Trotter transition in BCS pairing dynamics

Abstract

We study universal aspects of thermalization induced by Trotterization, a procedure routinely used in gate-based quantum computation. We use the reduced-BCS model -- quantum integrable with a classically integrable mean-field limit -- where the effects of Trotter chaos are expected to be particularly stark. The resulting Trotterized chaotic dynamics is characterized by its Lyapunov spectrum and rescaled Kolmogorov-Sinai entropy. The chaos quantifiers depend on the Trotterization time step $\tau$. We observe a Trotter transition at a finite step value ${\tau }_c\approx \sqrt{N}$. While the dynamics is weakly chaotic for time steps $\tau \ll {\tau }_c$, the regime of large Trotterization steps is characterized by short temporal correlations. We derive two different scaling laws for the two different regimes by numerically fitting the maximum Lyapunov exponent data. The scaling law of the large \(τ\) limit agrees well with the one derived from the kicked top map. Beyond its relevance to current quantum computers, our work opens new directions -- such as probing observables like the Loschmidt echo, which lie beyond standard mean-field description -- across the Trotter transition we uncover.

17 pages, 13 figures; revised abstract and introduction; added references; expanded explanations; improved figures with larger legends

References in corpus (46)

• Quantum Simulation
• Efficient quantum algorithms for simulating sparse Hamiltonians
• Equilibrium states of generic quantum systems subject to periodic driving
• Long-time behavior of periodically driven isolated interacting lattice systems
• Floquet-Magnus Theory and Generic Transient Dynamics in Periodically Driven Many-Body Quantum Systems
• Exponentially slow heating in periodically driven many-body systems
• Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems
• Effective Hamiltonians, prethermalization and slow energy absorption in periodically driven many-body systems
• The Lyapunov Characteristic Exponents and their computation
• Periodically driven ergodic and many-body localized quantum systems
• High order symplectic integrators for perturbed Hamiltonian systems
• A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems
• Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time
• Real- and imaginary-time evolution with compressed quantum circuits
• Exponentially more precise quantum simulation of fermions I: Quantum chemistry in second quantization
• Quantum localization bounds Trotter errors in digital quantum simulation
• Nonequilibrium Cooper pairing in the nonadiabatic regime
• Solution for the dynamics of the BCS and central spin problems
• Observation of a Transition Between Dynamical Phases in a Quantum Degenerate Fermi Gas
• Explicit symplectic approximation of nonseparable Hamiltonians: algorithm and long time performance
• Digital Quantum Simulation, Trotter Errors, and Quantum Chaos of the Kicked Top
• Numerical integration of variational equations
• Photonic Simulation of Entanglement Growth After a Spin Chain Quench
• Weakly nonergodic dynamics in the Gross--Pitaevskii lattice
• Absence of dynamical localization in interacting driven systems
• Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure
• Simulating hydrodynamics on noisy intermediate-scale quantum devices with random circuits
• Finite Size Corrections for the Pairing Hamiltonian
• Reinforcement Learning for Digital Quantum Simulation
• Dynamical Glass and Ergodization Times in Classical Josephson Junction Chains
• Dynamical glass in weakly non-integrable Klein-Gordon chains
• Statistical mechanics of an integrable system
• Optimal compression of quantum many-body time evolution operators into brickwall circuits
• Nonlinear dynamics and quantum chaos of a family of kicked $p$-spin models
• Heating in integrable time-periodic systems
• Trotter errors from dynamical structural instabilities of Floquet maps in quantum simulation
• Lyapunov spectrum scaling for classical many-body dynamics close to integrability
• Chaos and localization in the Discrete Nonlinear Schrödinger Equation
• Krylov complexity and Trotter transitions in unitary circuit dynamics
• Thermalization slowing down in multidimensional Josephson junction networks
• Geometric Floquet theory
• Fragile Many Body Ergodicity
• Robust effective ground state in a nonintegrable Floquet quantum circuit
• Dynamical chaos in the integrable Toda chain induced by time discretization
• Nonlocality as the source of purely quantum dynamics of BCS superconductors
• Non-linearity and chaos in the kicked top