relative geometric phases and the exchange property of MZMs in the corresponding KCM can then be deduced from the information of U_{ex}. The mapping between the threeparticle KCM and TFIM is further shown in Fig. 1e.
Local noise immunity
The robustness of the information encoded in the ground space of the KCM with Hamiltonian can also be studied in our system. We note that the degeneracy of the ground states of H_{0} is not topologically protected, as it can be lifted by a local operator that breaks Z_{2} symmetry, whereas the ground state of KCM is topologically protected^{31}. However, owing to the equivalence of the spectra of the and H_{0}, the information encoded in the ground state of H_{0} is protected by the same excitation gap against local noises with Z_{2} symmetry, which correspond to the noises in fermionic system. It is worth noting that, in our work, the ground states obtained from the ITE process are algorithmically encoded similar to that in quantum error correction^{32}. This nonlocal encoding of states in a larger number of physical qubits protects them against local flip and phase errors. In other words, the Ising model is a good memory with some of the robustness characteristics of the Majorana^{32}. As a result, the immunity of the local noises in KCM can be faithfully investigated, which establishes it as an attractive medium for storing and processing quantum information^{15}.
Without loss of generality, we take the local error D_{j} acting on a certain site j of the KCM. To verify the robustness against this local error, we should confirm that the operator D_{j} has a trivial effect on any state in the groundstate space of , that is, 〈φ_{f}D_{j}φ_{f}〉 is constant for all the states φ_{f}〉 in the groundstate space of . There are two independent types of local noise: flip error (X_{j}) and phase error (Z_{j}). According to the argument presented in ref. 5, the single local flip error is impossible due to the fermionic parity conversation in the ideal KCM (we do not consider the quasiparticle poisoning effect in practical systems^{33}). Therefore, the local flip error can only happen at two sites simultaneously, such as (c_{j} and are the annihilation and creation operators of the jth fermions, respectively). In terms of the TFIM the X_{1} error operator becomes through the JW transformation. Meanwhile, the local phase error on site 1, , corresponds to the operator of the spin system. Hence, to confirm the robustness of local errors in KCM, we need to check that and are constant for all states φ_{3s}〉 in the groundstate space of H_{0} (corresponding to all states φ_{3f}〉 in the ground state space of ).
Experimental results on simulating the braiding evolution
In our experiment the ITE operator plays a central role. For our current purpose, a useful method to implement the ITE is to design appropriate dissipative evolution^{30}, in which the groundstate information of the corresponding Hamiltonian is preserved but the information of the other states is dissipated (see Methods). Taking advantage of the commutation between and , and , and and in equation (2), the operator can be expressed as
where we set t=5 in our analysis. Owing to the fact that the eigenvectors and eigenvalues of each local operator, such as , can be obtained easily, the operation of can be implemented by the basis rotation (local unitary gate operation) and dissipation straightforwardly (see Supplementary Note 6 for more details). We therefore introduce an environmental degree of freedom and couple it to the system, which can be simply written as , where denotes the states that are orthogonal to the ground state φ_{g}〉. The environmental basis of 1_{e}〉 is dissipated during the evolution and only 0_{e}〉 is preserved. Therefore, the ground state of the corresponding Hamiltonian is obtained in the digital quantum simulator^{34}. It is worth to emphasize that this process is only dependent on the Hamiltonian. The general algorithm of ITE for a general Hamiltonian has been demonstrated in investigating algorithmic quantum cooling^{35}.
Here we employ ITE in optical systems, which provide ideal platforms for quantum simulations. They offer good control of their external parameters and they are largely isolated by their environment^{36}. As a proofofprinciple demonstration we map the threefermion system to the spin model and encode their information in spatial modes. In this way we can experimentally probe the braiding matrix in an alloptical system.
Figure 2 shows the experimental setup for the investigation of the MZMs (see Methods). Our quantum simulator is realized by single photons. The states of three spin1/2 sites correspond to a 2^{3}dimensional Hilbert space, which are encoded in the optical spatial modes of single photons (the image of the eight optical modes can be found in Supplementary Note 7). The spatial propagation of the photon in one direction is equivalent to the imaginary time evolution of (lowerdimensional) system. It is worth noting that during the mapping between the manybodytosingleparticle Hilbert spaces the notion of locality is lost: excitations that are nearby in the system, which is simulated, do not correspond to nearby spatial modes in the photonic system. Nevertheless, there are intriguing similarities. The basis of the spatial modes of photons is the same as that of the corresponding Hilbert space of threespin system (see Supplementary Note 6 for more details).
