Quantum Simulation of Coupled Harmonic Oscillators: From Theory to Implementation

TL;DR

The simulation results show that the complex initial state preparation proposed by Babbush et al. can be circumvented at least in the linear-chain case, and an efficient alternative is proposed that combines the sparse state-preparation routine of the first approach with the oracle and block-encoding-based simulation pipeline of the second.

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.

Figures (17)

• Figure 1: Four-mass oscillator system corresponding to the tridiagonal stiffness matrix $\mathbf{F}$. Each mass is connected to its neighbors and to ground through springs of identical strength $k=1$. Masses $m_1$ and $m_2$ have value $1$; masses $m_3$ and $m_4$ are heavier with value $4$.
• Figure 2: Circuit depth and total gate count for initial state preparation versus system size $N$.
• Figure 3: Ratio of experimental gate count to the scaling $f(N)\log^2(N)$, showing empirical upper bound $C=3.07$.
• Figure 4: (a) Total Kinetic energy from the quantum simulation compared with the exact classical result for $N = 2$ oscillators, over the interval $t \in [0, 5]$ with time step $\Delta t = 0.1$. (b) Absolute error between the quantum and classical results. The quantum circuit uses a second-order Suzuki–Trotter decomposition with $r_{st} = 20$ steps.
• Figure 5: Resource scaling and estimate for end-to-end implementation with second-order Trotterization. (a) Circuit Width: Comparison of circuit width with and without synthesis optimization as a function of system size $N$. (b) Circuit Depth and Total Gate Count: Circuit depth and total gate count as functions of system size $N$.