Direct Equivalence between Dynamics of Quantum Walks and Coupled Classical Oscillators
Lilith Zschetzsche, Refik Mansuroglu, András Molnár, Norbert Schuch
TL;DR
The paper establishes a direct, efficient correspondence between continuous-time quantum walks on exponentially large, sparse graphs and dynamics of classical harmonic oscillators on related graphs. It develops a sign-split embedding to transform sign patterns and constructs two reciprocal reductions: (i) quantum walks to harmonic oscillators by embedding into a doubled system and choosing a suitable $\gamma$, and (ii) harmonic oscillators to quantum walks by representing $A$ as $B^\dagger B$ and embedding to yield a quantum-walk evolution. The mappings preserve the underlying graph geometry, initialization, and read-out, are compatible with oracle access, and apply to both complete and restricted subclasses, providing new perspectives on BQP-hardness and enabling direct translation of algorithms between these paradigms. The framework yields an explicit relationship between output distributions via a sparse transition matrix $C$, supports efficient read-out from oscillator energies, and demonstrates a robust, general methodology for converting problems across quantum and classical-analog paradigms with broad applicability in quantum simulation and complexity.
Abstract
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).