Dynamic creation of a topologically-ordered Hamiltonian using spin-pulse control in the Heisenberg model

Hamiltonian engineering is an important approach for quantum information processing, when appropriate materials do not exist in nature or are unstable. So far there is no stable material for the Kitaev spin Hamiltonian with anisotropic interactions on a honeycomb lattice, which plays a crucial role in the realization of both Abelian and non-Abelian anyons. Here, we show two methods to dynamically realize the Kitaev spin Hamiltonian from the conventional Heisenberg spin Hamiltonian using pulse-control techniques based on the Baker-Campbell-Hausdorff (BCH) formula. In the first method, the Heisenberg interaction is changed into Ising interactions in the first process of the pulse sequence. In the next process of the first method, we transform them to a desirable anisotropic Kitaev spin Hamiltonian. In the second more efficient method, we show that if we carefully design two-dimensional pulses that vary depending on the qubit location, we can obtain the desired Hamiltonian in only one step of applying the BCH formula. As an example, we apply our methods to spin qubits based on quantum dots, in which the effects of both the spin-orbit interaction and the hyperfine interaction are estimated.


I. DETAILED EXPLANATION OF THE DIRECT METHOD
Let us show a simple example of the derivation process of the direct method for the six qubits in Fig. 1. In Fig. 1, the x and y inside the circles show the applications of π/2-pulses around the x and y axes to the qubit of the corresponding sites, respectively. More specifically, the x inside the circle at site i indicates H → exp[−i(π/2)X i ]H exp[i(π/2)X i ].
(1) Similar operations for the y and z rotations are also applied. The Heisenberg Hamiltonian of the six qubits is given by H S = x 12 + y 12 + z 12 + x 23 + y 23 + z 23 + x 34 + y 34 + z 34 + x 45 + y 45 + z 45 + x 56 + y 56 + z 56 + x 61 + y 61 + z 61 , where x jk ≡ J jk X j X k , y jk ≡ J jk Y j Y k , and z jk ≡ J jk Z j Z k . The process H x r1 = P x † 1 H S P x 1 , shown in Fig. 1(a), rotates the directions of the qubits 2, 4, and 6, and we obtain Pulse mapping to select only the x-link of the Kitaev Hamiltonian from the Ising Hamiltonian of (a). These figures are part of Fig.2 (a,b)

of the main text
In the next process, P x 2 rotates the directions of the qubits 1, 2, 4, and 5 [ Fig. 1(b)] around the y-axis. The transformed Hamiltonian H x r2 is given by In this second process, the rotation can be carried out around the z-axis, instead of the y-axis. Now, we have These are x-links in the ten qubits of Fig. 1 of the main text. Similarly, we can obtain y-and z-links.

II. DETAILED EXPLANATION OF THE EFFICIENT METHOD
In the efficient method to create the Kitaev Hamiltonian, the honeycomb lattice sites are divided into small units, as shown in Fig. 2 (a) and (b) in this supplemental material. In Fig. 2, the x, y and z inside the circles show the applications of π/2-pulses around the x, y and z axes to the qubit of the corresponding sites, respectively. H eff R is produced by applying the appropriate pulses, given by Here P eff means the product of these operations, such as The qubit sites without the rotations are the boundary qubits of the unit of Fig. 2(b). Let us look at the twelve qubits in Fig. 2

(b). The original Heisenberg Hamiltonian is expressed as
When (1) π/2-pulses around the z axis are applied to the qubits 1 and 4, (2) π/2-pulses around the y axis are applied to the qubits 2 and 5, and (3) π/2-pulses around the x axis are applied to the qubits 3 and 6, the Heisenberg spin Hamiltonian H S is changed into the rotated Hamiltonian H R given by By applying Eq. (45), we obtain the Kitaev Hamiltonian of the unit cell, shown in Fig. 2(b), given by such as

III. UNWANTED TERMS
The BCH formula generates the unwanted terms H BCH uw . Here we show the concrete form of this term of the unit of Fig. 2 where The 'lu ', 'lm', 'ld', 'ru', 'rm' and 'rd' show the relative positions of the honeycomb lattices around the center green honeycomb lattice in Fig. 2(a). This equation can be written in a more compact manner when using equations such as The unwanted terms which includes the qubits of the single honeycomb lattice are given by For example, the unwanted terms of a single honeycomb lattice [upper part of Eqs. (21)] are given by H (l+1;y,z) a1 H (l+2;x,y) a1 for l = 1, 4.

IV. EFFECT OF SPIN-ORBIT INTERACTION
Here we show a detailed derivation of the effect of the spin-orbit (SO) interactions. The spin-orbit interaction is described by where σ j = (X j , Y j , Z j ), and the magnitudes of the spin-orbit vectors c so = (c x , c y , c z ) and d so = (d x , d y , d z ) are 10 −2 smaller than J z 2 . Let us first consider the effective Hamiltonian of the spin-orbit interaction in the center honeycomb lattice of Fig. 2(b). We express the spin-orbit interaction after the single-qubit rotations by V R so = ∑ jk V R,so jk . Then V so and V R so are given by Thus, the effective Hamiltonian of the spin-orbit interaction in the center honeycomb lattice is given by