Variational Multi-Gaussian Quantum Trajectories

What it claims

The method is named sVMG (stochastic variational multi-Gaussian), built on a stochastic time-dependent variational principle (sTDVP). Its distinguishing capabilities over the semiclassical baselines it names: it converges controllably to exact single-trajectory dynamics by increasing the number of Gaussian components, and it can evaluate nonlinear state functionals such as Rényi-2 entanglement entropy along individual trajectories — something inaccessible to methods that operate on the Lindblad (density-matrix) level. It reaches parameter regimes (large photon population per mode) beyond brute-force exact simulation.

How it works

The Lindblad master equation is unraveled into quantum trajectories via heterodyne / quantum-state-diffusion unraveling, giving an Itô stochastic Schrödinger equation. The key move: the Itô chain rule is invalid for tangent-space projection, so they convert to the Stratonovich convention (adding a drift correction) to recover the ordinary chain rule, then project the exact increment onto the tangent space, yielding dθ = Re(S⁻¹)Re(V dt + Σ Y⁽ᵐ⁾∘dZ*) where S is the quantum geometric tensor.

The ansatz is a coherent superposition of N_c Gaussian wavepackets in position representation, each with complex weight, mean position/momentum, and a complex symmetric covariance matrix. N_c directly controls the captured non-Gaussianity. For large systems the covariance uses a low-rank decomposition A_k = diag(λ) + Λ_kΛ_kᵀ with rank K, giving N_p = 2N_c(1 + 2N + NK) real parameters. Because the equations reduce to polynomial-weighted Gaussian integrals, S, V and Y are assembled fully analytically via Wick's theorem, and integrated as a standard Stratonovich SDE using the Euler-Heun method.

Evidence

• Bose-Hubbard dimer (U=2γ, J=−2.5γ, ∆=−1.4γ), benchmarked against exact Fock-truncated simulation: sVMG with N_c=4 reproduces exact single trajectories at γδt=10⁻⁴; relative error decreases monotonically with N_c. GTA (= sVMG with N_c=1) and TWA both fail to capture the dynamics; TWA cannot access entanglement at all.
• Rescaled dimer (F→F√N, U→U/N) run up to N=1024; N_c=4 suffices for convergence at all N, consistent with non-Gaussianity decreasing as N grows in the infinite-range model.
• Lattices (U=γ, zJ=−2.5γ, ∆=−1.4γ), N_c=4, K=2: normalized π-mode population ⟨n̂_π⟩/N vanishes as N grows in 1D (up to 32 sites; domain walls wash out order) but stays finite and converges to a dome in 2D (up to 6×6), indicating the symmetry-broken phase survives only in 2D.

Limits

This is an arXiv v1 preprint (quant-ph, dated Sep 2026); no code or repository is mentioned. It targets interacting bosonic open systems specifically; generalization to spin systems and measurement-induced dynamics is stated as future work, not demonstrated. Lattice results top out at 6×6 (2D) and 32 sites (1D), and the 2D thermodynamic-limit transition is inferred from finite-size convergence rather than proven. The dimension-dependent criticality is offered as an experimental prediction for photonic platforms, not yet observed.