Long-range entanglement for spin qubits via quantum Hall edge modes

We propose and analyze a scheme for performing a long-range entangling gate for qubits encoded in electron spins trapped in semiconductor quantum dots. Our coupling makes use of an electrostatic interaction between the state-dependent charge configurations of a singlet-triplet qubit and the edge modes of a quantum Hall droplet. We show that distant singlet-triplet qubits can be selectively coupled, with gate times that can be much shorter than qubit dephasing times and faster than decoherence due to coupling to the edge modes. Based on parameters from recent experiments, we argue that fidelities above $99\%$ could in principle be achieved for a two-qubit entangling gate taking as little as 20 ns.

Figure 1

Diagrammatic representation of driven oscillations of the state-dependent electric dipole moment. A gate voltage defining the quantum dots can bias the potential wells in such a way as to favor tunneling of the left spin into the right dot. Pauli spin blockade prevents this tunneling if the spins are in a triplet state. If this gate voltage is modulated, the result is an oscillating qubit-state-dependent force on the edge modes of the nearby quantum Hall droplet.

Figure 2

Schematic diagram of the system for qubit-qubit coupling: a disk of radius $R$ is formed from a 2DEG engineered in a GaAs/AlGaAs heterostructure. A singlet-triplet qubit, of interdot separation $\mathrm{\Delta }y$, is formed radially on either side. The qubits are electrostatically coupled to the edge modes of the QH disk. By driving oscillations in the qubits, we can effectively couple multiple qubits to one another using the QH disk as a mediator.

Figure 3

Qubit-edge coupling given by Eq. (12) as a function of disk radius, $R$, with qubit-edge separation and interdot separation fixed at $y_1=150\mathrm{nm}$ and $\mathrm{\Delta }y=250\mathrm{nm}$, respectively; (inset) qubit-edge separation $y_1$ (with disk radius and interdot separation fixed at $R=8\mu \mathrm{m}$ and $\mathrm{\Delta }y=250\mathrm{nm}$, respectively) and interdot separation (for $R=8\mu \mathrm{m}$ and $y_1=150\mathrm{nm})$.

Figure 4

Simulation of the gate for a fixed detuning $\mathrm{\Delta }/\kappa =300$ using both the polaron shifted (red) and unshifted (blue) descriptions using a square pulse (a) and a shaped pulse (b), with the chosen equation parameters: $g_0=7.29\times {10}^8$ rad/s (see Sec. 3), $\kappa =17.2\times {10}^6$ rad/s, and qubit dephasing times of $T_2=7\mu \mathrm{s}$ and $T_2^{*}=700\mathrm{ns}$, from Ref. [16]. The shaped pulse was chosen to achieve maximum fidelity for this value of detuning.

Figure 5

Average fidelity as a function of detuning $\mathrm{\Delta }$ for high-frequency noise using the analytical expression of Eq. (19a) (solid blue line), and numerical simulations using the polaron shifted master equation of Eq. (4) (red diamonds) and the unshifted master equation from Eq. (3) (yellow squares). Parameters are chosen as in Fig. 4: $g_0=7.29\times {10}^8$ rad/s, $\kappa =17.2\times {10}^6$ rad/s, and qubit dephasing time of $T_2=7\mu \mathrm{s}$.

Figure 6

Average fidelity as a function of detuning $\mathrm{\Delta }$ for high-frequency noise (blue), low-frequency noise (red), and for all sources of noise (yellow), using both the analytic expressions (solid lines) of Eqs. (19a), (20b), and (21) and the full numerical simulations in the polaron shifted frame (diamonds, squares, circles). Parameters are chosen as in Fig. 4: $g_0=7.29\times {10}^8$ rad/s, $\kappa =17.2\times {10}^6$ rad/s, and qubit dephasing times $T_2=7\mu \mathrm{s}$ and $T_2^{*}=700$ ns.

Figure 7

Average infidelity as a function of qubit-edge coupling $g$, at optimal detuning, with the optimal detuning chosen for high-frequency noise (blue) and the low-frequency noise (red).