Compressed Optimization of Device Architectures for Semiconductor Quantum Devices

Recent advances in nanotechnology have enabled researchers to manipulate small collections of quantum-mechanical objects with unprecedented accuracy. In semiconductor quantum-dot qubits, this manipulation requires controlling the dot orbital energies, the tunnel couplings, and the electron occupations. These properties all depend on the voltages placed on the metallic electrodes that define the device, the positions of which are fixed once the device is fabricated. While there has been much success with small numbers of dots, as the number of dots grows, it will be increasingly useful to control these systems with as few electrode voltage changes as possible. Here, we introduce a protocol, which we call the “compressed optimization of device architectures” (CODA), in order both to efficiently identify sparse sets of voltage changes that control quantum systems and to introduce a metric that can be used to compare device designs. As an example of the former, we apply this method to simulated devices with up to 100 quantum dots and show that CODA automatically tunes devices more efficiently than other common nonlinear optimizers. To demonstrate the latter, we determine the optimal lateral scale for a triple quantum dot, yielding a simulated device that can be tuned with small voltage changes on a limited number of electrodes.

Figure 1

Example implementations of the CODA protocol, including comparisons of using the $L_1$ norm (the sum of absolute values) and the $L_2$ norm (the Euclidian length) of the voltage changes needed to achieve the desired values of the quantities of interest (dot occupations, tunnel couplings, etc.). (a) A simulated quantum-dot device maps electrode voltages to quantities of interest. Generically, there are more electrodes in a device than quantities of interest. (b) There are many combinations of electrode voltages that result in a target system state (gray dashed line). By choosing the solution associated with the minimum $L_1$ norm of voltage changes, described in Eq. (1) (purple), we simultaneously ensure that voltages are changed by small amounts on a small number of electrodes. Minimization of the $L_2$ norm (orange) does not ensure that voltage changes will be applied to a small number of electrodes. (c)–(f) A depiction of the CODA-algorithm tuning voltages to obtain $q_{\mathrm{target}}=1$ in a simple system. The voltages that yield the target quantity of interest are indicated with the circular segment in panels (c) and (e). Starting at $\delta v_1=\delta v_2=0$ mV (green circle), we calculate the Jacobian $\mathbf{J}$ and find all of the solutions to the linear equation $q_{\mathrm{target}}-q_{\mathrm{current}}=\mathbf{J}\cdot \delta \mathbf{v}$, shown as a green dashed line. We then find the voltage changes on this line that minimize the $L_1$ and $L_2$ norms, the blue circles in panels (c) and (e), respectively. Using the derivatives at this new point, we again estimate the voltage changes required to hit the target (blue dashed line) and again find the solution that minimizes the appropriate norm (black circle). The solution found here converges on $q_{\mathrm{target}}$. Additionally, we obtain the voltage changes associated with the minimum of the appropriate norm, indicated by the purple color scale in panel (c) and the orange color scale in panel (e). (g)–(i) CODA tuning of an eight-dot device. (g) A schematic of a simulated eight-dot device, with metal electrodes colored yellow or orange (lower level) and green (upper level). Here, the objective is to increase the occupation of the rightmost quantum dot (underneath the orange electrode) by one electron, keeping all other dot occupations and tunnel couplings unchanged. (h),(i) Visualization of the voltage changes obtained by the optimization protocol, plotted on a logarithmic color scale (electrodes with voltage changes less than 0.5 $\mu \mathrm{V}$ are colored white), minimizing the (h) $L_1$ and (i) $L_2$ norms of the voltage changes. As expected, minimization of the $L_1$ norm ensures that a limited number of electrode voltages are changed, whereas minimization of the $L_2$ norm does not.

Figure 2

A demonstration of the extensibility of the CODA protocol using a simple model involving up to 100 quantum dots. (a) A diagram of the simulated device. The quantities of interest in this device are the occupations $n_i$ of the quantum dots (dashed circles) and the tunnel rate ${\tau }_i$ between the dots (dashed double arrows). Each quantum dot has three electrodes in close proximity: one located directly above the quantum dot (blue square) and two located above and to either side of the quantum dot (red squares). Additionally, there is an electrode separating each pair of dots (thin blue rectangle). The dependence of $n_i$ and ${\tau }_i$ on the electrode voltages is defined phenomenologically in Eqs. (2) and (3), respectively. (b)–(d) Given a device with $m$ quantum dots, we use a variety of nonlinear optimizers (including CODA) to find the changes in electrode voltages that add one electron to the leftmost quantum dot, keeping all other occupations and tunnel rates constant. We consider devices with $m$ ranging from 2 to 100. The results corresponding to CODA are shown in green and the results corresponding with the other nonlinear optimizers are shown in hues of purple. In panel (b), we show the average over all considered devices of the maximum voltage change applied to any of the electrodes. The standard deviation is smaller than the points used in this plot. The uniformity of the results here indicates that all of the optimizers find solutions with similar voltage moderation. In panel (c), we show the number of electrodes used in each solution. The CODA procedure consistently requires fewer electrodes than any of the other nonlinear optimizers considered. In panel (d), we show the number of function calls used by each optimizer. CODA requires roughly an order of magnitude fewer function calls than any of the other nonlinear optimizers considered.

Figure 3

The protocol used to compare the “tunability” of device designs. Given multiple simulated devices, we use the CODA protocol to find the minimum $L_1$ norm of voltage variations needed to induce a common change in each device (e.g., to change the dot occupations in device #1, device #2, etc.). The device with the minimum norm can simultaneously provide voltage moderation and sparsity and should therefore be regarded as the most “tunable” device.

Figure 4

The use of CODA to optimize the triple-quantum-dot designs shown in Fig. 3. (a) Every device in the series has an identical electrode layout, save for the lateral scale of the device, which we characterize in terms of the width of the paddle electrode. The $\mathrm{Si}$/$\mathrm{Si}\mathrm{Ge}$ heterostructures for all devices considered are identical. (b) For each device in the series, we use the CODA protocol to lower all three dots from an occupation of 30 electrons to one electron, keeping the transmission coefficients fixed at 0.01. Additionally, the single-electron dots are required to have orbital excitation energies greater than 1 meV. (c)–(e) A visualization of the optimized voltage variations required to tune each device. Voltage changes are shown in red and blue and the resulting electron density distributions are shown in black. For smaller devices [e.g., panel (c)], fewer electrodes are required to tune the device. However, larger voltage changes must be applied to those electrodes, resulting in a high-voltage $L_1$ norm. For larger devices [e.g., panel (e)], quantum dots no longer form underneath the paddle electrodes (e.g., the rightmost dot, indicated with purple arrows), so that many electrodes are required to tune the device. The balancing of these effects leads to a local minimum in the voltage $L_1$ norm, corresponding to the device shown in panel (d).