Quantum Simulation and Energy Estimation for Discretized Anharmonic oscillator

TL;DR

The paper addresses the challenge of simulating anharmonic quantum oscillators beyond harmonic approximations on classical computers by implementing a discretized QAHO on IBM Quantum Experience using a 3-qubit circuit. It combines a filter-based, Toffoli-enabled quantum time-evolution framework with VQE and VQD to estimate ground and excited energies, achieving an energy accuracy of about $1.11\%$ relative to exact diagonalization. The authors demonstrate scalability to higher qubit counts and benchmark against perturbation theory and WKB, finding VQE to be more accurate in this setting. This work underscores the potential of hybrid quantum-classical approaches for quantum chemistry and materials science in the NISQ era and lays groundwork for more complex, multi-dimensional simulations.

Abstract

Anharmonic potential quantum system play crucial role in physics as they provide a more realistic description of oscillatory phenomena, which often deviate from the idealized harmonic model. However, simulating such system on classical computers is highly challenging due to nonlinear interactions, large state spaces, and the exponential scaling of memory and computational resources. In this work, quantum simulation is employed to model a quantum anharmonic oscillator (QAHO) using a 3-qubit system implemented on IBM's Quantum Experiences platform. A quantum circuit with a filter-based design and Toffoli gates is constructed to track quantum state evolution, capturing key phenomena like quantum revival. The framework is further extended to n-qubit system to enhance resolution and scalability. For energy estimation, the Variational Quantum Eigensolver (VQE) with a TwoLocal ansatz and variational Quantum Deflation (VQD), are used to compute ground and excited state energies. The proposed approach achieves high accuracy with an error of only 1.11% compared to exact methods. Notably, VQE outperforms classical approximations such as perturbation theory (error 6.71%) and the Wentzel-Kramers-Brillouin (WKB) approximation(error 5.36%), yielding more precise energy values. These results highlight the potential of quantum simulation and VQD as effective tools for investigating complex quantum system, paving the way for future application in quantum chemistry and materials science as quantum hardware continues to advance.

Figures (9)

• Figure 1: Potential surface of a 2D QAHO.
• Figure 2: Quantum circuit depicting (a) QFT and (b) IQFT.
• Figure 3: Filter circuit for state $\ket{001}$.
• Figure 4: Quantum circuit for $U_{\hat{p}}(t)$ implementation via QFT sandwiching and filter-based design.
• Figure 5: Probability amplitude from $0$ to $6\pi$.