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Lie-Trotter-Suzuki decompositions are an efficient way to approximate operator exponentials $\exp(t H)$ when $H$ is a sum of $n$ (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution…
Universal quantum simulation may provide insights into those many-body systems that cannot be described classically, and that cannot be efficiently simulated with current technology. The Trotter formula, which decomposes a desired unitary…
Nonequilibrium time evolution of large quantum systems is a strong candidate for quantum advantage. Variational quantum algorithms have been put forward for this task, but their quantum optimization routines suffer from trainability and…
We provide a quantum method for simulating Hamiltonian evolution with complexity polynomial in the logarithm of the inverse error. This is an exponential improvement over existing methods for Hamiltonian simulation. In addition, its scaling…
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the…
In this paper we study the convergence of a Lie-Trotter operator splitting for stochastic semi-linear evolution equations in a Hilbert space. The abstract Hilbert space setting allows for the consideration of convergence of the…
Among the single-trajectory Gaussian-based methods for solving the time-dependent Schr\"{o}dinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller's original thawed Gaussian approximation, it…
Prior to the recent development of symplectic integrators, the time-stepping operator $\e^{h(A+B)}$ was routinely decomposed into a sum of products of $\e^{h A}$ and $\e^{hB}$ in the study of hyperbolic partial differential equations. In…
Hamiltonian simulation represents an important module in a large class of quantum algorithms and simulations such as quantum machine learning, quantum linear algebra methods, and modeling for physics, material science and chemistry. One of…
In this work we introduce a quantum sorting algorithm with adaptable requirements of memory and circuit depth, and then use it to develop a new quantum version of the classical machine learning algorithm known as k-nearest neighbors (k-NN).…
We empirically study sorting in the evolving data model. In this model, a sorting algorithm maintains an approximation to the sorted order of a list of data items while simultaneously, with each comparison made by the algorithm, an…
Many HPC applications that solve differential equations rely on the Runge-Kutta family of methods for time integration. Among these methods, the fourth-order accurate RK4 scheme is especially popular. This time integration scheme requires…
We show that a recently discovered fourth order symplectic algorithm, which requires one evaluation of force gradient in addition to three evaluations of the force, when iterated to higher order, yielded algorithms that are far superior to…
We investigate tensor-train approaches to the solution of the time-dependent Schr\"{o}dinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using efficient low-rank tensor train representations,…
By decomposing the important sampled imaginary time Schr\"odinger evolution operator to fourth order with positive coefficients, we derived a number of distinct fourth order Diffusion Monte Carlo algorithms. These sophisticated algorithms…
We propose to explore the potential advantages of a new class of tracking algorithms loosely inspired by the Hough transform concept and where we include the time of arrival of each hit as an additional coordinate to be treated in the same…
The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the…
We study a variation of the Trotter-Suzuki decomposition, in which a Hamiltonian exponential is approximated by an ordered product of two-qubit operator exponentials such that the Trotter step size is enhanced for a small number of terms.…
Simulation of continuous time evolution requires time discretization on both classical and quantum computers. A finer time step improves simulation precision, but it inevitably leads to increased computational efforts. This is particularly…
We consider quadrature formulas of high order in time based on Radau-type, L-stable implicit Runge-Kutta schemes to solve time dependent stiff PDEs. Instead of solving a large nonlinear system of equations, we develop a method that performs…