Exchange-based two-qubit gate for singlet-triplet qubits

We analyze a simple exchange-based two-qubit gate for singlet-triplet qubits in gate-defined semiconductor quantum dots that can be implemented in a single exchange pulse. Excitations from the logical subspace are suppressed by a magnetic field gradient that causes spin-flip transitions to be non-energy-conserving. We show that the use of adiabatic pulses greatly reduces leakage processes comapred to square pulses. We also characterize the effect of charge noise on the entanglement fidelity of the gate both analytically and in simulations; demonstrating high entanglement fidelities for physically realistic experimental parameters. Specifically we find that it is possible to achieve fidelities and gate times that are comparable to single-qubit states using realistic magnetic field gradients.

Figure 1

Schematic diagram of a two-singlet-triplet-qubit configuration. The four quantum dots are indexed by $\{1,2,3,4\}$, and are slanted to indicate that a specific physical arrangement is unimportant. The intraqubit exchange couplings required for single-qubit operations are shown in dashed green $(J_{12}$ and $J_{34})$; and the interqubit couplings required for our two-qubit gate operation are shown in solid blue $(J_{14}$ and $J_{23})$. Important magnetic field gradients are depicted by thin broken grey lines joining solid discs, and labeled with the appropriate symbols: $\Delta B=\frac{1}{2}\left(B_1+B_2-B_3-B_4\right)$ and ${\Delta }_{ij}=B_i-B_j$.

Figure 2

Schematic showing the alignment of the single-qubit Bloch sphere. The magnetic field gradient $\Delta B$ rotates an arbitrary qubit state $|\psi \rangle$ about the $z$ axis of the Bloch sphere, while the exchange interaction $J_{12}$ rotates it about the $x$ axis.

Figure 3

(a) The square, linear, sinusoidal, and xsinusoidal adiabatic pulse profiles discussed in the text. Amplitudes are chosen to preserve gate operation time, and hence the average value of $J$. We have here chosen $J_{\mathrm{avg}}=0.18\mu \text{eV}$, which corresponds to a gate time of roughly $11.5\text{ns}$. (b) Leakage from the logical subspace during the operation of the gate for each of the square, linear, sinusoidal, and xsinusoidal pulses in (a). Note that the use of adiabatic profiles can significantly reduce leakage errors at the end of the gate operation.

Figure 4

A log-log plot of leakage immediately after a gate operation as a function of $J/\mu \Delta B$. Upper bound predictions from adiabatic perturbation theory (dashed lines) are compared to data from simulations (solid traces) for the square (blue), linear (green), sinusoidal (red), and xsinusoidal (purple) adiabatic pulses, demonstrating reasonable agreement.

Figure 5

Simulated leakage and entanglement fidelity immediately after a single two-qubit gate operation as a function of $J_{\mathrm{avg}}$ and $\Delta B$, with gate time corresponding to each labeled $J_{\mathrm{avg}}$ shown on the far right axes. Panels (a) and (c) show the leakage results with and without charge noise respectively, and panels (b) and (d) show the entanglement fidelity results with and without charge noise respectively. The color maps for leakage and entanglement fidelity are shown in the corresponding column, and are chosen such that comparable colors indicate comparable errors. The color maps are white at $1\%$ error, with larger error shown in shades of grey, and lesser error in shades of blue. Contours corresponding to 0.01–0.10% error are drawn at 0.01% intervals.