Quantum dynamical systems typically deal with huge num-bers of degrees of freedom, particularly functional integrals, which makes them very difficult to simulate using deter-ministic methods. Modeling such systems with stochas-tic differential equations (SDEs) is an attractive alterna-tive to Markov chain Monte-Carlo, but there remain stabil-ity problems. An example of a multi-dimensional complex SDE arises from an anharmonic oscillator with a one-mode Bose-Einstein condensate (BEC) Hamiltonian given in [1], for which the numerical solution becomes unreliable after approximatively t = 0.3 for all tested numerical methods. Splitting of drift methods as introduced in [3] have been tried on this problem and seemed to improve the stability of the results, but not to the desired extent (see [1]). To improve the results, new numerical splitting methods have recently been developed. However, it seems that the choice of the numerical scheme alone might not guarantee the reliability of the results. The use of carefully chosen stochastic gauges as in [1] improves the accuracy of the simulation greatly by taking into account some aspects of the dynamics of the system. Non-vanishing boundary terms from partial integration can be problematic and have to be taken care of, for example considering them as constraints and using projected SDEs techniques [2]. 1. Simulation of Bose-Einstein condensates Consider the case of an anharmonic oscillator with one-mode BEC Hamiltonian treated in [1]. From the master equation ∂ρ̂
Quantum dynamical systems typically deal with huge num-bers of degrees of freedom, particularly functional integrals, which makes them very difficult to simulate using deter-ministic methods. Modeling such systems with stochas-tic differential equations (SDEs) is an attractive alterna-tive to Markov chain Monte-Carlo, but there remain stabil-ity problems. An example of a multi-dimensional complex SDE arises from an anharmonic oscillator with a one-mode Bose-Einstein condensate (BEC) Hamiltonian given in [1], for which the numerical solution becomes unreliable after approximatively t = 0.3 for all tested numerical methods. Splitting of drift methods as introduced in [3] have been tried on this problem and seemed to improve the stability of the results, but not to the desired extent (see [1]). To improve the results, new numerical splitting methods have recently been developed. However, it seems that the choice of the numerical scheme alone might not guarantee the reliability of the results. The use of carefully chosen stochastic gauges as in [1] improves the accuracy of the simulation greatly by taking into account some aspects of the dynamics of the system. Non-vanishing boundary terms from partial integration can be problematic and have to be taken care of, for example considering them as constraints and using projected SDEs techniques [2]. 1. Simulation of Bose-Einstein condensates Consider the case of an anharmonic oscillator with one-mode BEC Hamiltonian treated in [1]. From the master equation ∂ρ̂
First principles simulations of the quantum dynamics of interacting Bose gases using the stochastic gauge representation are analysed. In a companion paper, we showed how the positive-P representation can be applied to these problems using stochastic differential equations. That method, however, is limited by increased sampling error as time evolves. Here, we show how the sampling error can be greatly reduced and the simulation time significantly extended using stochastic gauges. In particular, local stochastic gauges (a subset) are investigated. Improvements are confirmed in numerical calculations of single-, double- and multi-mode systems in the weak-mode coupling regime. Convergence issues are investigated, including the recognition of two modes by which stochastic equations produced by phase-space methods in general can diverge: movable singularities and a noise-weight relationship. The example calculated here displays wave-like behaviour in spatial correlation functions propagating in a uniform 1D gas after a sudden change in the coupling constant. This could in principle be tested experimentally using Feshbach resonance methods.
Stochastic differential equations (SDEs) model noisy dynamical systems across quantitative finance, chemistry, physics, and climate science. Classical solvers-Euler-Maruyama schemes, Monte Carlo methods, and finite-difference PDE solvers for Fokker-Planck equations-suffer from exponential growth in computational cost with dimension and accuracy requirements. We present a family of quantum algorithms for simulating and solving SDEs with improved scaling. Our approach combines Hamiltonian simulation for drift terms, quantum random number generation for noise increments, and quantum amplitude estimation for computing expectations. For linear and semilinear SDEs with Lipschitz coefficients, we prove that relevant quantities (option prices, expected hitting times, moments) can be estimated to accuracy ε in time Õ(poly(d)polylog(1/ε)), versus classical O(poly(d)/ε 2). For nonlinear SDEs, we introduce variational quantum schemes leveraging parameterized circuits, with a priori error bounds and trainability analysis avoiding barren plateaus. We analyze multi-asset Black-Scholes models, Heston stochastic volatility, and Langevin dynamics for molecular coarse-graining, quantifying resource requirements (qubits, gate depth, oracle calls). Numerical simulations on small instances demonstrate polynomial-to-super-polynomial empirical speedups in accuracy for fixed runtime, establishing SDE solving as a promising quantum-accelerated task with practical applications.
