When driving a car, pushing the brakes is a simple action for the driver; however, it is a complex process involving a series of mechanical and electrical devices. Some advanced, green automobile technologies — such as electric or hybrid models — are equipped with a system that can turn the energy released during braking into electric energy. This energy is stored by the so-called supercapacitor, which consists of a large array of capacitors.

A capacitor is a device that stores energy in the form of electric charge. In vehicles, this property makes it suitable for braking and accelerating. What is happening in the car is that the supercapacitor “steals” energy from its movement, slowing it down. Conversely, it can release energy to accelerate.

The simplest capacitor is the parallel-plate capacitor, which consists of two metallic plates separated by an insulator that can be a piece of plastic or air. When a voltage difference is established between the metallic plates, charges accumulate on their surfaces, polarizing the capacitor: positive charges accumulate in one plate and negative in the other. The electric charge distribution on the plates creates an electric field.

Schematic representation of a parallel-plate capacitor.

The capacitor is a fundamental device for many industries — including energy plants, proximity sensors, and automobile design. Its ubiquitous presence, in combination with its limitations, makes its optimization more pressing.

Computer simulation of such devices can help in designing better versions. However, such simulations can be extremely compute-demanding, specially under realistic conditions and for complex cases. Hence, there is a need to consider promising technologies such as quantum computing.

PASQAL developers use the differentiable quantum circuit algorithm (DQC) to simulate a parallel-plate capacitor in a grounded box. They calculated the potential and the electric field in the box for different distances separating the plates and using the vacuum as the insulator between them.

The electric potential for a chosen distance.

**Simulating the parallel plates capacitor with quantum algorithms**

DQC is an algorithm created at PASQAL to solve partial differential equations using quantum computing. Inspired by the main ideas of physics-informed machine learning, it allows us to leverage quantum computers to solve partial differential equations executable in any current quantum processor. Physics-informed machine learning is a classical computing method that aims to integrate data and the laws of physics governing the system under study.

To simulate the electric field generated by the charge distribution on the capacitor plate, we solve a differential equation known as Gauss’s law. When solving differential equations, the main object under attention is the derivatives, the rate of change of the physical process we want to find. To solve the differential equation for this system, we first fix the value of the potential on the grounded box and the capacitor’s plates; that is, we define the boundary conditions. With the DQC algorithm, we calculate derivatives exactly and incorporate boundary conditions analytically, which is a distinctive feature of this method.

The laws of nature dictate that if we divide the grounded box with the capacitor into four equal regions (with imaginary lines), the solution on the left side is the same as on the right side (it is symmetrical), but that it is different in the top than in the bottom (it is anti-symmetrical).

For the simulation presented in this post (see the movie below), our developers leverage the symmetry properties to calculate the electric field. In addition, the DQC method allows them to solve the differential equation in terms of the physical domain of interest, but also in terms of the distance between the plates as a variable instead of a fixed number. The latter allows to visualize solutions for any distance and help engineers to optimize the device according to the simulation.

In the following simulation of the DQC method (video), we illustrate how the electric field and the potential vary for different distances between the parallel plates.

**DQC vs conventional computational methods**

Conventional numerical techniques typically tackle differential equations by breaking down the space into a large number of small pieces. These approaches can be successfully used in many cases. However, they do not allow for the incorporation of symmetries directly and prevent the use of the distance between the plates as a symbol; instead, the distance must be fixed to a particular value.

Moreover, the accuracy of these numerical techniques is limited by the number of pieces that can be handled by the available computing resources, such as time and memory. On top of that, errors accumulate with each order of the derivative. Instead, DQC enables us to handle high-order derivatives without cumulative loss of accuracy. This property and the fact that DQC leverage the underlying physical laws of the system, implementing the boundary conditions and symmetries directly not only improves the quality of the solution but reduces the computational burden. Such a reduction of computational load can provide significant advantages in terms of speed and a distinct advantage for applications with a stringent limit on computational resources, ready to be executed on currently available quantum computers.

Capacitors are one of the main building blocks of most electronic devices, such as televisions, laptops, and mobile phones; we can find them everywhere in our everyday life. Solving simple but realistic models, such as the parallel plates capacitor, help improve its design, significantly impacting many industrial sectors, such as the automobilist, where advanced technologies will give us the power to stop and accelerate.

While here we showcased a simple example, PASQAL’s DQC algorithm is currently being explored to allow the users of quantum computers to simulate more realistic, complex engineering systems with high accuracy and speed, harnessing the power of classical and quantum computing.

**References**

Kyriienko, O., Paine, A. E., & Elfving, V. E. (2021). Solving nonlinear differential equations with differentiable quantum circuits. *Physical Review A*, *103*(5). https://doi.org/10.1103/physreva.103.052416.

Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. *Nature Reviews Physics*, *3*(6), 422–440. https://doi.org/10.1038/s42254-021-00314-5.

*Writer: Alexandra de Castro. Scientific contributors: Evan Philip and Vincent Elfving.*