Quantum Harmonic Oscillator Simulated on IBM Quantum Chip

For the first time, researchers have successfully simulated a quantum harmonic oscillator on IBM's quantum hardware, demonstrating particle dynamics under time-varying forces that were previously impossible to observe experimentally. This breakthrough opens new pathways for simulating complex physical systems on quantum computers, moving beyond theoretical proposals to practical implementation.

The research team from National Institute of Technology and Indian Institute of Science Education and Research Kolkata developed novel quantum circuits to simulate both one-dimensional and two-dimensional quantum harmonic oscillators (QHO) on IBM's ibmqx4 quantum processor. The quantum harmonic oscillator represents a fundamental physical system that governs dynamics across numerous scientific domains, yet studying its transient behavior under time-dependent force fields has remained challenging.

Using first-order Trotter decomposition, the researchers constructed quantum circuits that implemented the Hamiltonian for force-driven harmonic oscillators. For the single-qubit case, they simulated two states (ground state |0⟩ and first excited state |1⟩), while the two-qubit system modeled four states (|00⟩, |01⟩, |10⟩, and |11⟩). The circuits incorporated various quantum gates including U1, U3, CNOT, and Hadamard gates with carefully chosen parameters to represent the oscillator dynamics.

Experimental results from 1024 shots on the quantum chip revealed clear probability distributions showing how particles transition between states over time. When the oscillation frequency (ω) of the periodic driving force was set to 1 unit, the ground state probability dropped from 98.45% to 15.61% over seven time steps, while the first excited state probability rose from 1.55% to 84.39%. At ω=2, the transitions showed different patterns, with ground state probability fluctuating between 96.58% and 9.52%.

In the two-qubit system, the researchers observed more complex probability distributions across four states. At ω=1, probabilities ranged from 20.02% to 32.52% across different states and time steps, while at ω=5, the fluctuations became more pronounced with probabilities varying from 14.84% to 35.84%. These experimental data, presented in detailed graphs, clearly illustrate the dynamics of quantum harmonic oscillators under time-dependent force fields.

The successful implementation demonstrates that quantum computers can now simulate fundamental physical systems that were previously limited to theoretical study or classical approximation. Since many physical systems can be mapped to two-level systems, this work provides a foundation for simulating more complex quantum phenomena on existing quantum hardware.

This experimental verification of quantum harmonic oscillator dynamics on real quantum hardware represents a significant step toward practical quantum simulation. The researchers acknowledge that while quantum algorithms for harmonic oscillators have existed theoretically, this marks the first circuit implementation and experimental demonstration on quantum processors.

Reference: Baishya, A., Kumar, L., Behera, B.K., & Panigrahi, P.K. (2019). Experimental Demonstration of Force Driven Quantum Harmonic Oscillator in IBM Quantum Computer. arXiv:1906.01436.