Quantum Computing Concepts for Math Professionals

This week’s IBM Quantum blog shows how researchers are using group theory to guide the design of quantum algorithms. Read more: https://lnkd.in/eC9ku6qR There’s a tight link between physics, math, and information. When quantum mechanics was first discovered, mathematicians like Hermann Weyl found a new utility for group theory, which offered a natural framework to describe quantum mechanics. Today, quantum computers have emerged as tools to scale the problems we can solve by leveraging group theory and its description of the symmetries in quantum physics. On the IBM Quantum blog, we tell the story of how theorists at IBM uncovered a quantum algorithm that efficiently approximates notoriously difficult mathematical quantities known as Kronecker coefficients. These coefficients are common in representation theory, a branch of mathematics that describes symmetries, which is fundamental in fields like quantum physics and data science. The breakthrough came by revisiting a long-overlooked tool: the non-Abelian quantum Fourier transform. Previous attempts at applying this method to quantum computing applications have often fallen short, but our researchers found a way to use it to compute multiplicities in symmetric group representations—a challenging task for classical algorithms. The algorithm provides a meaningful polynomial advantage compared to the best classical algorithm known so far. More importantly, it opens a new bridge between quantum computing and mathematics, offering fresh tools to tackle long-standing open problems. Very proud of the team behind this work, which exemplifies how algorithm discovery is driving quantum computing forward by expanding the kinds of problems we can solve. Visit the link at the top of this post to read the full story.

Quantum computing feels confusing not because it is magic, but because the mathematics is rarely shown intuitively. I put together this PDF as part of a series on the mathematics of quantum computing, where I focus on building intuition instead of memorization. This post is the second one in the series. If you have not seen the first, you can find it on my feed. In this part, I break down the math of quantum gates in a visual and geometric way rather than relying on heavy notation. Inside this, I walk through how Quantum states behave like vectors Quantum gates act as reversible transformations Operations like X, Z, H, and CNOT are rotations and correlations, not logic tricks Entanglement and teleportation emerge naturally from linear algebra and measurement The goal is simple. If someone with a computer science background or even a curious beginner looks at this, they should be able to see what the gate is doing. This series is about understanding quantum computing from first principles, not memorizing formulas. If you want to explore these ideas interactively, I am also building a free quantum state visualizer to help develop intuition around Bloch spheres and quantum gates. You can try it here: https://lnkd.in/gwcXHruy More posts in this series coming soon.

Quantum computing for financial mathematics A key paper published in 2023 by Jack Jacquier, Oleksiy Kondratyev, Gordon Lee, and Mugad Oumgari reviews the state of quantum computing in financial mathematics and leaves a clear message: the value is not in waiting for the perfect machine, but in how we manage the transition with what we already have. Three application lines highlighted by the authors - Portfolio optimization with variational algorithms (QAOA, VQE), where hybrid approaches already help explore scenarios that scale poorly in the classical world. - Quantum Machine Learning, with generative and discriminative models (QGANs, QNNs, Quantum Circuit Born Machines) applied to market data generation, credit scoring, and detection of distribution shifts. - Quantum Monte Carlo, with algorithms achieving a quadratic speedup in expectation estimation, useful for high-dimensional derivative pricing. Other areas mentioned The paper also points to the potential of Quantum Semidefinite Programming (QSDP) for robust risk management and portfolio optimization under uncertainty. The key takeaway The authors emphasize: it’s not just about speed, it’s about thinking differently. - Use quantum algorithms to accelerate critical steps of classical pipelines. - Develop hybrid and quantum-inspired schemes. - Prepare data structures and methodologies that can scale once hardware matures. Ultimately: the real race lies in turning current limitations into opportunities for integration and new value models, while technological acceleration follows its own path. Link https://lnkd.in/d-CPDkN9 Imperial College London Abu Dhabi Investment Authority (ADIA) Lloyds Banking Group