Jordan-Wigner transformations for tree structures
aa r X i v : . [ c ond - m a t . o t h e r] O c t Jordan–Wigner transformations for tree structures
Stefan Backens
Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany
Alexander Shnirman
Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany andInstitute of Nanotechnology, Karlsruhe Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany
Yuriy Makhlin
Condensed-matter physics Laboratory, National Research University Higher School of Economics, 101000 Moscow, Russia andLandau Institute for Theoretical Physics, acad. Semyonov av. 1a, 142432, Chernogolovka, Russia
The celebrated Jordan–Wigner transformation provides an efficient mapping between spin chainsand fermionic systems in one dimension. Here we extend this spin–fermion mapping to arbitrary tree structures, which enables mapping between fermionic and spin systems with nearest-neighborcoupling. The mapping is achieved with the help of additional spins at the junctions between one-dimensional chains. This property allows for straightforward simulation of Majorana braiding inspin or qubit systems.
I. INTRODUCTION
The well-known Jordan–Wigner transformation relates spin- operators to fermionic creation and annihilation op-erators. Thereby, it allows for mapping between spin and fermionic systems. It was originally used by Jordan andWigner to define fermionic operators in the second quantization [1]. The Jordan–Wigner transformation introducesnon-local “string operators” to transform commuting operators of different spins into anti-commuting fermionic op-erators, and, in general, does not preserve locality. Nevertheless, it maps even-parity local fermionic Hamiltoniansto local spin Hamiltonians; moreover, certain spin Hamiltonians in 1D are mapped to free fermionic Hamiltonians,which are readily solvable [2].Generalizations to higher dimensions were discussed in recent decades [3–8]. They map (even-parity) fermionicHamiltonians to spin Hamiltonians, but even local quadratic fermionic terms are mapped onto operators which involvemany, in principle infinitely many, spin operators (though in some cases [3, 5] the weight of the involved spins mayslowly decay with distance). One may also consider introducing ancillary degrees of freedom. For instance, Verstrateand Cirac [7] suggested doubling the number of degrees of freedom for a 2D lattice to achieve local, but not necessarilysimple Hamiltonians in the spin language.Thus, free fermionic Hamiltonians are often mapped to complicated operators in the spin language. In Ref. [9], amodified Jordan–Wigner transformation was proposed, such that a three-leg star graph of free fermions (with nearest-neighbor hopping) could be mapped to a three-leg star graph of spin chains (with nearest-neighbor couplings). Themapping required the introduction of an extra spin- in the vertex of the spin graph, coupled to the three spin chainslocally via a specific 3-spin coupling. Furthermore, in Ref. [9] an alternative scenario was described, in which a three-leg spin graph with exclusively 2-spin interactions was mapped to a Kondo-like system of fermionic chains coupledby one spin (cf. application in Refs. [10, 11]). Here, we demonstrate that these transformations can be generalisedto binary-tree structures of 1D chains, i. e. connected, acyclic graphs with no more than three edges at each vertex.Furthermore, we argue that this result can be directly generalized to generic, non-binary trees.This kind of transformation is of special interest in particular since it can be used to simulate the physics and,notably, non-abelian statistics and braiding of fermionic Majorana modes [12, 13] in a (topologically non-protected)spin system. For the case of a T -junction geometry with a single topological segment in the chain providing twoMajorana modes, this implementation was explicated in Ref. [14]. Here we describe braiding operations betweenMajorana modes belonging to different topological segments in a system where the number of segments is arbitrary.A binary-tree structure may be viewed as consisting of many T -junctions; such structures may be useful for imple-mentation (physical simulation) of the Majorana braiding operation [15, 16] with applications in topological quantumcomputing. We also argue that such spin systems mimic fermionic quantum computers [17], which can be efficient,e. g., in quantum-chemistry simulations. Namely, braiding or other logic gates between remote qubits naturally in-clude Jordan-Wigner string operators, making these qubits fermionic. Encoding the population of molecular orbitalsin such qubits (see e.g., Ref. [18]) thus brings a considerable advantage for the computing algorithms.An explicit description of a Majorana braiding operation between two topological segments, implemented in thecorresponding spin system, is given in the Appendix. II. GEOMETRY AND NOTATIONS
