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arXiv:2603.05479

Quantum Simulation of Coupled Harmonic Oscillators: From Theory to Implementation

Viraj Dsouza, Weronika Golletz, Dimitrios Kranas, Bakhao Dioum, Vardaan Sahgal, Eden Schirman

quant-phSubmitted Mar 5, 2026• Announced Mar 6, 2026• 0 citations
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Abstract

We investigate the quantum algorithm of Babbush et al. (arXiv:2303.13012v3) for simulating coupled harmonic oscillators, which promises exponential speedups over classical methods. Focusing on linearly connected oscillator chains, we bridge the gap between theory and implementation by developing and comparing three concrete realizations of the algorithm. First, we implement a sparse initial state preparation combined with product-formula (Suzuki-Trotter) Hamiltonian simulation. Second, we implement a fully quantum, oracle-based framework in which classical data are accessed via oracles, the Hamiltonian is block-encoded, and time evolution is performed using QSVT-based Hamiltonian simulation. Third, we propose an efficient alternative that combines the sparse state-preparation routine of the first approach with the oracle and block-encoding-based simulation pipeline of the second. We provide these implementations on Classiq, a high-level quantum design platform and provide appropriate resource benchmarks. Our simulation results show that the complex initial state preparation proposed by Babbush et al. can be circumvented at least in the linear-chain case. Finally, we illustrate two physical applications-extracting normal modes and simulating coarse-grained energy propagation-demonstrating how the algorithm connects to measurable observables. Our results clarify the resource requirements of the algorithm and provide concrete pathways toward practical quantum advantage.

Comments
32 pages, 17 figures

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We present a quantum algorithm for simulating the classical dynamics of 2 n coupled oscillators (e.g., 2 n masses coupled by springs).Our approach leverages a mapping between the Schrödinger equation and Newton's equation for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators.When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in n, almost linear in the evolution time, and sublinear in the sparsity.As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time.We show that any classical algorithm solving this same problem is inefficient and must make 2 ΩðnÞ queries to the oracle, and when the oracles are instantiated by efficient quantum circuits, the problem is bounded-error quantum polynomial time complete.Thus, our approach solves a potentially practical application with an exponential speedup over classical computers.Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with 2 n modes.
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Simulating the Hamiltonian dynamics of quantum systems is one of the most promising applications of digital quantum computers. In this dissertation, we develop an understanding of quantum simulation algorithms concerning their design, analysis, implementation, and application. We implement three leading simulation algorithms, employing diverse techniques to tighten their error analyses and optimize circuit implementations. We produce concrete resource estimates for simulating a Heisenberg spin system, a problem arising in condensed matter physics that is otherwise difficult to solve on a classical computer. The resulting circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry, suggesting the simulation of spin systems as a promising candidate for an early demonstration of practical quantum computation. We design new simulation algorithms by using classical randomness. We show that by simply randomizing how the terms in the Hamiltonian are ordered, one can prove stronger bounds for product formulas and thereby give more efficient quantum simulations. We also develop a classical sampler for time-dependent Hamiltonians, using which we give a simulation algorithm that substantially improves over previous approaches when the Hamiltonian varies significantly with time. We propose a general theory to analyzing product formulas, an approach to quantum simulation widely used in experimental demonstrations but whose error scaling was poorly understood. Our approach directly exploits the commutativity of Hamiltonian, overcoming the limitations of prior error analyses. We prove new speedups of product formulas for simulating many quantum systems, including simulations of nearest-neighbor lattice systems, second-quantized plane-wave electronic structure, kkk-local Hamiltonians, rapidly decaying power-law interactions, and clustered Hamiltonians, nearly matching or even outperforming the best previous results in quantum simulation. We accompany our analysis with numerical calculation, which suggests that the bounds also have nearly tight constant prefactors. We identify applications of quantum simulation to designing other quantum algorithms and improving quantum Monte Carlo methods. We develop an algorithmic framework ``quantum singular value transformation'' using techniques from quantum simulation and apply it to implement principal component regression. We also apply our new analysis of product formulas and obtain improved quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets.
Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using O(log⁡M)O(\log M)O(logM) qubits with MMM is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schrödinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schrödinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.