The classical stochastic differential equation given as dX(t,ω)=a(t,X)dt+b(t,X)dQ(t),X(t0)=X0,t∈[t0,T], where the driving process Q(t) is a martingale, and a and b are adapted functionals has several numerical schemes developed for it, where each exhibits distinct stability and convergence properties depending on the nature of the driving process and the functional space of the solution. A generalization of this equation is the quantum stochastic differential equation introduced by Hudson and Parthasarathy, expressed as dX(t)=E(t,X(t))dΛπ(t)+F(t,X(t))dAf†(t)+G(t,X(t))dAg(t)+H(t,X(t))dt, where E,F,G, and H are stochastic operator-valued functions defined on a Hilbert space. In 1992, Ekhaquere reformulated this equation in an equivalent form involving the non-classical differential equation stated as; dtd⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0)=X0,t∈[t0,T], which represents an ordinary differential equation of non-classical type. The stochastic process X(t) acts as a densely defined linear operator on a tensor product of two Hilbert spaces, one of which is the Boson Fock space. The vectors η and ξ belong to a dense subset of this tensor product Hilbert space, and the mapping (η,ξ)↦P(t,X)(η,ξ) defines a sesquilinear form for each fixed pair (t,X). We analyze the numerical approximation of the QSDE using its equivalent inner-product form. The numerical investigation employs three principal methods: the Lagrangian Quadrature Method, the Linear Multistep Scheme, and the Newton–Cotes Method. Each method is carefully designed to ensure the consistency, stability, and convergence of the approximate solution X(t) toward the exact stochastic process. These schemes are implemented using Python programming language, which provides a flexible and efficient computational environment for testing the proposed algorithms. Our computational results show that the developed schemes generalize Taylor-type approximations of Itô processes under weak convergence criteria. The proposed algorithms exhibit numerical stability and yield results that compare favorably with analytical benchmarks. Furthermore, the implementation avoids direct approximation of the driving quantum processes, thereby preserving the natural adaptation of the schemes to quantum dynamical models.
The quantum dynamics and grand canonical thermodynamics of many-mode (one-, two-, and three-dimensional) interacting Bose gases are simulated from first principles. The model uses a lattice Hamiltonian based on a continuum second-quantized model with two-particle interactions, external potential, and interactions with an environment, with no further approximations. The interparticle potential can be either an (effective) delta function as in Bose-Hubbard models, or extended with a shape resolved by the lattice. Simulations are of a set of stochastic equations that in the limit of many realizations correspond exactly to the full quantum evolution of the many-body systems. These equations describe the evolution of samples of the gauge P distribution of the quantum state, details of which are developed. Conditions under which general quantum phase-space representations can be used to derive stochastic simulation methods are investigated in detail, given the criteria: 1) The simulation corresponds exactly to quantum mechanics in the limit of many trajectories. 2) The number of equations scales linearly with system size, to allow the possibility of efficient first-principles quantum mesoscopic simulations. 3) All observables can be calculated from one simulation. 4) Each stochastic realization is independent to allow straightforward use of parallel algorithms. Special emphasis is placed on allowing for simulation of open systems. In contrast to typical Monte Carlo techniques based on path integrals, the phase-space representation approach can also be used for dynamical calculations. Two major (and related) known technical stumbling blocks with such stochastic simulations are instabilities in the stochastic equations, and pathological trajectory distributions as the boundaries of phase space are approached. These can (and often do) lead to systematic biases in the calculated observables. The nature of these problems are investigated in detail. Many phase-space distributions have, however, more phase-space freedoms than the minimum required for exact correspondence to quantum mechanics, and these freedoms can in many cases be exploited to overcome the instability and boundary term problems, recovering an unbiased simulation. The stochastic gauge technique, which achieves this in a systematic way, is derived and heuristic guidelines for its use are developed. The gauge P representation is an extension of the positive P distribution, which uses coherent basis states, but allows a variety of useful stochastic gauges that are used to overcome the stability problems. Its properties are investigated, and the resulting equations to be simulated for the open interacting Bose gas system are derived. The dynamics of the following many-mode systems are simulated as examples: 1) Uniform one-dimensional and two-dimensional Bose gases after the rapid appearance of significant two-body collisions (e.g. after entering a Feshbach resonance). 2) Trapped bosons, where the size of the trap is of the same order as the range of the interparticle potential. 3) Stimulated Bose enhancement of scattered atom modes during the collision of two Bose-Einstein condensates. The grand canonical thermodynamics of uniform one-dimensional Bose gases is also calculated for a variety of temperatures and collision strengths. Observables calculated include first to third order spatial correlation functions (including at finite interparticle separation) and momentum distributions. The predicted phenomena are discussed. Improvements over the positive P distribution and other methods are discussed, and simulation times are analyzed for Bose-Hubbard lattice models from a general perspective. To understand the behavior of the equations, and subsequently optimize the gauges for the interacting Bose gas, single- and coupled two-mode dynamical and thermodynamical models of interacting Bose gases are investigated in detail. Directions in which future progress can be expected are considered. Lastly, safeguards are necessary to avoid biased averages when exponentials of Gaussian-like trajectory distributions are used (as here), and these are investigated.
Quantum-Inspired Stochastic Differential Equation (Q-SDE) Optimization addresses the challenges of optimizing complex, stochastic systems by integrating quantum-inspired stochastic differential equation modeling. This research presents a novel optimization algorithm, the Q-SDE-based method, designed to effectively handle highly non-linear, stochastic optimization problems characterized by significant noise. We develop a framework that utilizes the inherent stochasticity of Q-SDEs, coupled with differential equation principles, to provide a robust and adaptive approach to finding optimal solutions. The core mechanism involves a dynamic adjustment of the Q-SDE's parameters to minimize the error function across a wider range of solutions, mitigating the impact of noise and enhancing the algorithm's performance in challenging scenarios. This work demonstrates the effectiveness of this approach through extensive numerical simulations and comparisons with established optimization techniques, highlighting its potential for significant improvements in the optimization of stochastic systems.
The paper deals with the numerical solution of the nonlinear Itô stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrödinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.