The basis of a spin state can be expressed as the eigenstates of σ^{z} (denoted by z〉, with an eigenvalue of 1, and , with an eigenvalue of −1), σ^{y} (denoted by y〉, with an eigenvalue of 1, and with an eigenvalue of −1) or σ^{x} (denoted by x〉, with an eigenvalue of 1, and , with an eigenvalue of −1). In our experiment, we employ beam displacers (BDs) of various lengths to prepare the initial states. A BD is a birefringent crystal, in which light beams with horizontal and vertical polarizations are separated by a certain displacement (depending on the length of the crystal)^{37}. Two types of BDs are employed, BD30 (with a beam displacement of 3.0 mm) and BD60 (with a beam displacement of 6.0 mm). The polarization of the photon can be rotated using halfwave plates (HWPs) and the relative amplitudes of the different spatial modes can be conveniently adjusted. To obtain the ground states of the corresponding Hamiltonian, the polarization of the photons is used as the environmental degree of freedom. The coupling between the spatial mode and the polarization is achieved using HWPs, which rotate the polarization in the corresponding paths. Dissipative evolution is accomplished by passing the photons through a polarization beam splitter (PBS), which transmits the horizontal component and reflects the vertical component. In our case, only photons in the optical modes that have horizontal polarizations are preserved; these modes correspond to the ground states of the Hamiltonian. The components with vertical polarizations are completely dissipated. The precision of the dissipative evolution is dependent on the ratio between the reflected and transmitted parts of the vertical polarization after the PBS, which can be higher than 500:1. To ensure dissipative dynamics using a PBS, we must express the input states in the basis of the eigenstates of the corresponding Hamiltonian, which are referred to basis rotations. The eigenstates of can be expressed as , which are represented by the eight spatial modes of the single photon and are shown in the state preparation pane labelled by Pre in Fig. 2. The corresponding crosssections of the spatial modes are shown in the right column in Fig. 2. Only the terms of xxx〉 and are preserved after the ITE operation of H_{0}, which corresponds to two spatial modes in the dissipative evolution of DE0. For the ITE operation of , we should only consider the term of (the term is the same as that in H_{0}). Two HWPs with the angles setting to be 45° in the basis rotation BR1 are implemented in modes of xxx〉 and , in which the basis of the first spin is rotated to z〉 and . The two spatial modes are then separated to four by a BD60 with the spatial modes represented as , zxx〉, and . After the dissipative evolution of DE1, only the terms of and are preserved. The ITE operation of H_{2} can be implemented by the same way with the details given in Supplementary Note 6.
To clearly show the twomode output states, the corresponding density matrices, ρ, are expressed in the basis of {I,σ^{x},σ^{y},σ^{z}} as follows: . Here I represents the identity operator and p_{1}, p_{2} and p_{3} are the corresponding real amplitudes, which uniquely identify density matrices on a Bloch sphere. The initial states after the ITE operation of DE0 is shown in Fig. 3a. The corresponding final states after the operator U_{ex} are illustrated in Fig. 3b. It is shown that the final state is the same as the initial state when the initial state is 0_{3s}〉 (, denoted as dot 4) or 1_{3s}〉 (, denoted as dot 6), which suggests that there is no offdiagonal elements in the braiding matrix, U_{ex}. In addition, the relative geometric phase between 0_{3s}〉 and 1_{3s}〉 during the exchanging can be directly measured. Indeed, the final states are obtained by rotating the initial states counterclockwise along the X axis through an angle of π/2 when the initial state is a superposition state of 0_{3s}〉 and 1_{3s}〉, that is, the basis 0_{3s}〉 obtains an extra phase factor of −i relative to 1_{3s}〉. Thus, the braiding matrix can be determined by the relative geometric phase giving U_{ex}=diag(1,e^{iπ/2}). The braiding matrix determined here agrees well with the theoretical result^{21}. This implies that any input state , with the complex coefficients of α and β (α^{2}+β^{2}=1), would be transformed to , that is, the state would be changed to in the basis of 0_{3s}〉 and 1_{3s}〉.
