Abhinav Muraleedharan

In this work we introduce the quantum springs and sticks (QSS) model, a quantum machine learning model that performs supervised learning with an exponential amount of degrees of freedom. The model is based on the springs and sticks model, a classical mechanical system that uses coupled oscillators and dissipation to perform non-linear regression. We map the dynamics to a time-dependent Hamiltonian, which uses the Caldirola–Kanai parametrization to simulate dissipation through unitary evolution. Then, the solution to the supervised learning problem can be extracted via a spectral decomposition. Finally, we simulate our algorithm on simple regression tasks using time-dependent Trotterization, and compare it to the classical Langevin-dynamics version of such algorithm.

We present quantum algorithms for simulating the dynamics of a broad class of classical oscillator systems containing 2^n coupled oscillators (Eg: 2^n masses coupled by springs), including those with time-dependent forces, time- varying stiffness matrices, and weak nonlinear interactions. This generaliza- tion of the Harmonic oscillator simulation algorithm is achieved through an approach that we call “Nonlinear Schrödingerization”, which involves reduction of the dynamical system to a nonlinear Schrödinger equation and then reduced to a time-independent Schrodinger Equation through perturbative techniques. The linearization of the equation is performed using an approach that allows the dynamics of a nonlinear Schrödinger equation to be approximated as a linear Schrödinger equation in a higher dimensional space. This allows Hamiltonian Simulation algorithms to be applied to simulate the dynamics of resulting sys- tem. When the properties of the classical dynamical systems can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in n, and almost linear in evolution time for most dynamical systems. Our work extends the applicability of quan- tum algorithms to simulate the dynamics of non-conservative and nonlinear classical systems, addressing key limitations in previous approaches