Long-distance entangling gates between quantum dot spins mediated by a superconducting resonator

Recent experiments with silicon qubits demonstrated strong coupling of a microwave resonator to the spin of a single electron in a double quantum dot, opening up the possibility of long-range spin-spin interactions. We present our theoretical calculation of effective interactions between distant quantum dot spins coupled by a resonator, and propose a protocol for fast, high-fidelity two-qubit gates consistent with experimentally demonstrated capabilities. Our simulations show that, in the presence of noise, spin-spin entangling gates significantly outperform cavity-mediated gates on charge qubits.

Figure 1

Diagram of a DQD system. An electron in a Si/SiGe heterostructure is trapped in the double-well potential and allowed to tunnel between the two dots. The double-well potential is defined by aluminum gates on top of the heterostructure, shown in red and gold. The plunger gate PR above the right dot is capacitatively coupled to a probe in the microwave resonator. Nearby Co micromagnets create different magnetic fields $\overrightarrow{B_L}$ and $\overrightarrow{B_R}$ at the left and right dots, respectively.

Figure 2

A plot of the analytically (solid lines) and numerically (markers) calculated spin splitting ${\omega }_{z1}^{\prime \prime }$ as ${\epsilon }_1$ is adjusted for various fixed values of ${\mathrm{\Omega }}_1$. Here, we have set ${\omega }_r/2\pi =6\mathrm{GHz}, {\omega }_{z1}/2\pi ={\omega }_{z2}/2\pi =5.85\mathrm{GHz}, {\mathrm{\Omega }}_2/h=7.5\mathrm{GHz}, g_{x1}/h=g_{x2}/h=200\mathrm{MHz}, g_{AC1}/h=g_{AC2}/h=40\mathrm{MHz}$, and ${\theta }_2=\pi /2$.

Figure 3

Plot of effective coupling strength $J$ vs the spin splittings ${\omega }_{z1}={\omega }_{z2}=\omega$. Here, we have set ${\omega }_r/2\pi =6\mathrm{GHz}, {\mathrm{\Omega }}_1/h={\mathrm{\Omega }}_2/h=7.5\mathrm{GHz}, g_{x1}/h=g_{x2}/h=200\mathrm{MHz}, g_{AC1}/h=g_{AC2}/h=40\mathrm{MHz}$, and ${\theta }_1={\theta }_2=\pi /2$. Near $\omega /2\pi =6.02\mathrm{GHz}$, the spins are brought into resonance with the cavity, and our approximations break down.

Figure 4

For this simulation, we set ${\omega }_r/2\pi =6\mathrm{GHz}, {\omega }_{z1}/2\pi ={\omega }_{z2}/2\pi =5.86\mathrm{GHz}, {\mathrm{\Omega }}_1/h=7.3\mathrm{GHz}, {\mathrm{\Omega }}_2/h=7.5\mathrm{GHz}, g_{AC1}/h=45\mathrm{MHz}, g_{AC2}/h=40\mathrm{MHz}, g_{x1}=200\mathrm{MHz}$, and $g_{x2}=230\mathrm{MHz}$. Here, we plot the ${\epsilon }_i$ used to generate an entangling gate, along with the average gate fidelity of the spins' time evolution relative to a $\sqrt{\text{i}swap}$, maximized at each time step over local operations. Beginning at $t=100\mathrm{ns}$, we smoothly decrease both detunings to bring the spins into resonance. Then, starting at $t=550\mathrm{ns}$, we smoothly transition back. Total leakage does not exceed $3\%$ after $t=600\mathrm{ns}$.

Figure 5

Average infidelity of $\sqrt{\text{i}swap}$ gates for various values of ${\sigma }_{\epsilon }$. For each noise instance, at each time step we compute the average gate fidelity maximized over local operations. We then average over noise instances the minimum of the gate infidelity as a function of time. Here, we have set ${\omega }_r/2\pi =6\mathrm{GHz}, {\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=\mathrm{\Omega }, g_{AC1}/h=g_{AC2}/h=40\mathrm{MHz}$, and ${\theta }_1={\theta }_2=\pi /2$. (a) A plot of the infidelity for the spin qubit, where we have set ${\omega }_{z1}={\omega }_{z2}=\omega$ and $g_{x1}/h=g_{x2}/h=200\mathrm{MHz}$. To ensure that spins remain dispersively coupled to the resonator, at each $\mathrm{\Omega }$, we choose $\omega$ such that ${\omega }_r^{\prime }-{\omega }_{zi}^{\prime }=30g_{xi}^{\prime }$ (definitions given in the Supplemental Material [27]). Average gate times increase from $830\mathrm{ns}$ at $\mathrm{\Omega }/h=8\mathrm{GHz}$ to $4.3\mu \mathrm{s}$ at $\mathrm{\Omega }/h=20\mathrm{GHz}$. (b) A plot of the infidelity for the charge qubit, where we have set $g_{x1}=g_{x2}=0$. Average gate times for the charge qubit are shorter than for the spin qubit, ranging from $180\mathrm{ns}$ at $\mathrm{\Omega }/h=8\mathrm{GHz}$ to $2.4\mu \mathrm{s}$ at $\mathrm{\Omega }/h=20\mathrm{GHz}$.