We consider spins on a tree-like lattice of the type depicted in Fig. 1. Each edge of the tree is a one-dimensionalspin chain. The chains are connected at the vertices, and the whole structure indicates the notion of locality (in fact,we focus on nearest-neighbor couplings). In binary trees, they are connected in triples and, in general, interactionsbetween boundary spins from all three chains are allowed, so-called ∆ -junctions [10], indicating all three pairwisecouplings. In the particular case when one of the three couplings in the junction vanishes, we obtain a T -junction,where all three chains have a common boundary spin. We also consider fermions on the same tree and discuss methodsto convert between spin and fermionic systems.A priori, the tree structures do not have a distinctive root and the edges do not have orientation. For the purposes ofthe transformation, however, we choose an arbitrary vertex as a root and assign to each edge (i. e. chain) an orientationaway from the root. Based on this hierarchy, we introduce a notation for our further discussion by assigning a nameto each vertex and chain in the tree: The root is denoted “0” and the three outgoing chains acquire numbers 1, 2, and3. Then, step by step, each other vertex acquires a name α , identical with the incoming chain, while the two outgoingchains are assigned a longer name, αβ , with β = 1 or , see Fig. 1.According to the orientation, the spins or fermions in each chain α are numbered from 1 to its length L α ; they arerepresented by the Pauli matrices σ x,y,zα ( j ) and the fermionic creation/annihilation operators c † α ( j ) /c α ( j ) , respectively.To construct a fermion–spin transformation, we shall need ancillary spin operators, one per chain, which we assignto the vertex at the beginning of the chain. The corresponding Pauli matrices S βα are labelled with the vertex index α and the chain number 1, 2, or 3. An example is depicted in Fig. 1. The spin operators S , , α at each vertex α arespin components of the ancillary spin at this vertex.To describe a fermion-spin transformation, we use separate Jordan–Wigner transformations for each chain α , c α ( j ) = η α " j − Y k =1 σ zα ( k ) σ − α ( j ) (1a) c † α ( j ) = η α " j − Y k =1 σ zα ( k ) σ + α ( j ) , (1b)where σ ± α ( j ) = [ σ xα ( j ) ± i σ yα ( j )] . The Klein factors η α , with η α = 1 , are to be chosen to ensure proper (anti-)commutation relations between operators in different chains; they are discussed later. Similar to the standard Jordan-Wigner transformation, these relations ensure that a local quadratic fermionic Hamiltonian is also a local operator inthe spin language. In particular, a useful corollary of these definitions, σ zα ( j ) = 2 c † α ( j ) c α ( j ) − − c α ( j ) c † α ( j ) , (2)shows that a (magnetic) field in z -direction corresponds to a local chemical potential at a fermionic site. S β S β
11 12 S β III. FREE FERMIONS AND 3-SPIN COUPLINGS
To complete the description of the transformation, we need to define the operators η α . For the chains directly atthe root, β = 1 , , , we define the transformation exactly like in Ref. [9]: η β = S β . (3a)For any other chain, denoted by αβ with the parent chain α and β = 1 , , the following definition applies: η αβ = η α " L α Y k =1 σ zα ( k ) S βα . (3b)These definitions satisfy the conditions stated in the previous section.Let us now consider various nearest-neighbor quadratic fermionic couplings and their spin counterparts underthe constructed transformation. Within any one-dimensional chain, the Jordan–Wigner transformation is known toconvert local quadratic fermionic Hamiltonians into local quadratic spin Hamiltonians; the factors η α = 1 in Eq. (1)do not affect this. Therefore we will examine only the couplings at the vertices between different chains. There aretwo kinds of vertex couplings: those between a parent and a descendant chain and those between two descendantchains