Entanglement generation via power-of-swap operations between dynamic electron-spin qubits

Surface acoustic waves (SAWs) can create moving quantum dots in piezoelectric materials. Here we show how electron-spin qubits located on dynamic quantum dots can be entangled. Previous theoretical and numerical models of quantum-dot entanglement generation have been insufficient to study quantum dynamics in realistic experimental devices. We utilize state-of-the-art graphics processing units to simulate the wave-function dynamics of two electrons carried by a SAW through a two-dimensional semiconductor heterostructure. We build a methodology to implement a power-of-swap gate via the Coulomb interaction. A benefit of the SAW architecture is that it provides a coherent way of transporting the qubits through an electrostatic potential. This architecture allows us to avoid problems associated with fast control pulses and guarantees operation consistency, providing an advantage over static qubits. For interdot barrier heights where the double occupation energy is sufficiently greater than the double-dot hopping energy, we find that parameters based on experiments in GaAs/AlGaAs heterostructures can produce a high-fidelity root-of-swap operation. Our results provide a methodology for a crucial component of dynamic-qubit quantum computing.

Figure 1

(a) Cross section of the potential layout in the region of high barrier alongside a trace of the initial state of the wave function along the $x$ dimension. (b) Schematic of a SAW-based power-of-swap device. Electrons are carried by SAWs from bottom to top (positive $y$ coordinate) along two channels, undergoing a power-of-swap operation in the central gate region. Dotted lines show the path electrons can take through the device.

Figure 2

Two-particle spatial wave functions. (a) Ground state $|{\mathrm{\Psi }}^{\text{S}}\rangle$ ($y_1=0$ and $y_2=0$). (b) First excited state $|{\mathrm{\Psi }}^{\text{A}}\rangle$ ($y_1=0$ and $y_2=0$). (c) Combination of the ground state and first excited state $|{\mathrm{\Psi }}^{\text{LR}}\rangle$. The first particle is localized in the left channel and the second particle is localized in the right channel ($y_1=0$ and $y_2=0$). (d) Gaussian spread of both particles in the $y$ dimension ($x_1=0$ and $x_2=0$). All four panels show the wave function divided by its extremum, with the $z$ axis in arbitrary units.

Figure 3

Entanglement generation using the Coulomb tunneling method. Top and middle row: trace over the $x$ and $y$ dimensions, respectively, for the initial state (left), root-of-swap state (center), and swap state (right) of the wave function. Bottom row: trace over the second particle for the initial state (left), root-of-swap state (center), and swap state (right) of the wave function. Coordinates are chosen to be in the SAW frame of reference, with $y=0$ corresponding to a SAW minimum and $x=0$ the peak of the tunnel barrier.

Figure 4

Probability of swap as a function of tunnel barrier height for fixed interaction duration. Time evolution simulation results (circles) are fit using Eq. (9) (solid line). The parameters $J_0=2.888{\text{ps}}^{-1}$ and $b=0.933{\text{meV}}^{-1}$ were found numerically. The inset figure illustrates the occupation of the computational basis states as well as the double-occupancy states. In this example, the input state ${|\uparrow \downarrow \rangle }_{\text{LR}}$ undergoes a root-of-swap operation with finite tunnel barrier potential ramps.

Figure 5

Comparison to analytical models. Power-of-swap frequency as a function of effective wave-function spread in the $z$ dimension. Coulomb softening accounts for the finite $z$ dimension and plays an important role in determining the rate of the exchange interaction.

Figure 6

Entanglement generation via the collision of two electrons. Top and middle row: trace over the $x$ and $y$ dimensions, respectively, for the initial state (left), root-of-swap state (center), and swap state (right) of the wave function. Bottom row: trace over the second particle for the initial state (left), root-of-swap state (center), and swap state (right) of the wave function. Coordinates are chosen to be in the SAW frame of reference, with $y=0$ corresponding to a SAW minimum and $x=0$ the middle of the harmonic channel.