Controllable Anisotropic Exchange Coupling between Spin Qubits in Quantum Dots

The exchange coupling between quantum dot spin qubits is isotropic, which restricts the types of quantum gates that can be formed. Here, we propose a method for controlling anisotropic interactions between spins arranged in a bus geometry. The symmetry is broken by an external magnetic field, resulting in $XXZ$-type interactions that can efficiently generate maximally entangled Greenberger-Horne-Zeilinger states or universal gate sets for exchange-only quantum computing. We exploit the $XXZ$ couplings to propose a qubit scheme, based on double dots.

Figure 1

Typical architectures for generating $XXZ$ couplings. (a) A spatially varying magnetic field applied to a long spin bus, with a small local field on the qubits. (b) Coupled two-spin qubits, in a uniform magnetic field.

Figure 2

Energy spectra for two different spin buses of size (a) $N=10$ and (b) $N=11$, as a function of uniform magnetic field $B_b$. Note that energies and magnetic fields are expressed in units of the bus spin coupling $J_b$. For each plot, we show just the ten lowest energy levels.

Figure 3

Plots of effective coupling parameters $\tilde{J}$ (solid black line) and $\tilde{\Delta }$ (dashed red line), or the coupling anisotropy ratio $\tilde{J}/\tilde{\Delta }$, between two qubits coupled through an even-size bus, with $J_j=0.02J_b$ and a uniform magnetic field. (a) The two qubits are coupled to opposite ends of a bus of size $N=10$, with a variable field. (b) The two qubits are coupled to opposite ends of a bus with constant field $B_b=0.1J_b$ and a variable bus size. (c),(d) Qubit 1 is attached to node $j=1$ on a bus of size $N=10$, with $B_b=0.2J_b$. Qubit 2 is attached to node $j^{\prime }$. Here, $j^{\prime }=1$ refers to the case where both qubits are attached to node 1.