Jordan–Wigner transformations for tree structures

The celebrated Jordan–Wigner transformation provides an efficient mapping between spin chains and 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-neighbor coupling. 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 in spin or qubit systems.

in the vertex of the spin graph, coupled to the three spin chains locally 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 coupled by one spin (cf. application in refs 10,11 ). Here, we demonstrate that these transformations can be generalised to binary-tree structures of 1D chains, 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 two Majorana modes, this implementation was explicated in ref. 14 . Here we describe braiding operations between Majorana 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 implementation (physical simulation) of the Majorana braiding operation 15,16 with applications in topological quantum computing. 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 include Jordan-Wigner string operators, making these qubits fermionic. Encoding the population of molecular orbitals in such qubits (see e.g., refs 18,19 ) thus brings a considerable advantage for the computing algorithms.

Results
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-dimensional spin 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, interactions between boundary spins from all three chains are allowed, so-called Δ-junctions 10 , indicating all three pairwise couplings. 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 methods to 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 of the transformation, however, we choose an arbitrary vertex as a root and assign to each edge (chain) an orientation away from the root. Based on this hierarchy, we introduce a notation for our further discussion by assigning a name to each vertex and chain in the tree: The root is denoted "0" and the three outgoing chains acquire numbers 1, 2, and 3. Then, step by step, each other vertex acquires a name α, identical with the incoming chain, while the two outgoing chains are assigned a longer name, αβ, with β = 1 or 2, 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 are represented by the Pauli matrices σ α j ( ) x y z , , and the fermionic creation/annihilation operators α α † c j c j ( )/ ( ), respectively. To construct a fermion-spin transformation, we shall need ancillary spin operators, one per chain, which we assign to the vertex at the beginning of the chain. The corresponding Pauli matrices α β S 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 α S 1,2,3 at each vertex α are spin components of the ancillary spin at this vertex.
To describe a fermion-spin transformation, we use separate Jordan-Wigner transformations for each chain α, x y 1 2 . The Klein factors η α , with η = α 1 2 , are to be chosen to ensure proper (anti-) commutation relations between operators in different chains 20,21 ; 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 in the spin language (this does not hold if cycles are present). In particular, a useful corollary of these definitions, z shows that a (magnetic) field in z-direction corresponds to a local chemical potential at a fermionic site.

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 at the root, β = 1, 2, 3, we define the transformation exactly like in ref. 9 : www.nature.com/scientificreports www.nature.com/scientificreports/ For any other chain, denoted by αβ with the parent chain α and β = 1, 2, the following definition applies: These definitions satisfy the conditions stated in the previous section.
Let us now consider various nearest-neighbour quadratic fermionic couplings and their spin counterparts under the constructed transformation. Within any one-dimensional chain, the Jordan-Wigner transformation is known to convert local quadratic fermionic Hamiltonians into local quadratic spin Hamiltonians; the factors η = α 1 2 in eq. (1) do not affect this. Therefore we will examine only the couplings at the vertices between different chains. There are two kinds of vertex couplings: those between a parent and a descendant chain and those between two descendant chains of the same parent. A coupling term of the first kind between chains α and αβ (with β = 1, 2) has the general form , which is transformed, using the relation (1), into , S 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 empty and β γ = , 1, 2, 3 with ancillary spin operators β S 0 ) has the general form , which is similarly mapped to where ε βγν is the Levi-Civita symbol. 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 chains outside of this figure: In that case, the internal chains in Fig. 1 are of length zero and do not contribute products to the Klein factors, but coupling terms involving more than three spins may appear.
XY spin system and fermionic Kondo model. In this section, we consider a tree structure of spins with local XY couplings and use the Jordan-Wigner transformation backwards in order to find the corresponding fermionic problems. For a single 1D chain, the Jordan-Wigner transformation maps these to free fermions. In order to find the corresponding fermionic Hamiltonian for a tree structure, we use the generalized Jordan-Wigner transformation defined in eqs (1) and (3). These involve ancillary spin operators α β S , which commute with local spins σ(j), but not with the fermions c(j). We show below that the original XY spin model is 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 inner vertices, for the fermionic parity of chain α. As the products consist of Pauli matrices σ z only, operators α β S inherit the commutation relations of α β  S . In other words,  S are spin-1/2 operators, and one can verify that they commute with the fermionic operators.
Let us illustrate this with the example of Fig. 1: The string (parity) operators guarantee that α β S 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 we only have to examine the two kinds of vertex couplings, as we did in the preceding section. They result in Kondo-like couplings of the fermionic chains: Majorana braiding and the spin representation. In this section, we are interested in spin implementation of free-fermion models on tree structures. These can be applied, in particular, to realize (physically simulate) Majorana qubits and quantum logical operations using ordinary quantum bits.
Majorana modes arising in the topological phase of the Kitaev chain 22 , a one-dimensional fermionic system, can be braided in a T-junction geometry by local tuning of the chemical potential 15 . One can see from the discussion above that similar to refs 9,14 , the corresponding spin model involves Ising couplings within the chains, the ancillary-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 the transverse field in the x direction. Furthermore, we use fermionic Majorana operators γ α (m), which are connected to the usual fermionic creation and annihilation operators c † , c in the following way: They satisfy the anti-commutation relations, mn and allow us to express the transformation in a convenient form: www.nature.com/scientificreports www.nature.com/scientificreports/ 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 Methods). Now we can simply translate into the spin system the unitary operator produced by, e.g., counter-clockwise braiding of Majorana modes γ A , γ B