of the same parent. A coupling term of the first kind between chains α and αβ (with β = 1 , ) has the generalform H α,αβ = u c α ( L α ) c αβ (1) + t c † α ( L α ) c αβ (1) + h. c. , (4a)which is transformed, using the relation (1), into H S α,αβ = S βα h u σ − α ( L α ) σ − αβ (1) − t σ + α ( L α ) σ − αβ (1) + h. c. i . (4b)A coupling of the second kind between chains αβ and αγ (here β = γ ; β and γ can be 1 or 2; at the root, α is emptyand β, γ = 1 , , with ancillary spin operators S β ) has the general form H αβ,αγ = u c αβ (1) c αγ (1) + t c † αβ (1) c αγ (1) + h. c. , (5a)which is similarly mapped to H S αβ,αγ = S βα S γα h u σ − αβ (1) σ − αγ (1) + t σ + αβ (1) σ − αγ (1) i + h. c. = S να ǫ βγν h i u σ − αβ (1) σ − αγ (1) + i t σ + αβ (1) σ − αγ (1) + h. c. i . (5b)Let us note that the transformation described can be generalized to arbitrary tree structures, beyond binary trees.Indeed, any higher-order vertex (with more that three edges) can be thought of as built out of three-edge vertices.For instance, Fig. 1 can be viewed as a five-edge vertex, which allows us to define the Klein factors for all chainsoutside of this figure: In that case, the internal chains in Fig. 1 are of length zero and do not contribute products tothe Klein factors, but coupling terms involving more than three spins may appear. IV. XY SPIN SYSTEM AND FERMIONIC KONDO MODEL
In this section, we consider a tree structure of spins with local XY couplings and use the JW transformation back-wards in order to find the corresponding fermionic problems. For a single 1D chain, the Jordan–Wigner transformationmaps these to free fermions. In order to find the corresponding fermionic Hamiltonian for a tree structure, we usethe generalized Jordan–Wigner transformation defined in Eqs. (1) and (3). These involve ancillary spin operators S βα ,which commute with local spins σ ( j ) , but not with the fermions c ( j ) . We show below that the original XY spin modelis equivalent to a Kondo-type model on the same tree with one impurity spin per vertex.To simplify the resulting fermionic Hamiltonians, we introduce, instead of S , other spin operators at the innervertices, ˜ S βα . We define S β = ˜ S β Y chain labels γ not beginningwith β P γ and S βα = ˜ S βα Y chain labels αγ not beginningwith αβ P αγ , (6)where we introduced the notation P α = L α Y k =1 σ zα ( k ) (7)for the fermionic parity of chain α . As the products consist of Pauli matrices σ z only, operators S βα inherit thecommutation relations of ˜ S βα . In other words, ˜ S are spin-1/2 operators, and one can verify that they commute withthe fermionic operators.Let us illustrate this with the example of Fig. 1: S = ˜ S P P (8a) S = ˜ S [ P P ( P P P )] P (8b) S = ˜ S [ P P ( P P P )] P (8c)(the grouping highlights the tree structure), S = ˜ S ( P P P ) (9a) S = ˜ S P (9b) S = ˜ S P ( P P P ) (9c)and S = ˜ S P (10a) S = ˜ S P (10b) S = ˜ S P P . (10c)The string (parity) operators guarantee that S βα commute with the fermionic operators of all chains.Again, the Jordan–Wigner transformation is known to map XY-coupled spins in a 1D chain to free fermions, so weonly have to examine the two kinds of vertex couplings, as we did in the preceding section. They result in Kondo-likecouplings of the fermionic chains: H α,αβ = u σ − α ( L α ) σ − αβ (1) + t σ + α ( L α ) σ − αβ (1) + h. c. (11a) −→ H F α,αβ = S βα (cid:2) u c α ( L α ) c αβ (1) − t c † α ( L α ) c αβ (1) + h. c. (cid:3) (11b)and H αβ,αγ = u σ − αβ (1) σ − αγ (1) + t σ + αβ (1) σ − αγ (1) + h. c. (12a) −→ H F αβ,αγ = S βα S γα h u c αβ (1) c αγ (1) + t c † αβ (1) c αγ (1) i + h. c. = S να ǫ βγν h i u c αβ (1) c αγ (1) + i t c † αβ (1) c αγ (1) + h. c. i . (12b)Thus, inter-chain couplings are controlled by the ancillary spins. V. MAJORANA BRAIDING AND THE SPIN REPRESENTATION
In this section, we are interested in spin implementation of free-fermion models on tree structures. These can beapplied, in particular, to realize (physically simulate) Majorana qubits and quantum logical operations using ordinaryquantum bits.Majorana modes arising in the topological phase of the Kitaev chain [19], a one-dimensional fermionic system, canbe braided in a T -junction geometry by local tuning of the chemical potential [15]. One can see from the discussionabove that similar to Refs. [9, 14], the corresponding spin model involves Ising couplings within the chains, theancillary-spin-controlled Ising couplings at the junctions as well as a transverse magnetic field.In the following, the spin indices are swapped for convenience, to ensure the resulting zz Ising couplings and thetransverse field in the x direction. Furthermore, we use fermionic Majorana operators γ α ( m ) [20] satisfying theanti-commutation relations, { γ α ( m ) , γ β ( n ) } + = 2 δ αβ δ mn , (13)to express the transformation in a convenient form: γ α (2 j −
