Characterization of an exchange-based two-qubit gate for resonant exchange qubits

Resonant exchange qubits are a promising addition to the family of experimentally implemented encodings of single qubits using semiconductor quantum dots. We have shown previously that it ought to be straightforward to perform a CPHASE gate between two resonant exchange qubits with a single exchange pulse. This approach uses energy gaps to suppress leakage rather than conventional pulse sequences. In this paper we present analysis and simulations of our proposed two-qubit gate subject to charge and Overhauser field noise at levels observed in current experiments. Our main result is that we expect implementations of our two-qubit gate to achieve high fidelities, with errors at the percent level and gate times comparable to single-qubit operations. As such, exchange-coupled resonant exchange qubits remain an attractive approach for quantum computing.

Figure 1

Bloch sphere schematic depicting the effect of intraqubit exchange couplings on qubit states. Under $J_{12}$ and $J_{23}$ couplings, states rotate about the same axes as those of the rotations indicated in green and red, respectively. If both $J_{12}$ and $J_{23}$ are equal to $J_z$, states rotate about the $z$ axis as indicated by the emboldened blue rotation. Qubit states are defined with respect to this rotating reference frame. Resonant oscillatory pulses involving variations of the $J_{12}$ and $J_{23}$ couplings allow arbitrary rotations, and hence provide universal single qubit control. As these intraqubit couplings are controllable using gate voltages, this allows for universal electronic control of qubits.

Figure 2

Physical arrangements of quantum dots (or geometries) considered in this work. Dashed green lines indicate large intraqubit exchange couplings, with solid blue lines indicating the weaker exchange coupling between qubits that can be used to effect two-qubit gates. We will focus mainly on the (a) butterfly and (b) linear geometries, with the (c) rectangular geometry being another alternative.

Figure 3

Energy level spectrum of the $m_z=1$ subspace of a two resonant exchange qubit system. Energy levels are each labeled, with starred levels indicating logical eigenstates. The eigenstates corresponding to each label are explicitly written out in the Supplemental Material [28]. Under butterfly coupling $J_c$, the logical states couple to like colored/dashed states leading to a minimal energy gap of $\sim 3J_z/2$. Note that the like colored/dashed levels form distinct subspaces. Under linear coupling, all colored/thick energy levels are coupled (including logical states), leading to a minimal energy gap of $\sim J_z/2$.

Figure 4

(a) Leakage $\mathcal{L}$ and (b) infidelity $1-\mathcal{F}$ (bottom) at the end of a single two-qubit gate operation for several different adiabatic profiles in both the butterfly and linear geometries. The $x$ axis is shared between the plots, and is over $J_c/\mathrm{\Delta }E$: the ratio of interqubit coupling and the geometry-dependent minimum energy gap. The upper $x$ axis provides a conversion from $J_c/\mathrm{\Delta }E$ to $J_c/J_z$ for the butterfly geometry (an expression in terms of the controllable parameters of the model). Colors indicate the adiabatic pulse profile used, while solid (dashed) lines denote that the butterfly (linear) geometry is being considered. Crucially, these plots demonstrate that use of adiabatic pulses can improve suppression of leakage by several orders of magnitude, provided that $J_c$ is small compared to the energy gap, and that this leads to a corresponding increase in gate fidelities.

Figure 5

Contribution to fidelity (${\mathcal{F}}_{\mathrm{noisy}}-{\mathcal{F}}_{\mathrm{noiseless}}$) from (a) pseudostatic (dc) and (b) white (hf) charge noise on each of the five exchange couplings active during a single gate operation using the butterfly geometry. The color map used is divergent at a fidelity of 0.9, as shown above, with blue colors indicating performance in excess of 0.9, and gray colors indicating performance below 0.9. Since dc noise is essentially dependent only on the ratio $J_c/J_z$, we plot in (a) the infidelity contribution as a function of $J_c/J_z$ and ${\sigma }_{\varepsilon }$. The value of ${\sigma }_{\varepsilon }$ from Table 3 is indicated by a dashed line at $15.8\mu \text{V}$. In (b) we plot the fidelity contribution from white charge noise for $D=0.244\mu {\text{V}}^2\text{ns}$ as in Table 3. The qualitative behavior of both plots is predicted by Eq. (4).

Figure 6

Contribution to fidelity (${\mathcal{F}}_{\mathrm{noisy}}-{\mathcal{F}}_{\mathrm{noiseless}}$) from a pseudostatic Overhauser field with standard deviation ${\sigma }_B$. In (a) the fidelity contribution is plotted as a function of $J_c/J_z$ and $J_z$ for ${\sigma }_B=2\text{mT}$, revealing that Overhauser noise contributes to infidelity in a manner somewhat proportional to gate time (contours of constant gate time are indicated by gray lines), except at large $J_c/J_z$ corresponding to large leakage. In (b) the fidelity contribution is plotted as a function of $J_c/J_z$ (with fixed $J_z=1.65\mu \text{eV}$) and ${\sigma }_B$. The radial contours of (b) (in regions where leakage does not dominate) imply that infidelity is proportional to ${\sigma }_B/J_c$ which is roughly $\propto \tau {\sigma }_B$, thus corroborating that the infidelity grows roughly as gate time. The color map used is the same as in Fig. 5.

Figure 7

Entanglement fidelity for the butterfly geometry subject to charge and Overhauser noise, as described in the main text, with noise levels calibrated to correspond to experiment (see Table 3). Performance is found to be limited predominantly by leakage (from above), intraqubit dc charge noise (from below), and Overhauser noise (from the left), leading to an optimal ratio of $J_c/J_z\sim 0.6$ where fidelity reaches $\sim 95\%$ for large enough $J_z$. The color map used is the same as in Fig. 5.

Figure 8

Entanglement fidelity for the linear geometry subject to charge and Overhauser noise, as described in the main text, with noise levels calibrated to correspond to experiment (see Table 3). A single ${\sigma }_x^A{\sigma }_y^B$ echo pulse is applied at $\tau /2$ in order to echo out both intraqubit psuedostatic charge noise and non-${\sigma }_z^A{\sigma }_z^B$ two-qubit phase, which restores maximum fidelities to $\sim 97\%$. Fidelities are limited predominantly by leakage (from above), Overhauser (from left and below), and high frequency noise (from below). The color map used is the same as in Fig. 5.