The real and imaginary parts of the experimentally determined operator of the exchange process (the computation basis are represented as xxx〉 and ), as determined via the quantum process tomography (represented by Pauli operators {I,X(σ^{x}),Y(σ^{y}), Z(σ^{z})})^{38}, are presented in Fig. 3c,d. The fidelity is calculated to be 94.13±0.04% (the errors are deduced from the Poissonian photon counting noise and the detailed method is shown in Supplementary Note 8).
If the statistics between the MZMs were Abelian then the braiding matrix between these two degenerate states would have been the identity matrix without any relative phase between the degenerate basis states. The fact that we obtain a relative phase factor is in agreement with the nonAbelian character of MZMs.
Experimental results on simulating local noises immunity
We further investigate local noises immunity of the information encoded in the ground space of the KCM. The output state after the dissipative evolution DE0, , is treated as the initial state. After the operation of (fliperror operator), the two output modes would become eight. The final state is the same as the initial one by projecting the state back to the ground space of H_{0} (that is, the operation of DE0 and we omit the unobservable overall phase). Similarly, the initial two output modes () become four modes after the local phase noise operation of . The final state is projected back to the initial one with the ITE operation of (see Supplementary Note 6 for more details). In our experiment, the system is subjected to a projection back onto the initial state after applying the errors. By doing so, the success probability is small, as the preserved state is just a fraction of the total state. However, it serves well for establishing the technology and methodology for the inprinciple demonstration of the anyonic physics. Figure 4a,b show the real and imaginary parts of the fliperror protection operator with a fidelity of 97.91±0.03%. Figure 4c,d show the real and imaginary parts of the phaseerror protection operator with a fidelity of 96.99±0.04%. The errors of the fidelity values are deduced from the Poissonian photon counting noise. This fact reveals the protection from the local flip error and phase error in the KCM. The total operation behaves as an identity, thus demonstrating immunity against noise. More experimental results can be found in Supplementary Note 9.
In general, the fidelity of protection operators are also affected by the noises without Z_{2} symmetry and the evolution time in the ITE. Owing to the fact that the environment can be well controlled in our optical simulator, the setup established here can also be used to investigate the effect of the noise without Z_{2} symmetry in the ground space of the TFIM. Our simulator offers the possibility to study the effect of other types of noise, which can be presented in the real system supporting MZMs (superconductorbased system). A central advantage of the optical simulation is that all the implementations can be realized with high precision.
Discussion
In this study we experimentally investigated the physics of MZMs emerging in the KCM by simulating it with an equivalent photonic system. The photonic quantum simulator, which employs a set of projective measurements, represents a general and powerful approach to directly study quantum evolutions. We were able to transport two MZMs at the endpoints of a chain and measure the geometric phases resulting from their exchange. Moreover, we experimentally demonstrated that the degenerate subspace of the MZMs is resilient against local perturbations.
Our simulation is based on a nonlocal JW transformation that maps the fermionic KCM to an equivalent spin model. This transformation, although it changes the states of the system, does not alter the associated unitary evolutions, thus allowing us to measure directly the braiding matrix of MZMs. The appeal of our photonic quantum simulator is based on the fact that it allows for the transport of MZMs. It offers the exciting future possibility to investigate the braiding character of mutiMZM systems and demonstrate their nonAbelian statistics.
Applications towards topological quantum computation may be possible by employing the versatile and wellcontrolled optical quantum simulator presented here. MZMs and their braiding cannot be used to perform universal quantum computations^{15}. However, one can envision implementing specifically tailored quantum algorithms, such as the Deutch–Josza algorithm^{39} where the information will be topologically protected during the whole process.
Owing to the fact that we encode the multisite spin1/2 system in the spatial modes of a single photon, the scalability of this method is limited. Nevertheless, it serves well for demonstrating the fundamental statistical properties and robustness of MZMs with such a small system size. Our main focus is on the physics and establishing the technology and the methodology. The gained knowhow (investigating the braiding of anyonic quasiparticles through the ITE) can be picked up by other technologies that offer scalability, such as the trapped ions^{13} or Josephson junctions^{14} system.