1) = η α " j − Y k =1 σ xα ( k ) σ zα ( j ) (14a) γ α (2 j ) = η α " j − Y k =1 σ xα ( k ) σ yα ( j ) (14b) ⇒ σ xα ( j ) = i γ α (2 j − γ α (2 j ) . (14c)The Klein factors η α are those defined in Equations (3).The transformation relates the topological (nontopological) phase in the fermionic chains to the ferromagnetic(paramagnetic) phase of the spin system (for more details see Appendix). Now we can simply translate into the spinsystem the unitary operator produced by, e. g., counter-clockwise braiding of Majorana modes γ A , γ B [15]: U = exp (cid:0) π γ A γ B (cid:1) (15)(how this can be implemented may depend on the tree structure and the initial positions of γ A and γ B ). In the caseof two Majorana modes that are provided by one topological segment located in a single chain before and after thebraiding, the Klein factors cancel in the spin representation, so the additional spin mediating the coupling at thejunction does not influence the result of the operation and is left unaffected at the end [14].Braiding neighbouring Majorana modes from two topological segments in different chains corresponds to a morecomplicated operation in the spin system. By choosing, e. g., γ A = γ (2 m − and γ B = γ (2 n − , we obtain: U , = exp π · S x m − Y j =1 σ x ( j ) · σ z ( m ) · S z n − Y k =1 σ x ( k ) · σ z ( n ) = exp − i π S y · σ z ( m ) m − Y j =1 σ x ( j ) n − Y k =1 σ x ( k ) · σ z ( n ) effectively −−−−−−→ exp (cid:2) − i π S y σ z ( m ) σ z ( n ) (cid:3) , (16)if the spins outside the ferromagnetic intervals are polarised in the x -direction. When expressed in terms of the Paulimatrices τ for the two ’topological’ qubits involved (two ferromagnetic intervals, cf. Ref. [14]), this gives U (1 ,
3) = exp (cid:2) − i π S y τ z τ z (cid:3) . (17)A detailed description of this operation in the spin language is given in the Appendix.However, for two ‘topological’ intervals in the same chain we obtain a similar expression, but without the interme-diate ancillary spin: U = exp (cid:2) − i π τ y τ z (cid:3) . (18)Thus one obtains a two-qubit operation.In a more general situation with arbitrary initial position of two distant braided boundaries (and associated Ma-jorana modes), the brading operation involves, apart from these two qubits, the ancillary spins at all intermediatevertices as well as the parity (qubit-flip) operators Q σ x for all intermediate qubit intervals. Thus, the braidingimplements not a two-qubit operation but a multi-qubit operation (and also entangles qubits with the ancillas).Here a few comments are in order: First, to achieve direct two-qubit gates between distant ‘topological’ qubitintervals, one can complement the described braiding operation with further operations involving intermediate qubits.However, for the purposes of quantum computation one does not necessarily need two-qubit logic gates betweendistant qubits since two-qubit gates between neighbors are sufficient, as they form a universal set of gates togetherwith single-qubit operations. Furthermore, one can also view this subtlety from a different perspective. Instead ofthinking in terms of the qubit description, one can describe the operations in terms of the fermionic (Majorana)modes involved. Then the braiding operations implement two-fermion gates, and one deals with fermionic quantumcomputation . This viewpoint may be useful for simulations of fermionic Hamiltonians (see e.g. Ref [18]), includingmany-body solid-state models and complex individual molecules.A further remark concerns the symmetry and the braiding procedure: Each of the chains considered belongs tothe BDI symmetry class [21], with a time-reversal-type symmetry T such that T = 1 . In a single chain (A.1b), aMajorana zero mode appears at each boundary between topological and non-topological regions. A vertex connectingthree chains can be viewed as an edge of a 1-D system. Here the symmetry becomes crucial [22, 23]. If the T symmetryis preserved by the chain coupling at the vertex, the edge (vertex) carries an integer ( Z ) ‘topological’ charge. In ourcase this allows for configurations with more than one Majorana zero mode at the vertex and an unwanted extradegeneracy when during the braiding procedure this vertex connects two or three ‘topological’ regions. A T -breakingchain coupling, however, places the system to the D class with a Z invariant, and typically one (or no) Majoranazero mode exists at the vertex (cf. Ref. [15]). In this case, no extra degeneracies arise during the braiding operations.In particular, this is the case for the coupling considered in Ref. [14]. VI. DISCUSSION
