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Variational Multi-Gaussian Phase-Space Bosonic Dynamics via Automatic Differentiation

Jacopo Tosca, Francesco Carnazza, Luca Giacomelli, and Cristiano Ciuti

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Phys. Rev. X 16, 031077 – Published 24 September, 2026

DOI: https://doi.org/10.1103/q3m5-q44b

Abstract

We introduce a variational method for simulating the dynamics of interacting open quantum bosonic systems deep in the quantum regime. The method is based on a multidimensional Wigner phase-space representation and employs a variational multi-Gaussian Ansatz, whose accuracy is systematically controlled by the number of Gaussian components. The variational equations of motion are derived from the Dirac-Frenkel principle and evaluated efficiently by combining the analytical structure of Gaussian functions with automatic differentiation. As a key physical application, we study a driven-dissipative two-dimensional Bose-Hubbard lattice with two-boson coherent driving and two-body losses. Using our dynamical approach, we compute the finite-size scaling of the Liouvillian spectral gap—extracted from the relaxation dynamics—which vanishes in the thermodynamic limit. Our results reveal critical slowing down with dynamical exponents of the 2D quantum Ising universality class, demonstrating the power of our method to capture complex quantum dynamics in large open systems.

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Predicting the dynamics of interacting open quantum bosonic systems is notoriously difficult because the complexity of their quantum state grows exponentially with the number of bosonic modes. Instead of representing the quantum state directly in Hilbert space, we adopt a different perspective based on the Wigner function, a phase-space representation that maps the quantum state onto a quasiprobability distribution in the space of generalized positions and momenta. We describe this distribution using a variational ansatz consisting of a sum of Gaussian functions. Each Gaussian is characterized by a weight, a complex center, and a covariance matrix, and the quantum dynamics is entirely determined by the time evolution of these variational parameters according to the Dirac-Frenkel variational principle. The analytical properties of Gaussian functions allow the multidimensional integrals entering the variational equations to be evaluated exactly, while modern automatic differentiation computes the required high-order derivatives efficiently. This unique combination of analytical mathematics and modern computational techniques yields a remarkably scalable variational phase-space approach for simulating the dynamics of large open quantum bosonic systems. We demonstrate its power by uncovering the universal critical dynamics of a large driven-dissipative Bose-Hubbard lattice, well beyond the reach of previous approaches.

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Phys. Rev. X 16, 031077– Published 24 September, 2026
  • Received 22 July 2025
  • Revised 19 February 2026
  • Accepted 9 July 2026
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