Simulation of realistic classical mechanical systems is of great importance to many areas of engineering such as robotics, dynamics of rotating machinery and control theory. In this work, we develop quantum algorithms to estimate quantities of interest such as the kinetic energy in a given classical mechanical system in the presence of friction or damping as well as forcing or source terms, which makes the algorithm of practical interest. We show that for such systems, the quantum algorithm scales polynomially with the logarithm of the dimension of the system. We cast this problem in terms of Hamilton's equations of motion (equivalent to the first variation of the Lagrangian) and solve them using quantum algorithms for differential equations. We then consider the hardness of estimating the kinetic energy of a damped coupled oscillator system. We show that estimating the kinetic energy at a given time of this system to within additive precision is BQP hard when the strength of the damping term is bounded by an inverse polynomial in the number of qubits. We then consider the problem of designing optimal control of classical systems, which can be cast as the second variation of the Lagrangian. In this direction, we first consider the Riccati equation, which is a nonlinear differential equation ubiquitous in control theory. We give an efficient quantum algorithm to solve the Riccati differential equation well into the nonlinear regime. To our knowledge, this is the first example of any nonlinear differential equation that can be solved when the strength of the nonlinearity is asymptotically greater than the amount of dissipation. We then show how to use this algorithm to solve the linear quadratic regulator problem, which is an example of the Hamilton-Jacobi-Bellman equation.
Simulation of realistic classical mechanical systems is of great importance to many areas of engineering such as robotics, dynamics of rotating machinery and control theory. In this work, we develop quantum algorithms to estimate quantities of interest such as the kinetic energy in a given classical mechanical system in the presence of friction or damping as well as forcing or source terms, which makes the algorithm of practical interest. We show that for such systems, the quantum algorithm scales polynomially with the logarithm of the dimension of the system. We cast this problem in terms of Hamilton's equations of motion (equivalent to the first variation of the Lagrangian) and solve them using quantum algorithms for differential equations. We then consider the hardness of estimating the kinetic energy of a damped coupled oscillator system. We show that estimating the kinetic energy at a given time of this system to within additive precision is BQP hard when the strength of the damping term is bounded by an inverse polynomial in the number of qubits. We then consider the problem of designing optimal control of classical systems, which can be cast as the second variation of the Lagrangian. In this direction, we first consider the Riccati equation, which is a nonlinear differential equation ubiquitous in control theory. We give an efficient quantum algorithm to solve the Riccati differential equation well into the nonlinear regime. To our knowledge, this is the first example of any nonlinear differential equation that can be solved when the strength of the nonlinearity is asymptotically greater than the amount of dissipation. We then show how to use this algorithm to solve the linear quadratic regulator problem, which is an example of the Hamilton-Jacobi-Bellman equation.
We develop quantum algorithms for optimal control over a prescribed finite time interval (finite horizon), focusing on continuous-time linear-quadratic regulation (LQR). This foundational control problem often requires solving a nonlinear matrix Riccati equation, which can be costly for large systems and long control durations on conventional classical solvers. We instead work directly with the state-costate optimality equations, reformulating the LQR problem as a multiple-shooting linear system. Importantly, we prove a condition-number bound for the system that is independent of the horizon, under stabilizability and detectability assumptions. By incorporating the initial and final conditions into a single linear system, we can solve the subsequent linear system more efficiently with a quantum linear systems subroutine. For a given initial state, our quantum algorithms return classical estimates of the optimal cost and the optimal control inputs at a selected time, without constructing a Riccati solution operator. Given sparse-matrix block-encodings, an initial-state preparation oracle and a fixed number of control inputs, the query and gate complexities scale polylogarithmically with system size and control duration, provided that sparsity, relevant norms, and stability parameters are bounded. The dependence on the target error is inverse-linear up to logarithmic factors. We establish the required bounds uniformly in system size and duration for a family of damped mass-spring chains with two endpoint controls. Numerical experiments support the analysis.
