The Schrödinger Equation Gets Practical: New Quantum Tool Simulates the Physics of the Real World

By US Department of Energy April 5, 2025

Collected at: https://scitechdaily.com/the-schrodinger-equation-gets-practical-new-quantum-tool-simulates-the-physics-of-the-real-world/

A new quantum algorithm accelerates the simulation of coupled oscillator dynamics.

Quantum computers have the potential to solve certain problems far more efficiently than classical computers. In a recent development, researchers have designed a quantum algorithm to simulate systems of coupled masses and springs, known as coupled oscillators. These systems are fundamental in modeling a wide range of physical phenomena, from molecules to mechanical structures like bridges.

To simulate these systems, the researchers first translated the behavior of the coupled oscillators into a form of the Schrödinger equation, which describes how the quantum state of a system evolves over time. They then used advanced Hamiltonian simulation techniques to model the system on a quantum computer.

Hamiltonian methods provide a framework for understanding how physical systems evolve, connecting principles of classical mechanics with those of quantum mechanics. By leveraging these techniques, the researchers were able to represent the dynamics of N coupled oscillators using only about log(N) quantum bits (qubits), a significant reduction compared to the resources required by classical simulations.

Researchers have developed very few new classes of provable exponential speedups such as the one described here. This new research shows that a quantum computer acting on a number n of quantum bits can be simulated using 2n coupled harmonic oscillators.

This algorithm results in an exponentially faster simulation of coupled oscillators compared to ordinary algorithms. The approach also demonstrates a novel and subtle link between quantum dynamics and harmonic oscillators. This work could prove useful to a wide range of real-world problems involving coupled oscillators. These applications range from engineering to neuroscience to chemistry.

Two Quantum Algorithms and a New Simulation Approach

Researchers provide two quantum algorithms for simulating coupled harmonic oscillators that provide exponential speedup. Using energy conservation and the fact that the Hamiltonian is quadratic, they represented the dynamics of the displacements and the momenta as a unitary evolution.

They then simulated the unitary dynamics using Hamiltonian simulation methods including a novel approach for computing fractional queries. This quantum algorithm provides an exponential advantage over classical algorithms by finding a set of coupling constants such that the classical coupled Harmonic oscillators can simulate an arbitrary quantum computation.

Additionally, the researchers found further query lower bounds and demonstrated that if a classical algorithm existed for simulating oscillators that matched the performance of the researchers’ new algorithm, then the classical algorithm would violate query lower bounds on the number of times that the labels of the vertices on a graph need to be accessed for a walker to move from the entrance to exit of a maze described by that graph. This means that the algorithm provably provides an exponential advantage for a very important set of problems and reveals that quantum computing has even more potential impact than previously thought.

Reference: “Exponential Quantum Speedup in Simulating Coupled Classical Oscillators” by Ryan Babbush, Dominic W. Berry, Robin Kothari, Rolando D. Somma and Nathan Wiebe, 4 December 2023, Physical Review X.
DOI: 10.1103/PhysRevX.13.041041

This work was supported by the Department of Energy Office of Science, National Quantum Information Science Research Center, Co-design Center for Quantum Advantage (C2QA). This work was also supported by grants from Google Quantum AI and the Australian Research Council.