Heat conduction in low-dimensional systems exhibits strong deviations from Fourier behavior due to anharmonicity and long-lived vibrational correlations, challenging conventional computational approaches. The
β-Fermi--Pasta--Ulam--Tsingou (
β-FPUT) chain provides a minimal nonlinear lattice model for studying anomalous transport, yet its quantum real-time dynamics remain difficult to access with classical methods. We develop a first-quantized digital quantum-simulation framework for the quantum
β-FPUT lattice, targeting fault-tolerant quantum computers. By working directly with discretized lattice displacements rather than truncated phonon occupation spaces, the approach captures anharmonic interactions while avoiding bosonic encoding overheads. We construct Trotterized circuit blocks for real-time evolution and introduce a Hermitian quadrature decomposition of Fourier-mode displacement operators that enables shallow quantum circuits for mode-resolved displacement correlators. We analyze the quantum resources required for the full simulation and measurement workflow, providing qubit counts, gate complexity, circuit-depth and resource estimates as functions of system size and resolution within a fault-tolerant workflow. These results establish a concrete algorithmic blueprint for simulating quantum transport dynamics in nonlinear low-dimensional lattice models on fault-tolerant quantum hardware.
We study the transition from integrability to chaos for the three-particle Fermi–Pasta–Ulam–Tsingou (FPUT) model. We can show that both the quartic β-FPUT model (α=0) and the cubic one (β=0) are integrable by introducing an appropriate Fourier representation to express the nonlinear terms of the Hamiltonian. For generic values of α and β, the model is non-integrable and displays a mixed phase space with both chaotic and regular trajectories. In the classical case, chaos is diagnosed by the investigation of Poincaré sections. In the quantum case, the level spacing statistics in the energy basis belongs to the Gaussian orthogonal ensemble in the chaotic regime, and crosses over to Poissonian behavior in the quasi-integrable low-energy limit. In the chaotic part of the spectrum, two generic observables obey the eigenstate thermalization hypothesis.
The Fermi--Pasta--Ulam--Tsingou (FPUT) problem provides a paradigmatic framework for understanding thermalization in weakly nonlinear many-body Hamiltonian systems. This focused review summarizes major developments in near-integrable dynamics, wave resonances, and phonon kinetic theory, emphasizing recent results on thermalization-time scaling in nonlinear lattices. A coherent picture emerges in the thermodynamic limit based on the eigenmode properties of an appropriate integrable reference system. If the reference system has extended eigenmodes and the leading resonant or quasi-resonant processes form a sufficiently connected network, the thermalization time follows
Teq∝g−2, where
g measures the effective deviation from integrability. This scaling is found broadly in ordered and weakly disordered lattices and is robust to dimensionality, interaction potential, integrability-breaking mechanism, and multimode initial conditions. Identifying the correct integrable reference is essential; in some one-dimensional FPUT-type lattices with cubic interactions, the nearby Toda lattice, rather than the harmonic chain, provides the proper reference. If all reference eigenmodes are localized, spatial-overlap constraints progressively fragment low-order resonance networks as
g decreases, and thermalization becomes controlled by higher-order processes. Numerical studies reveal successive regimes
Teq∝g−γ with
γ=2,4,6, together with weak system-size dependence. Whether this hierarchy persists asymptotically and whether a finite thermalization threshold exists remain open questions. We also discuss finite-size effects, strongly nonintegrable dynamics, heat transport, localization, and higher-dimensional lattices.
Abstract We investigate the equilibration process of the strongly coupled quartic Fermi–Pasta–Ulam–Tsingou model by adding Langevin baths to the ends of the chain. The time evolution of the system is investigated by means of extensive numerical simulations and shown to match the results expected from equilibrium statistical mechanics in the time-asymptotic limit. Upon increasing the nonlinear coupling, the thermalization of the energy spectrum displays an increasing asymmetry in favor of small-scale, high-frequency modes, which relax significantly faster than the large-scale, low-frequency ones. The global equilibration time is found to scale linearly with system size and shown to exhibit a power-law decay with the strength of the nonlinearity and temperature. Nonlinear interaction adds to energy distribution among modes, thus speeding up the thermalization process.
Abstract A new model of vibrational energy transfer in molecular systems taking into account anharmonic (third order) interactions of localized vibrations with a chain of harmonic oscillators is developed. The role of the energy spectrum of the chain and of the magnitude of the non-linear coupling is discussed in detail by an exact numerical solution of the quantum dynamical problem based on the tensor-train (matrix product state) representation of the vibrational wave function. Results show that the type of wave packet motion is determined by the eigen-spectrum of the chain and by its excitation time. It is found that when the excitation of the chain takes place on a much shorter timescale than the energy transfer along the chain the vibrational wave packet moves in a ballistic way independently of the length of the chain. On the other hand when the excitation of the chain takes place on the timescale of the energy transfer along the chain the overall motion becomes superballistic. These findings shed new light on recent observations of ballistic energy transfer along polymethylene chains.
We propose the algorithm for determining vibrational quantum eigenstates of periodic linear chain of atoms coupled by harmonic and third order anharmonic interactions (Fermi-Ulam-Pasta
α problem) in the long wavelength limit within the resonant approach. Eigenstates can be encoded by the sequence of integer numbers determining their energies and wavefunctions. Using these eigenstates we described a single phonon state time evolution showing coherent oscillations. The applications of theory to vibrational energy transport and quantum informatics are discussed.