We present a quantum algorithm for simulating classical oscillator networks characterized by non-Markovian dissipation and time-varying material properties, extending recent speedups for undamped harmonic systems to viscoacoustic and viscoelastic media. We embed the history-dependent dynamics into a Markovian state space governed by a non-Hermitian operator by approximating memory kernels through a Prony series. We then use linear combination of Hamiltonian simulation to estimate the instantaneous kinetic and potential energy for a subset of oscillators at time t within error ε using a number of queries to the oscillator system that scales as O(αtott/ε), where αtot is polynomial in the strength of the dissipation, spring constants, inverse masses, and sparsity of the connections in the network. We further show that this energy estimation task is in the worst-case classically hard (i.e., a corresponding decision problem is BQP-complete), even in the presence of strong dissipation. For time-dependent materials, we show that changes in material properties appear as effective dissipation or growth in the energy representation. Additionally, we prove the infeasibility of exponential quantum advantages in locally coupled topologies through a novel form of Lieb-Robinson-like bounds that apply to differential equations. This allows our quantum algorithms to provide quartic speedups for locally coupled damped oscillator systems in three dimensions, raising the possibility of practical quantum speedups for realistically damped oscillator networks and approximated wave equations.
Important nonlinear dynamics, such as those found in plasma and fluid systems, are typically hard to simulate on classical computers. Thus, if fault-tolerant quantum computers could efficiently solve such nonlinear problems, it would be a transformative change for many industries. In a recent breakthrough [Liu et al., PNAS 2021], the first efficient quantum algorithm for solving nonlinear differential equations was constructed, based on a single condition R<1, where R characterizes the ratio of nonlinearity to dissipation. This result, however, is limited to the class of purely dissipative systems with negative log-norm, which excludes application to many important problems. In this work, we correct technical issues with this and other prior analysis, and substantially extend the scope of nonlinear dynamical systems that can be efficiently simulated on a quantum computer in a number of ways. Firstly, we extend the existing results from purely dissipative systems to a much broader class of stable systems, and show that every quadratic Lyapunov function for the linearized system corresponds to an independent R-number criterion for the convergence of the Carleman scheme. Secondly, we extend our stable system results to physically relevant settings where conserved polynomial quantities exist. Finally, we provide extensive results for the class of non-resonant systems. With this, we are able to show that efficient quantum algorithms exist for a much wider class of nonlinear systems than previously known, and prove the BQP-completeness of nonlinear oscillator problems of exponential size. In our analysis, we also obtain several results related to the Poincaré-Dulac theorem and diagonalization of the Carleman matrix, which could be of independent interest.
We present an end-to-end quantum algorithm for simulating nonlinear dynamics described by a system of stochastic dissipative differential equations with a quadratic nonlinearity. The stochastic part of the system is modeled by a Gaussian noise in the equation of motion and in the initial conditions. Our algorithm can approximate the expected value of any correlation function that depends on O(1) variables with rigorous bounds on the approximation error. The runtime scales polynomially with logN, t, J, and λ1−1, where N is the total number of variables, t is the evolution time, J is the nonlinearity strength, and λ1 is the smallest dissipation rate. However, the runtime scales exponentially with a parameter quantifying inverse relative error in the initial conditions. To the best of our knowledge, this is the first rigorous quantum algorithm capable of simulating strongly nonlinear systems with J≫λ1 at the cost poly-logarithmic in N and polynomial in t. The considered simulation problem is shown to be BQP-complete, providing a strong evidence for a quantum advantage. We benchmark the quantum algorithm via numerical experiments by simulating a vortex flow in the 2D Navier Stokes equation.
Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming [Formula: see text], where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity [Formula: see text], where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for [Formula: see text] Finally, we discuss potential applications, showing that the [Formula: see text] condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.
We present a framework for efficient extraction of the viscosity solutions of nonlinear Hamilton-Jacobi equations with convex Hamiltonians. These viscosity solutions play a central role in areas such as front propagation, mean-field games, optimal control, machine learning, and a direct application to the forced Burgers' equation. Our method is based on an entropy penalisation method proposed by Gomes and Valdinoci, which generalises the Cole-Hopf transform from quadratic to general convex Hamiltonians, allowing a reformulation of viscous Hamilton-Jacobi dynamics by a discrete-time linear dynamics which approximates a linear heat-like parabolic equation, and can also extend to continuous-time dynamics. This makes the method suitable for quantum simulation. The validity of these results hold for arbitrary nonlinearity that correspond to convex Hamiltonians, and for arbitrarily long times, thus obviating a chief obstacle in most quantum algorithms for nonlinear partial differential equations. We provide quantum algorithms, both analog and digital, for extracting pointwise values, gradients, minima, and function evaluations at the minimiser of the viscosity solution, without requiring nonlinear updates or full state reconstruction.