High-fidelity singlet-triplet $S\text{−}{T}_{-}$ qubits in inhomogeneous magnetic fields

We propose an optimized set of quantum gates for a singlet-triplet qubit in a double quantum dot with two electrons utilizing the $S\text{−}{T}_{-}$ subspace. Qubit rotations are driven by the applied magnetic field and a field gradient provided by a micromagnet. We optimize the fidelity of this qubit as a function of the magnetic fields, taking advantage of “sweet spots” where the rotation frequencies are independent of the energy level detuning, providing protection against charge noise. We simulate gate operations and qubit rotations in the presence of quasistatic noise from charge and nuclear spins as well as leakage to nonqubit states. Our results show that, for silicon quantum dots, gate fidelities greater than $99\%$ should be realizable, for rotations about two nearly orthogonal axes.

Figure 1

(a) Illustration of a nonuniform magnetic field ${\mathbf{B}}^{\mathrm{m}}$ provided by a micromagnet (dark purple rectangle) fabricated above a double quantum dot, and a uniform external field ${\mathbf{B}}^{\mathrm{ext}}$ (blue arrow). Random, quasistatic Overhauser fields are also present, due to nuclear spins. (b) Singlet-triplet energy diagram, showing the dominant couplings between levels (arrows). A Bloch sphere representation of the $S\text{−}{T}_{-}$ qubit indicates the rotation axes associated with the different coupling terms. (c) (Top) Singlet-triplet energy diagram as a function of detuning $\epsilon$. $X$ rotations are performed at a detuning sweet spot (black circle at ${\epsilon }_X$) where the qubit energy levels are parallel and the splitting is set by $\mathrm{\Delta }{B}_x$. $Z^{\prime }$ rotations occur in the far-detuned regime $\left({\epsilon }_{Z^{\prime }}\ll 0\right)$, with a rotation axis $Z^{\prime }$ tilted slightly away from $Z$ on the Bloch sphere. (Bottom) Illustration of typical pulse sequences for implementing $X$ and $Z$ rotations. Measurement of the singlet probability is done at the detuning value ${\epsilon }_m>0$ in the $(0,2)$ charge state. The $Z$ protocol shows a Ramsey pulse sequence where the $Z$ rotation is implemented using a three-step sequence [25] to correct for the tilt of the $Z^{\prime }$ axis, as illustrated on the Bloch sphere.

Figure 2

Optimization of $X$ rotations. (a) Semilogarithmic plot of the state infidelity of an $X_{\pi }$ rotation from $|1\rangle$ to $|0\rangle ,1-F_s\left(X_{\pi }\right)$, as a function of the applied longitudinal field $B_z$, for several values of the field gradient $\mathrm{\Delta }{B}_x$, as indicated in the legend. (Inset) A similar plot showing the contributions to the state infidelity due to leakage, $P_{\mathrm{leak}}$, and the combined effect of detuning and Overhauser field fluctuations, $1-P_{-}^{\prime }$, for the case $g{\mu }_B\mathrm{\Delta }{B}_x=0.25 \mu \mathrm{eV}$. (b) A color density plot of $1-F_s\left(X_{\pi }\right)$ for an $X_{\pi }$ rotation, as a function of $B_z$ and $\mathrm{\Delta }{B}_x$. The red star indicates the optimal working point $g{\mu }_B(\mathrm{\Delta }{B}_x,B_z)=(0.25,0.75) \mu \mathrm{eV}$. (c) Larmor oscillations ($X$ rotations), and the corresponding Gaussian decay envelope, $[1\pm e^{-{(t/T_2\left(X\right))}^2}]/2$, obtained at the optimal working point, with $T_2^{*}\left(X\right)=\sqrt{2}\hslash /{\sigma }_h$.

Figure 3

Optimization of $Z^{\prime }$ rotations. (a) Semilogarithmic plot of the state infidelity of a $Z_{\pi }^{\prime }$ rotation, $1-F_s\left(Z_{\pi }^{\prime }\right)$, as a function of the detuning, for nearly optimal values of $g{\mu }_B\mathrm{\Delta }{B}_x$ and $g{\mu }_B{B}_z$. The red markers indicate the correspondence with curves in Fig. 2, with the star indicating the optimal working point. (b) $Z^{\prime }$ rotations performed at the starred point in (a), for ${\epsilon }_{Z^{\prime }}=-1.5$ meV.

Figure 4

Singlet-triplet energy diagrams, including the qubit states $|0\rangle$ and $|1\rangle$ and leakage states $|T_0\rangle$ and $|T_{+}\rangle$, as a function of the detuning, for two different tunnel coupling models. Here, state $|S^{\prime }\rangle$ lies outside the range of the plot. Solid lines: case (i), the constant tunnel coupling model, with $t_c=20\mu \mathrm{eV}$ and magnetic field values $g{\mu }_B(\mathrm{\Delta }{B}_x,B_z)=(0.25,0.75)\mu \mathrm{eV}$, which were optimized as described in the main text. On the right-hand side of the anticrossing, the qubit states correspond to $|0\rangle \simeq |S\rangle$ and $|1\rangle \simeq |T_{-}\rangle$. Dashed lines: case (ii), the detuning-dependent tunnel coupling model, $t_c\left(\epsilon \right)=t_{0}exp(\epsilon /{\epsilon }_0)$, with $t_0=20 \mu \mathrm{eV}$ and ${\epsilon }_0=1$ meV. Here too, the magnetic field parameters $g{\mu }_B(\mathrm{\Delta }{B}_x,B_z)=(0.3,0.7)\mu \mathrm{eV}$ were optimized to achieve a high $X_{\pi }$ gate fidelity. (Inset) The singlet mixing terms $\left(\right|cos\eta |,|sin\eta \left|\right)$, from Eq. (1). Note that $|sin\eta |\simeq J\left(\epsilon \right)/t_c$, the effective exchange energy, to a very good approximation.

Figure 5

Probability $P_{+}$ of leaking into state $|T_{+}\rangle$ as a function of time $t$ during $X$ rotations, for the optimal field values $g{\mu }_B(\mathrm{\Delta }{B}_x,B_z)=(0.25,0.75)\mu \mathrm{eV}$, plotted with the blue, solid line (the left hand axis). The envelope of the oscillations closely follows the probability $P_S$ of occupying state $|S\rangle$, as indicated by the maroon, dashed line (the right-hand axis).

Figure 6

The real and imaginary parts of the $\chi$ matrix elements for the fully optimized $X_{\pi }$ and $Z_{\pi }^{\prime }$ gates, obtained from Eq. (D2). Triangles indicate nonzero target values of the $\chi$ matrix elements.

Figure 7

The numerically computed probability $P_{-}$ of occupying the state $|T_{-}\rangle$ as a function of time $t$ during $X$ rotations in the absence of Overhauser field fluctuations, assuming the initial state $|S\rangle$. The dephasing envelope (dashed lines) is the Gaussian decay given by Eq. (E11, E12, E13), setting ${\sigma }_h=0$, with $T_{2\perp }^{*}\left(X\right)=4.7\mu \mathrm{s}$.

Figure 8

$Z^{\prime }$ rotations in the asymptotic regime $\epsilon \to -\infty$, obtained from analytic solutions for the qubit dynamics arising from Eq. (F1). Here, we plot the probability $P(-X)$ of being in the final state $|-X\rangle$ as a function of time $t$, for an initial state $|X\rangle$ and the optimal magnetic fields $(\mathrm{\Delta }{B}_x,B_z)=(0.25,0.75)\mu \mathrm{eV}$. The analytic solutions are numerically averaged over the Overhauser field fluctuations.