The Jordan–Wigner transformation maps free fermionic Hamiltonians to local spin Hamiltonians. Furthermore, anearest-neighbor hopping term is mapped to a local quadratic spin term. Some generalizations of the Jordan–Wignertransformation to higher-dimensional lattices were proposed [3–6], which, however, map a local hopping fermionicterm to a spin term involving many, often infinitely many, spins on distant sites. Other approaches (e. g., Ref. [7]) useancillary degrees of freedom but also map free fermionic terms to fourth- or higher-order spin terms.On one hand, this implies that only some unusual spin models may be analyzed with the help of such transformations.On the other hand, one could use a spin–fermion mapping to implement a fermionic model in a system built of spins(or qubits). With the motivation to implement fermionic (Majorana) degrees of freedom in a realistic qubit system,we have extended an earlier result of Ref. [9] to arbitrary tree structures. The resulting transformation maps nearest-neighbor fermionic terms to nearest-neighbor spin terms. Thus, it allows for an implementation of the Majoranaphysics in tree structures built out of qubit chains, extending the results of Ref. [14]. This transformation provides,e. g., a spin equivalent of Majorana braiding operations. We have further shown that this construction can begeneralized to arbitrary tree structures.It must be noted that these mappings involve an enlargement of the original Hilbert space, due to the addition ofspins ˜ S α to the system. Thus, the degeneracy of all states is multiplied by a factor of 2 to the power of the numberof inner vertices, but the accuracy of the mapping is not affected.Finally, we would like to mention that experimental realizations of the 3-spin interactions crucial for our mappingwere discussed in the literature (see, e. g., Refs. [24],[14]). VII. ACKNOWLEDGEMENTS
We are grateful to Guido Pagano for useful discussions. This research was financially supported by the DFG-RSFgrant No. 16-42-01035 (Russian node) and No. SH 81/4-1, MI 658/9-1 (German node).
Appendix: Spin representation of two-interval Majorana braiding
To obtain a spin representation for a fermionic T -junction of Kitaev chains, we use the transformation given inEqs. (14). This yields [14]: H = X α =1 H ,α + H S int (A.1a) H ,α = − L α X j =1 h α ( j ) σ xα ( j ) − J L α − X j =1 σ zα ( j ) σ zα ( j + 1) (A.1b) H S int = − X αβγ | ǫ αβγ | J αβ S γ σ zα (1) σ zβ (1) , (A.1c)a system of Ising spin chains with a local transverse magnetic field h α ( j ) , which corresponds to the locally tunablechemical potential in the fermionic system. Assuming J > , any interval of spins with h ≫ J in one of the chains isferromagnetic, whereas h ≪ J results in a trivial (paramagnetic) phase. The three chains are linked by the componentsof an additional central spin S via 3-spin couplings of strength J αβ = J βα . This structure is depicted in Fig. 2. S y S z S x σ z (1) σ z (2) · · · σ z ( L ) σ z (1) σ z (2) · · · σ z ( L ) σ z (1) σ z (2) ... FIG. 2. Ising spin chains in a T -geometry. A fermionic T -junction suitable for Majorana braiding [15] has a spin representationof this structure [14], which is described by the Hamiltonian in Eq. (A.1). The couplings between three Ising spin chains aremediated by the components of an additional spin S , cf. Ref. [9] and Section III. The system can be manipulated by tuningtransverse fields (not depicted here) that act on the individual spins σ α ( j ) of the three chains. Now we consider the spin equivalent of braiding Majorana modes from two different topological intervals in thefermionic system. The topological intervals and adiabatic shifts of their boundaries within a chain can be translatedto the ferromagnetic intervals in the