Low-Rank Variational Quantum Algorithm for the Dynamics of Open Quantum Systems

Sara Santos; Xinyu Song; Vincenzo Savona

doi:10.22331/q-2025-02-04-1620

Low-Rank Variational Quantum Algorithm for the Dynamics of Open Quantum Systems

Sara Santos, Xinyu Song, and Vincenzo Savona

Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Center for Quantum Science and Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

Published:	2025-02-04, volume 9, page 1620
Editor:	Di Fang
Eprint:	arXiv:2403.05908v3
Doi:	https://doi.org/10.22331/q-2025-02-04-1620
Citation:	Quantum 9, 1620 (2025).

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Abstract

The simulation of many-body open quantum systems is key to solving numerous outstanding problems in physics, chemistry, material science, and in the development of quantum technologies. Near-term quantum computers may bring considerable advantage for the efficient simulation of their static and dynamical properties, thanks to hybrid quantum-classical variational algorithms to approximate the dynamics of the density matrix describing the quantum state in terms of an ensemble average. Here, a variational quantum algorithm is developed to simulate the real-time evolution of the density matrix governed by the Lindblad master equation, under the assumption that the quantum state has a bounded entropy along the dynamics, entailing a low-rank representation of its density matrix. The algorithm encodes each pure state of the statistical mixture as a parametrized quantum circuit, and the associated probabilities as additional variational parameters stored classically, thereby requiring a significantly lower number of qubits than algorithms where the full density matrix is encoded in the quantum memory. Two variational ansatze are proposed, and their effectiveness is assessed in the simulation of the dynamics of a 2D dissipative transverse field Ising model. The results underscore the algorithm's efficiency in simulating the dynamics of open quantum systems in the low-rank regime with limited quantum resources on a near-term quantum device.

Popular summary

Open quantum systems—where quantum particles interact with their environment—are central to many fields, including physics, chemistry, and quantum technology. Simulating their behavior is essential but computationally demanding, even for modern classical computers. We develop a quantum algorithm offering a significant improvement in efficiency.

The low-rank variational quantum algorithm focuses on systems with low entropy, where the quantum state remains relatively simple over time. Instead of modeling the entire quantum state, the algorithm uses a reduced representation that captures only the most important components. This drastically lowers the number of quantum bits and computational resources required.

Combining quantum and classical computing, the algorithm represents individual quantum states with quantum circuits and handles probabilities with classical processing. Two strategies are proposed: one optimized for simplicity and resource efficiency, and the other for flexibility and accuracy. Tests on the transverse field Ising model—a standard benchmark in quantum physics—show the method’s ability to simulate real-world systems using shallow quantum circuits, suitable for today’s noisy quantum devices.

This approach not only reduces computational costs but also brings simulations of open quantum systems within reach of current quantum hardware. Future improvements, such as adaptive techniques and error mitigation, could make the algorithm even more versatile, paving the way for advances in the simulation of large-scale open quantum systems.

► BibTeX data

@article{Santos2025lowrankvariational,
  doi = {10.22331/q-2025-02-04-1620},
  url = {https://doi.org/10.22331/q-2025-02-04-1620},
  title = {Low-{R}ank {V}ariational {Q}uantum {A}lgorithm for the {D}ynamics of {O}pen {Q}uantum {S}ystems},
  author = {Santos, Sara and Song, Xinyu and Savona, Vincenzo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1620},
  month = feb,
  year = {2025}
}

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