When someone learns to play a string instrument, like a guitar, this person starts producing melodies very soon. However, creating melodies in drums is an incredibly hard task. Why does this happen? There is actually a profound mathematical explanation. Vibrations on a string are described by equations in one spatial dimension, while the membrane of a drumhead is a two-spatial dimensional situation. Yes, adding a dimension has such dramatic consequences.

Vibrations on strings with their ends fixed or attached to a point have frequencies that are multiples of one another, called modes or harmonics. The first mode corresponds to the simple vibration, with the string going up and down. The second mode, which has a point in the middle — where the string does not move at all, called a node — has twice the frequency of the first mode. The next mode has two nodes and three times the first node’s frequency. You got the point. Through these modes, we create the notes that we combine to produce melodies.

So, what happens when we hit the drums? In a drumhead — an elastic circular membrane rigidly attached at its perimeter — the modes are much more complicated. Their frequencies are not integer multiples of each other, affecting our ability to create melodies. Vibrations produced in strings and drums can be expressed in differential equations, mathematical formulas that predict the evolution and patterns of behaviors in physical systems.

a) First three modes on a string, b) First three modes on a circular membrane.

Differential equations are an essential tool in science and engineering. They play a central role in our understanding of a variety of phenomena, from how atoms combine to create molecules to how airplanes fly, from earthquake propagation to the expansion of the universe.

Scientists employ powerful classical computers to simulate the dynamics of complex systems using differential equations. However, in many cases finding the solution is beyond the reach of classical computers.

Quantum computing is a promising technology with the potential to largely outperform classical computing to address complex relevant problems, such as how waves propagate in an earthquake or how the eardrum works.

Quantum computer hardware and software are based on the principles of quantum mechanics, the laws of physics that describe how subatomic particles behave in the atom and how atoms combine to create molecules.

# The drumhead modes recreated with PASQAL’s quantum software

Our developers have created an algorithm called *differentiable quantum circuits* (DQC) to solve differential equations. They have been testing the potential of this algorithm successfully using a quantum computer simulator.

This is the first attempt to use the DQC method to simulate the evolution of a system — the drumhead — knowing its initial conditions (the status of the membrane at the beginning of the study) and its boundary conditions (the status of the membrane’s border).

In the movie, we show examples of how our technique reproduces some drumhead vibration patterns with fidelity.

The first simulation corresponds to the simplest mode that such a membrane supports; that is, the whole membrane going up and down, with no nodes.

In the second simulation, we appreciate a different vibration pattern. Unlike strings, the drumhead nodes are not points but lines, straight lines in the radial direction and concentric circles. For this second mode, we recognize a circular node.

Perhaps the most interesting case is the third, where we “hit” the drumhead to create a mixture of (a) and (b) modes. This situation produces a pattern of vibration that will never repeat itself, even if one waits indefinitely!

But how does this quantum algorithm work, and why is it promising in outperforming classical computers? Let’s dive in.

# Physics-informed approach to address real-world phenomena

Scientists have developed various techniques to understand how a complex system evolves. A widely used approach is to solve the equations governing the system using numerical methods. The drawback of many of these algorithms is that they are unable to exploit observational data.

Over the past decades, data availability has been key to tackling many complex problems with machine learning. Data-driven techniques have become a powerful tool when the system has no known equations or when addressing the laws of physics is beyond the computers’ capacity. However, it is not always possible to acquire enough data and scientists often face the risk of producing biased or inaccurate results.

In recent years, researchers have developed methods that blend the physics of a system, expressed by differential equations, with machine learning, called physics-informed machine learning.

Traditional data-driven methods use the *loss function —* a tool to evaluate the prediction. In physics-informed machine learning, the loss function evaluates the outcome, not only on how close it is to the data, but on its consistency with the differential equation governing the system and its initial and boundary conditions. The solution of the differential equation is represented by the *universal function approximator***,** which is adjusted through the loss function.

DQC is an algorithm inspired by the physics-informed machine learning paradigm. This is how it works:

First, we translate the data points into quantum states — via a *quantum feature map* — that can be prepared in the hardware. At PASQAL, we shine fine-tuned laser pulses into the atoms to manipulate their states to encode the data.

The quantum information unit is named the qubit, which can be represented by atomic properties, spin or energy. PASQAL’s technology uses two excited electronic energy states in a rubidium atom to represent a qubit. The average value of the energy of the quantum states is used to represent the *universal function approximator*.

The quantum universal function approximator depends on parameters that are tuned using a loss function. The idea is to measure the qubits’ energy and check if the value satisfies the equation, its boundary and initial conditions. We use classical computing to tune the parameters if the outcome is still not close enough to its goal. Then, we adjust the lasers accordingly and measure again the energy of the quantum state. This hybrid quantum-classical computing cycle repeats until we are satisfied with the solution.

In physics-informed methods, the universal function approximator must fulfill two requirements: expressivity and differentiability. Expressivity is the capacity to approximate the solution for a differential equation. Differentiability is the ability to calculate its rate of change in space and time, and with respect to the tunable parameters.

The quantum universal function approximator has extremely good expressivity. This happens because, in quantum computing, the number of possible combinations of quantum states grows exponentially with the number of qubits.

Expressivity can be achieved in the PASQAL platform fantastically because of its capacity to manipulate hundreds of atoms. Since we use electronic energy levels in rubidium atoms, and the number of electronic states in an atom is infinite, there are an extraordinary number of possible choices to represent qubits.

In classical computing, derivatives are calculated numerically. With PASQAL’s DQC method, we represent the derivatives of the solution analytically — without appealing to numerical methods — in terms of simple algebraic operations, such as addition and multiplication.

# Limitations and outlook

Quantum computing is in its infancy from the experimental point of view: no hardware is yet in full capacity to solve real-world problems.

However, there is no better moment to develop algorithms and emulate how this would work when the hardware is ready. Although DQC could be implemented in any present-day quantum computer, it is tailored for efficient execution in PASQAL’s architecture.

Moreover, PASQAL’s proprietary software allows seamless back-end switching from the current simulator to real quantum hardware.

“All this is using an emulator, yes; but it is proof-of-principle that we can address ‘almost’ industrially relevant problems in the near term,” says Evan Philip, quantum algorithm developer at PASQAL.

The drumhead is a simple example of the DQC algorithm applicability, where no numerical data is needed, only the initial and boundary conditions to solve its wave equation. However, this is the first example considering the evolution in time of a system, which adds a layer of complexity. This shows once again the potential of PASQAL’s DQC method to solve sophisticated differential equations, paving the way to solving industry-relevant problems.

Studying vibrations in membranes helps get closer to understanding how the human eardrum works, how we listen, the dynamics involved, and our relationship with sounds and music. Or address interesting long-standing puzzles, such as can we hear the shape of a drum?

Do you have any questions about our technology and applications? Don’t hesitate to reach out to us!

# 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

Henriet, L., Beguin, L., Signoles, A., Lahaye, T., Browaeys, A., Reymond, G. O., & Jurczak, C. (2020). Quantum computing with neutral atoms. Quantum, 4, 327. https://doi.org/10.22331/q-2020-09-21-327

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