spin representation exactly as for a single topological interval [14]: The fermionic-parity groundstates | i , | i of a topological interval correspond to linear combinations | i ≡ | ↑ ↑ ↑ i + | ↓ ↓ ↓ i√ , (A.2a) | i ≡ | ↑ ↑ ↑ i − | ↓ ↓ ↓ i√ (A.2b)of the ferromagnetic spin eigenstates. However, two-interval braiding cannot be effected in such a way that at mostone of the 3-spin couplings in Equation (A.1c) is relevant at each step. Therefore, the coupler spin S undergoes agenerally non-trivial rotation in the process, which we will examine in the following.Initially, the topological/ferromagnetic intervals have to be prepared, e. g., in the first and third chain at somedistance from the coupler spin S . We assume that J = J = J > for illustration. First, we consider an initialstate with the spins in both intervals and the coupler aligned in the + z -direction: | ψ i = | ↑ ↑ ↑ ⊙i ⊗ |⊙⊙i ⊗ |⊙ ↑ ↑ ↑ i ⊗ | S ↑i . (A.3)Here the indices denote the three spin chains; the corresponding arrows indicate the spin orientation in ferromagnetic( ↑ / ↓ ) and paramagnetic areas ( ⊙ ). They symbolize the locations of the ferromagnetic intervals in the T -junctiongeometry (Fig. 2), but the calculation does not depend on the specific interval lengths and distances to the coupler.The first step comprises shifting the right boundary of the first interval (i. e., one Majorana mode in the fermionicsystem) into the second chain, which results in the state | ψ i = |⊙ ↑ ↑ ↑ i ⊗ | ↑ ↑ i ⊗ |⊙ ↑ ↑ ↑ i ⊗ | S ↑i . (A.4)With | S ↓i as initial coupler state, the spins in the second chain in Eq. (A.4) would just be flipped compared to thosein the first chain.The non-trivial part begins when the second ferromagnetic interval is also shifted to the junction, while the spinorientation of the ferromagnetic intervals remains fixed. One can verify that for any initial state, including super-positions, the final state at this stage is always the same as in the case when the second ferromagnetic interval isadiabatically shifted towards the junction at J = J = 0 , and only then these couplings are slowly turned on. Thisobservation simplifies the further calculation. Indeed, at the end of this stage, the coupler spin rotates to adjust tothe change of its effective magnetic field from the z -direction to the space diagonal √ (cid:16) (cid:17) : | ψ i = |⊙ ↑ ↑ ↑ i ⊗ | ↑ ↑ i ⊗ | ↑ ↑ ↑ ⊙i ⊗ h cos ϕ | S ↑i + e i π sin ϕ | S ↓i i , (A.5)where < ϕ < π/ and cos ϕ = 1 / √ . Similarly, at the next stage, when the ferromagnetic interval in the first chainis shifted away from the junction, the coupler spin adjusts to the x -direction: | ψ i = | ↑ ↑ ↑ ⊙i ⊗ | ↑ ↑ i ⊗ | ↑ ↑ ↑ ⊙i ⊗ √ (cid:2) | S ↑i + | S ↓i (cid:3) . (A.6)Retracting the remaining ferromagnetic interval back to the third chain is a trivial step again: | ψ i = | ↑ ↑ ↑ ⊙i ⊗ |⊙⊙i ⊗ |⊙ ↑ ↑ ↑ i ⊗ √ (cid:2) | S ↑i + | S ↓i (cid:3) . (A.7)Unlike in the case of single-interval braiding [14], the coupler spin does not return to its intial state at the end of theoperation (see, however, discussion in Sec. V). In Eqs. (A.6), (A.7), we have dropped the overall phase factor thatcan be linked to the geometric phase of the spin evolution. It turns out to be the same for all states of interest to us(cf. Eq.(A.8) below) and will be omitted.For other initial conditions, the operation can be treated similarly, giving the complete result | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ (cid:7)(cid:7)(cid:7) i ⊗ | S ↑i −→ | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ (cid:7)(cid:7)(cid:7) i ⊗ √ (cid:2) | S ↑i + | S ↓i (cid:3) (A.8a) | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ ♦♦♦ i ⊗ | S ↑i −→ | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ ♦♦♦ i ⊗ √ (cid:2) | S ↑i − | S ↓i (cid:3) (A.8b) | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ (cid:7)(cid:7)(cid:7) i ⊗ | S ↓i −→ − | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ (cid:7)(cid:7)(cid:7) i ⊗ √ (cid:2) | S ↑i − | S ↓i (cid:3) (A.8c) | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ ♦♦♦ i ⊗ | S ↓i −→ | (cid:7)(cid:7)(cid:7) ⊙i ⊗ |⊙ ♦♦♦ i ⊗ √ (cid:2) | S ↑i + | S ↓i (cid:3) (A.8d)with placeholders { (cid:7) , ♦ } = { ↑ , ↓ } . The initial as well as final state of the second chain is always |⊙⊙i . Using theparity eigenstates (A.2), we can verify that Eqs. (A.8) indeed correspond to the Majorana braiding. 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