In this work, we propose new methods of parameter estimation using stochastic sampling quantum phase-space simulations. We show that it is possible to compute the quantum Fisher information (QFI) from semiclassical stochastic samples using the Truncated Wigner Approximation (TWA). This method extends the class of quantum systems whose fundamental sensitivity limit can be computed efficiently to any system that can be modelled using the TWA, allowing the analysis of more meteorologically useful quantum states. We illustrate this approach with examples, including a system that evolves outside the spin-squeezing regime, where the method of moments fails.
In this work, we propose new methods of parameter estimation using stochastic sampling quantum phase-space simulations. We show that it is possible to compute the quantum Fisher information (QFI) from semiclassical stochastic samples using the Truncated Wigner Approximation (TWA). This method extends the class of quantum systems whose fundamental sensitivity limit can be computed efficiently to any system that can be modelled using the TWA, allowing the analysis of more meteorologically useful quantum states. We illustrate this approach with examples, including a system that evolves outside the spin-squeezing regime, where the method of moments fails.
A phase-space method, suitable to study the many-body spin dynamics in generic spin lattice models is introduced, discussed, and tested. This technique, which we refer to as the dTWA, is based on the discrete sampling of the Wigner function instead of the continuous sampling in traditional Truncated Wigner Approximation (TWA) methods. The power of the method is demonstrated by comparisons with analytical and numerically exact calculations, which show a significant improvement compared to the TWA, e.g. by capturing revivals of spin-observables. The dTWA is demonstrated to not only correctly reproduce dynamics of one-particle observables but also two-point correlations and spin squeezing, a property relying on entanglement. This method thus opens the possibility to study the quantum dynamics in long-range spin models, accessible to recent state-of-the-art experiments, in regimes where other numerical techniques are inapplicable.
We review the truncated Wigner approximation (TWA) for spins as a computationally inexpensive numerical approximation method to describe interacting and / or dissipative many-body spin systems. Using the Wigner-Moyal mapping from Hilbert space to a suitable phase space, the many-body density matrix is represented by a c-number distribution, the Wigner function. The gauge freedom in continuous phase space can be exploited to find positive Wigner functions for a large class of spin states, including entangled ones. Employing different sets of correspondence rules, we derive equations of motion for the Wigner function, which, applying controlled approximations, can be mapped to stochastic differential equations. This allows a computationally inexpensive simulation of expectation values. Using a phase-space analog of the quantum regression theorem also multi-time correlations and spectra can be obtained. To illustrate the potential of the method, we benchmark the TWA for spins with some exactly solvable problems of interacting, dissipative spin systems, and then discuss its application to collective processes, such as the superradiant emission of light. Extending the TWA to imaginary time furthermore provides a tool to approximately calculate thermal and ground states of spin Hamiltonians. Finally, we show that the TWA stochastic equations can equivalently be derived within a path-integral approach, provided that the operator products in the dissipator are rigorously mapped onto the curved phase space.
We present a semiclassical phase-space method to calculate thermal and ground states of large interacting spin systems. To this end, we extend the recently developed truncated Wigner approximation for spins (TWA) to the imaginary time, termed iTWA. The evolution of the canonical density matrix in imaginary time is mapped to a partial differential equation of its Wigner function. Truncation at the Fokker-Planck level leads to a set of stochastic differential equations, which can be efficiently simulated even for large systems. We show that for general Ising Hamiltonians the approximation becomes exact for large imaginary times subject only to sampling errors. Thus the iTWA is ideal to determine the ground state of spin glasses or to find solutions to quadratic unconstrained binary optimization problems (QUBO) on a controlled approximation level. We illustrate this for MaxCut on random, unweighted 3-regular graphs, encoded in an anti-ferromagnetic Ising Hamiltonian, for which finding the exact ground state and even approximations to it beyond a certain accuracy is know to be NP hard. Furthermore, in order to assess the quality of the method also for general spin models, we analyze the ground-state quantum phase transition of the transverse-field Ising model in one and two spatial dimensions, finding reasonably good agreement with the exact behavior.
We introduce the parafermionic truncated Wigner approximation (pTWA), a semiclassical phase-space framework for simulating the nonequilibrium dynamics of lattice systems with fractional exchange statistics. The method extends truncated Wigner approaches developed for bosonic and fermionic systems to Zn Fock parafermions by expressing the Hamiltonian in terms of local Hubbard operators that form a closed Lie algebra. This representation leads to a Lie--Poisson phase-space formulation in which quantum dynamics is approximated by stochastic sampling of initial conditions followed by deterministic semiclassical evolution. We benchmark the approach in several settings, including single-site clock dynamics, the fully connected Zn clock model, long-range Z3 clock chains, and disordered Z3 Fock parafermion chains. The method reproduces key features of the exact dynamics, including excitation spreading, disorder-induced suppression of transport, and the emergence of long-time imbalance plateaus. Our results demonstrate that pTWA provides a practical tool for exploring the dynamics of parafermionic systems in regimes where exact numerical methods are limited by Hilbert-space growth.
Phase-space approaches such as the truncated Wigner approximation (TWA) provide an efficient semiclassical framework for performing approximate simulations of the dynamics of open quantum many-body systems outside the reach of exact numerical methods but beyond the mean-field level. For bosonic systems, TWA is known to be equivalent to a Keldysh path-integral formulation truncated at second order in the so-called quantum fluctuations. This semiclassical approach provides an alternative transparent route towards approximate stochastic equations of motion, which can be efficiently solved. Here we establish the corresponding path-integral formulation for interacting open spin-1/2 systems using the continuous SU(2) phase space. We show, in particular, that a consistent treatment of dissipation requires correctly mapping operator products onto the curved spin phase space, leading to stochastic equations that coincide with those obtained from the continuous TWA formulation and thus reproduce the exact dynamics of a single dissipative spin. Our results provide a unified field-theoretic foundation for the TWA to dissipative spin dynamics and offer a systematic starting point for extensions beyond the semiclassical approximation.
Abstract Numerical techniques to efficiently model out-of-equilibrium dynamics in interacting quantum many-body systems are key for advancing our capability to harness and understand complex quantum matter. Here we propose a new numerical approach which we refer to as generalized discrete truncated Wigner approximation (GDTWA). It is based on a discrete semi-classical phase space sampling and allows to investigate quantum dynamics in lattice spin systems with arbitrary S ≥ 1/2. We show that the GDTWA can accurately simulate dynamics of large ensembles in arbitrary dimensions. We apply it for S > 1/2 spin-models with dipolar long-range interactions, a scenario arising in recent experiments with magnetic atoms. We show that the method can capture beyond mean-field effects, not only at short times, but it also can correctly reproduce long time quantum-thermalization dynamics. We benchmark the method with exact diagonalization in small systems, with perturbation theory for short times, and with analytical predictions made for models which feature quantum-thermalization at long times. We apply our method to study dynamics in large S > 1/2 spin-models and compute experimentally accessible observables such as Zeeman level populations, contrast of spin coherence, spin squeezing, and entanglement quantified by single-spin Renyi entropies. We reveal that large S systems can feature larger entanglement than corresponding S = 1/2 systems. Our analyses demonstrate that the GDTWA can be a powerful tool for modeling complex spin dynamics in regimes where other state-of-the art numerical methods fail.