15
: A B (how this can be implemented may depend on the tree structure and the initial positions of γ A and γ B ). In the case of two Majorana modes that are provided by one topological segment located in a single chain before and after the braiding, the Klein factors cancel in the spin representation, so the additional spin mediating the coupling at the junction 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 more complicated operation in the spin system. By choosing, e.g., , we obtain: if the spins outside the ferromagnetic intervals are polarised in the x-direction. When expressed in terms of the Pauli matrices τ for the two 'topological' qubits involved (two ferromagnetic intervals, cf. ref. 14 ), this gives y z z 0 1 2 A detailed description of this operation in the spin language is given in the section Methods. However, for two 'topological' intervals in the same chain we obtain a similar expression, but without the intermediate ancillary spin: Thus one obtains a two-qubit operation.
In a more general situation with arbitrary initial position of two distant braided boundaries (and associated Majorana modes), the braiding operation involves, apart from these two qubits, the ancillary spins at all intermediate vertices as well as the parity (qubit-flip) operators ∏σ x for all intermediate qubit intervals. Thus, the braiding implements 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' qubit intervals, 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 between distant qubits since two-qubit gates between neighbours are sufficient, as they form a universal set of gates together with single-qubit operations. Furthermore, one can also view this subtlety from a different perspective. Instead of thinking 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 quantum computation. This viewpoint may be useful for simulations of fermionic Hamiltonians (see e.g. ref. 18 ), including many-body solid-state models and complex individual molecules.
A further remark concerns the symmetry and the braiding procedure: Each of the chains considered belongs to the BDI symmetry class 23 , with a time-reversal-type symmetry  such that = 1 2  . In a single chain (19b), a Majorana zero mode appears at each boundary between topological and non-topological regions. A vertex connecting three chains can be viewed as an edge of a 1-D system. Here the symmetry becomes crucial 24,25 . If the  symmetry is preserved by the chain coupling at the vertex, the edge (vertex) carries an integer () 'topological' charge. In our case this allows for configurations with more than one Majorana zero mode at the vertex and an unwanted extra degeneracy when during the braiding procedure this vertex connects two or three 'topological' regions. A  -breaking chain coupling, however, places the system to the D class with a  2 invariant, and typically one (or no) Majorana zero 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 .
a system of Ising spin chains with a local transverse magnetic field h α (j), which corresponds to the locally tunable chemical potential in the fermionic system. Assuming J > 0, any interval of spins with  h J in one of the chains is ferromagnetic, whereas  h J results in a trivial (paramagnetic) phase. The three chains are linked by the components of an additional central spin S 0 via 3-spin couplings of strength = αβ βα J J . This structure is depicted in Fig. 2. Now we consider the spin equivalent of braiding Majorana modes from two different topological intervals in the fermionic system. The topological intervals and adiabatic shifts of their boundaries within a chain can be translated to the ferromagnetic intervals in the spin representation exactly as for a single topological interval 14 : The fermionic-parity ground states | 〉 | 〉 0 , 1 of a topological interval correspond to linear combinations ≡ ↑↑↑ − ↓↓↓ 1 2 (20b) Figure 2. Ising spin chains in a T-geometry. A fermionic T-junction suitable for Majorana braiding 15 has a spin representation of this structure 14 , which is described by the Hamiltonian in eq. (19). The couplings between three Ising spin chains are mediated by the components of an additional spin S 0 , cf. ref. 9 and eqs (4, 5). The system can be manipulated by tuning transverse fields (not depicted here) that act on the individual spins σ α (j) of the three chains.