Double occupancy errors in quantum computing operations: Corrections to adiabaticity

We study the corrections to adiabatic dynamics of two coupled quantum dot spin qubits, each dot singly occupied with an electron, in the context of a quantum computing operation. Tunneling causes double occupancy at the conclusion of an operation and constitutes a processing error. We model the gate operation with an effective two-level system, where nonadiabatic transitions correspond to double occupancy. The model is integrable and possesses three independent parameters. We confirm the accuracy of Dykhne’s formula, a nonperturbative estimate of transitions, and discuss physically intuitive conditions for its validity. Our semiclassical results are in excellent agreement with numerical simulations of the exact time evolution. A similar approach applies to two-level systems in different contexts.

Figure 1

A realistic profile of the tunneling pulse (23), labeled with the characteristic duration $(T\approx 13\tau )$ and ramp time scales. The pulse shown has dimensionless strength $\delta =\frac{1}{2}$.

Figure 2

A profile of the instantaneous eigenvalues $\pm \epsilon \left(t\right)$ corresponding to $\lambda =2$ and the pulse shown in Fig. 1.

Figure 3

The analytic structure of the function $\epsilon \left(t\right)$ shown only in a segment of the upper half plane. The contour ${\mathcal{C}}_a$ is associated with transitions that occur due to the ramping on of the pulse, while contour ${\mathcal{C}}_b$ is associated with the ramping off. By Cauchy’s theorem, an integral on the contour ${\mathcal{C}}_a$ is equal to the integral on the contour ${\tilde{\mathcal{C}}}_a$. Bold lines represent branch cuts, dots represent branch points, and poles are denoted with a ×.

Figure 4

The probability for nonadiabatic transitions for $\lambda =2$ and $\eta =50$ as a function of $\delta$. We compare our semiclassical estimate according to expression (35) with results from numerical simulations of the exact quantum mechanical time evolution as done in Ref. 35. The results are in excellent agreement.

Figure 5

The probability for nonadiabatic transitions for $\lambda =4$ and $\eta =50$ as a function of $\delta$.

Figure 6

The probability for nonadiabatic transitions for $\lambda =1$ and $\eta =50$ as a function of $\delta$.

Figure 7

An example of a Riemann surface with two sheets, a branch point at $t=t_0$ and a contour $\mathcal{C}$ corresponding to a transition.

Figure 8

The contour $\mathcal{C}$ gives a subdominant correction to Dykhne’s formula.