We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.
We consider the dynamics of N interacting Bosons in three dimensions which are strongly confined in one or two directions. We analyze the mean field scaling and the nonlinear Schrödinger scaling of the interaction potential w and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit the dynamics of the N-particle system is described by a nonlinear equation in two or one dimensions. In the case of the mean field scaling this equation is the Hartree equation and for the second scaling the nonlinear Schrödinger equation. In both cases we obtain explicit bounds for the rate of convergence of the N-particle dynamics to the one-particle dynamics.
Mean-field Hartree theory is a central tool for reducing many-body dynamics to an effective one-particle evolution. This approximation has been employed when the Hamiltonian governing the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail.
First, we consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, for non-Hermitian Hamiltonian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at
t=0 in the thermodynamic limit. Then, we analytically solve a model of
N bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for
N→∞ of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit
N→∞.
Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.
Mean-field Hartree theory is a central tool for mapping interacting many-body dynamics into an effective nonlinear one-particle evolution. This approximation has been also employed when the Hamiltonian that governs the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations, where the nonlinear equation is mapped into a many-body Schr\"odinger equation, which may be non-Hermitian. In this application, the mean-field approximation can be seen as a linearization method, and at the end of the algorithm, the obtained linear system is solved using the Harrow-Hassidim-Lloyd (HHL) algorithm. The non-Hermitian mean-field Hartree approximation is far from being completely understood, and no rigorous results justify its applications. Instead, it is usually presented as a heuristic extension of the Hermitian case. In the first part of this thesis, we show that this approximation can fail. We consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, if the Hamiltonian is not Hermitian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at
t=0 in the limit of infinitely many particles. We construct a series of examples that we can solve analytically or numerically, which support the failure of the Hartree approximation and give further insight into the new phenomena that arise in non-Hermitian physics. Furthermore, we consider the mean-field evolution of open quantum systems and we show how to use the mean-field approximation of open quantum systems as a linearization method of nonlinear differential equations. In the second part of this thesis, we study the mean-field dynamics of a Hermitian system of
N interacting bosons starting from an initially condensed state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having
n particles outside the condensate decays exponentially in
n for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having
n excitations. Finally, we consider the HHL algorithm. Several circuit-level variants have been proposed to simplify its implementation, including constructions in which the clock register is factorized via Hadamard gates to reduce circuit depth and entanglement. We provide a rigorous analysis of the convergence of the output towards the ideal solution, and we show that this commonly deployed variant can fundamentally alter the asymptotic convergence of the algorithm. This phenomenon is not a finite-size artifact and compromises the correctness guarantees of the factorized-clock implementation. We provide an analytic characterization of the underlying mechanism and propose a minimal and experimentally feasible modification that suffices to restore convergence while preserving the practical advantages of the simplified circuit. Our results clarify an overlooked correctness issue in a widely used HHL implementation pattern and delineate the conditions under which simplified realizations faithfully reproduce the intended algorithmic behavior.