Methods
The ITE operator on states
For a given Hamiltonian H with a complete set of eigenstates e_{k}〉 and the corresponding eigenvalues E_{k}, any arbitrary pure state φ〉 can be expressed as
with q_{k} representing the corresponding complex amplitude. Here we focus on pure states, but the argument is also valid for mixed states. The corresponding ITE operator (U)^{28} on the state becomes
The evolution time t is chosen to be 5 in our analysis, which is long enough to drive the input state to the ground state of H. After the ITE, the amplitude q_{k} is changed to be q_{k}exp(−5E_{k}). We can see that the decay of the amplitude is strongly (exponentially) dependent on the energy: the higher the energy, the faster the decay of the amplitude. Therefore, only the ground states (with lowest energy) survives during the evolution. Furthermore, due to the fact that the Hamiltonian H_{0}, H_{1} and H_{2} can be divided into two commuting parts, we can separate each ITE operator to two ITE operators whose eigenvectors can be directly obtained (see equation (3)).
Experimental process in probing braiding characteristics
The experimental setup for probing braiding characteristics is shown in Fig. 2. The process follows the logical diagram presented in the pane enclosed in the black solid line, denoted by Logi. The setup used to prepare the input state is illustrated in the pane labelled Pre. For simplicity, at the beginning of the evolution, all three qubits are expressed in the basis of σ^{x}, in which the ground states of H_{0} are encoded in the two spatial modes xxx〉 and . The dissipative evolution of H_{0} with arbitrary input states is illustrated in the pane labelled DE0, the output state of which is treated as the initial state. The basis rotation BR1 is employed and the subsequent dissipative evolution (DE1) should drive the state to approximate the ground state of H_{1}.
After BR2 and DE2, the two input modes are expanded to four output modes, which are then sent to BR3, to implement the dissipative evolution DE0. In our experiment, we perform two types of measurements, that is, a twomode measurement (TM) and a fourmode measurement (FM). The initial states, which contain spatial information, are transformed into states that contain polarization information, on which standard quantumstate tomography of polarization states can be performed. The TM corresponds to the tomography of the polarization states of a single qubit, for which four types of polarization measurements are required: horizontal polarization (H), vertical polarization (V), righthand circular polarization () and diagonal polarization (). The polarizationanalysis setup is constructed using a quarterwave plate, a HWP and a PBS. The photon is then detected using a singlephoton detector. Similarly, the FM corresponds to the measurement of 2qubit polarization states and 16 measurements are required. In the TM and FM, beam splitters are used to send photons to different measurement instruments. The preparation of the initial states for the demonstration of noise immunity is the same as is shown in Pre and DE0. The disturbance operation is then implemented. After a second DE0, a TM is used to reconstruct the final states (see Supplementary Notes 10 and 11 for more details).
Data availability
The data that support the findings of this study are available from the authors on request.
Additional information
How to cite this article: Xu, J.S. et al. Simulating the exchange of Majorana zero modes with a photonic system. Nat. Commun. 7, 13194 doi: 10.1038/ncomms13194 (2016).
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Acknowledgements
This work was supported by the National Key Research and Development Program of China (Grants Number 2016YFA0302700), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant Number XDB01030300), National Natural Science Foundation of China (Grant Numbers 11274297, 11105135, 11474267, 11004185, 60921091, 11274289, 10874170, 61322506, 11325419 and 61327901), the Fundamental Research Funds for the Central Universities (Grant Number WK2470000020), Youth Innovation Promotion Association and Excellent Young Scientist Program CAS, Program for New Century Excellent Talents in University (NCET120508) and Science foundation for the excellent PHD thesis (Grant Number 201218).
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Y.J.H. constructed the theoretical scheme. J.S.X. and C.F.L. designed the experiment. K.S. and J.S.X. performed the experiment. J.K.P. contributed to the theoretical analysis. J.S.X., Y.J.H. and J.K.P. wrote the manuscript. Y.J.H. supervised the theoretical part of the project. C.F.L. and G.C.G. supervised the project. All authors read the paper and discussed the results.
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Supplementary Figures 113, Supplementary Table 1, Supplementary Notes 111 and Supplementary References. (PDF 5301 kb)
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Xu, JS., Sun, K., Han, YJ. et al. Simulating the exchange of Majorana zero modes with a photonic system. Nat Commun 7, 13194 (2016). https://doi.org/10.1038/ncomms13194
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