Quantum Walks and Swinging Pendulums: Two Sides of the Same Coin express
It turns out that two completely different languages describe one quantum reality. In one, a particle strolls through a city along all streets simultaneously. This is a quantum walk, and it’s linked to spectroscopy of that road network. In the other, coupled pendulums swap nudges. Scientists have built a bridge: any walking algorithm can be precisely turned into oscillations — and vice versa.
It’s like translating a symphony from the language of violins to the piano without losing a single note. The translation preserves the entire structure: the arrangement of nodes and the data input. Now you can choose the most convenient representation — like a map or a timetable. This simplifies proving complexity.
An unexpected conclusion: the chaotic trajectories of a wanderer and the measured rhythms of springs are mathematically indistinguishable. Ideas from one field instantly work in the other. Quantum computers will become better at modeling molecules and the evolution of black holes.
Richard Feynman dreamed of quantum machines that unveil nature’s secrets.
Original abstract
as written by the authors arXiv:2512.03681
Continuous time quantum walks on exponentially large, sparse graphs form a powerful paradigm for quantum computing: On the one hand, they can be efficiently simulated on a quantum computer. On the other hand, they are themselves BQP-complete, providing an alternative framework for thinking about quantum computing -- a perspective which has indeed led to a number of novel algorithms and oracle problems. Recently, simulating the dynamics of a system of harmonic oscillators (that is, masses and springs) was set forth as another BQP-complete problem defined on exponentially large, sparse graphs. In this work, we establish a direct and transparent mapping between these two classes of problems. As compared to linking the two classes of problems via their BQP-completeness, our mapping has several desirable features: It is transparent, in that it respects the structure of the problem, including the geometry of the underlying graph, initialization, read-out, and efficient oracle access, resulting in low overhead in terms of both space and time; it allows to map also between restricted subsets of instances of both problems which are not BQP-complete; it provides a recipe to directly translate any quantum algorithm designed in the quantum walk paradigm to harmonic oscillators (and vice versa); and finally, it provides an alternative, transparent way to prove BQP-completeness of the harmonic oscillator problem by mapping it to BQP-completeness construction for the quantum walk problem (or vice versa).
The harmonic oscillator is one of the most universal models in physics. It describes everything: from atomic vibrations in a crystal to ripples in spacetime.
Perhaps the sentient ocean in Lem’s Solaris is a giant network of interconnected oscillators, where every wave carries a thought.