Quantum Machine Learning: a quick overview

Climate change and the global energy crisis are two of the most pressing threats we face today. Accurate and efficient climate modelling is critical to tackling these issues in multiple ways. Naturally, climate models are needed to understand the impact of climate change on our planet and to predict extreme weather events that arise as a consequence. However, they are also needed to develop new renewable energy technologies and to optimally integrate their output with the power grid. For example, accurate predictions of the amount of sunlight and wind on specific days allow us to determine how much electricity will be generated by solar panels and wind turbines, enabling us to reduce our dependence on fossil fuels. Unfortunately however, existing classical simulation methods struggle with the sheer scale and complexity of the global climate and cannot model it sufficiently well to produce highly accurate predictions, as you likely already know if you ever rely on weather forecasts.

A natural question, then, is whether the dawn of quantum computing has anything promising to offer the field of climate modelling. At its heart, climate modelling relies on solving large sets of coupled partial differential equations (PDEs) that govern a wide range of physical processes, including fluid dynamics, energy transfer, and mass transport. At ColibriTD, we have developed a hybrid differential equation solver (H-DES) [1] for exactly such problems. H-DES combines the best of the classical and quantum computers that exist today to solve systems of PDEs. It has been tested on real quantum computers with good results, and you can even use the QUICK-PDE Qiskit function from IBM to try it out for yourself! [2]

So why should we expect our quantum algorithm to perform any better than classical algorithms for solving systems of PDEs? The answer mostly comes down to how the resources it uses scale with for increasingly complex problems. Classical methods for solving PDEs suffer from the famous ‘curse of dimensionality’ — the computational cost and time taken for calculations increase very steeply (often even exponentially) with the number of variables/dimensions of the problem considered or with the resolution required. This means many computations are simply too complex to be feasible, and problems must be artificially simplified to achieve results on reasonable timescales.

By cleverly exploiting the principles of quantum information and spectral methods, H-DES offers a resource-efficient alternative. Using only a few qubits, H-DES can determine solutions for multi-dimensional PDEs. The number of qubits required grows only linearly with the number of dimensions or variables, the potential complexity of the solution function grows exponentially with the number of qubits, and the desired resolution has almost no impact on the computational time required. This implies that as quantum hardware advances and H-DES is refined alongside it, H-DES should come to provide meaningful advantages over classical simulation methods in terms of time, cost or energy required. Furthermore, H-DES is a fully general PDE solver, meaning it can be used to model phenomena from any area of physics. This makes it an excellent foundation for climate modelling simulations, which require the simultaneous modelling of a broad range of physical processes in tandem.

Fig. 1. A three-dimensional (two spatial dimensions + time) simulation of a convection–diffusion process using H-DES. Only eight qubits were required to achieve this result, which shows excellent agreement with classical simulation benchmarks.

[1] Jaffali, Hamza, Jonas Bastos de Araujo, Nadia Milazzo, Marta Reina, Henri de Boutray, Karla Baumann, Frédéric Holweck, Youcef Mohdeb, and Roland Katz. 2024. “H-DES: A Quantum-Classical Hybrid Differential Equation Solver.” arXiv. https://doi.org/10.48550/arXiv.2410.01130

‍[2] https://quantum.cloud.ibm.com/docs/en/guides/colibritd-pde