Electrically controlled quantum gates for two-spin qubits in two double quantum dots

Exchange-coupled singlet-triplet spin qubits in two gate-defined double quantum dots are considered theoretically. Using charge density operators to describe the double-dot orbital states, we calculate the Coulomb couplings between the qubits and identify optimal bias points for single- and two-qubit operations, as well as convenient idle positions. The same intuitive formulation is used to derive dephasing rates of these qubits due to the fluctuating charge environment, thereby providing the main considerations for a quantum computation architecture that is within current experimental capabilities.

Figure 1

(a) Geometry of the two double-dot qubits. We denote the left qubit as the target and the right qubit as the control; (b) Two-qubit Coulomb couplings and exchange interaction vs. target qubit bias ${\epsilon }_t$, for a fixed control qubit bias ${\epsilon }_c=0$ and interqubit distance $R=300$ nm. Locations of zero effective exchange are marked by Gray circles. Interdot bias is normalized to QD confinement and detuning is measured from the singlet anticrossing point.

Figure 2

(a) Normalized target and control qubit bias at symmetric low-bias (solid red line) and high-bias (solid blue line) points vs. $R$. The dashed blue line depicts the high-bias position for the $\gamma$-corrected sweet-spot. The dotted black line marks the $S(1,1)$-$S(0,2)$ anticrossing bias point; (b) $\gamma$ coupling values at low-bias (red line) and high-bias (blue line) sweet spots.

Figure 3

Dephasing times of target qubit vs. control qubit bias measured from the low-bias sweet spot, at $R=300$ nm. The control qubit is prepared in a singlet (blue squares) or triplet (red plus symbols) state, with magnetic field gradient of: (a,c) $\delta h_c=0$, (b,d) $\delta h_c=1\mu eV$. In (c) and (d) the sweet spot is shifted, satisfying $J_q-2({\beta }_q+\gamma )=0$ in both qubits.

Figure 4

(a) $\pi$ rotation about the $Z$ axis using a three-pulse sequence. $\delta h_t=1\mu eV$ and $R=300$ nm. The first and third rotations are performed at a bias position satisfying ${\tilde{J}}_t=\delta h_t$ ($\theta =\pi /4$); (b) Gate error as a function of $\delta h$, evaluated from numerical simulation (solid blue line), from Eq. (11) (dashed red line), and keeping only the ${\gamma }_{\theta }$ terms in Eq. (11) (dotted green line).

Figure 5

(a) CPHASE gate errors vs. $\delta h$, for a target qubit initially at $\left(\right|S\rangle -|T_0\rangle )/\sqrt{2}$, and $R=300$ nm. The solid blue (dashed green) line depicts phase error $\sim {(\delta h/2\gamma )}^2$ for control in singlet (triplet) state, and the dotted red line shows population errors $\sim \delta h/2\gamma$, when control is in triplet. Population errors when control is in singlet are very small (not shown); (b) CNOT gate errors vs. $\delta h_t$, for an initial target singlet state (same results are obtained for an initial target triplet state). The solid blue (dashed green) line depicts population errors for control in singlet (triplet) state, and the dotted red line shows phase error when control is in triplet. Phase error when control is in singlet is very small (not shown); (c) CNOT gate time vs. $\delta h_t$. Note that ${\tau }_{\mathrm{cp}}=0.026$ ns is fixed by $\gamma$, and most of the CNOT gate time is spent on the Hadamard target gates.

Figure 6

Multipole expansion contributions to interqubit Coulomb coupling vs. interqubit distance. Both double dots are biased at the $S(1,1)$-$S(0,2)$ anticrossing point. (a) $\beta$ terms; (b) $\gamma$ terms.