Search

Although quantum computing holds promise to accelerate a wide range of computational tasks, the quantum simulation of quantum dynamics as originally envisaged by Feynman remains the most promising candidate for achieving quantum advantage. A less explored possibility with comparably far-reaching technological applicability is the quantum simulation of classical nonlinear dynamics. Attempts to develop digital quantum algorithms based on the Koopman von Neumann formalism have met with challenges because of the necessary projection step from an infinite-dimensional Hilbert space to the finite-dimensional subspace described by a collection of qubits. This finitization produces numerical artifacts that limit solutions to very short time horizons. In this paper we review continuous-variable quantum computing (CVQC), which naturally avoids such obstacles, and a CVQC algorithm for KvN simulation of classical nonlinear dynamics is advocated. In particular, we present explicit gate synthesis for product-formula Hamiltonian simulation of anharmonic vibrational dynamics., Comment: 11 pages

In this work, we develop a computational framework that aims at simultaneously optimizing the shape and the slip velocity of an axisymmetric microswimmer suspended in a viscous fluid. We consider shapes of a given reduced volume that maximize the swimming efficiency, i.e., the (size-independent) ratio of the power loss arising from towing the rigid body of the same shape and size at the same translation velocity to the actual power loss incurred by swimming via the slip velocity. The optimal slip and efficiency (with shape fixed) are here given in terms of two Stokes flow solutions, and we then establish shape sensitivity formulas of adjoint-solution that provide objective function derivatives with respect to any set of shape parameters on the sole basis of the above two flow solutions. Our computational treatment relies on a fast and accurate boundary integral solver for solving all Stokes flow problems. We validate our analytic shape derivative formulas via comparisons against finite-difference gradient evaluations, and present several shape optimization examples.

Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of time-dependent PDEs, the dominant paradigm involves casting the approximate solution in terms of stochastic minimization of an objective function given by the norm of the PDE residual, viewed as a function of the neural network parameters. Recently, advancements have been made in the direction of an alternative approach which shares aspects of nonlinearly parametrized Galerkin methods and variational quantum Monte Carlo, especially for high-dimensional, time-dependent PDEs that extend beyond the usual scope of quantum physics. This paper is inspired by the potential of solving Hamilton-Jacobi-Bellman (HJB) PDEs using Neural Galerkin methods and commences the exploration of nonlinearly parametrized trial functions for which the evolution equations are analytically tractable. As a precursor to the Neural Galerkin scheme, we present trial functions with evolution equations that admit closed-form solutions, focusing on time-dependent HJB equations relevant to finance.

Digital quantum simulation (DQS) of continuous-variable quantum systems in the position basis requires efficient implementation of diagonal unitaries approximating the time evolution operator generated by the potential energy function. In this work, we provide efficient implementations suitable for potential functions approximable by piecewise polynomials, with either uniform or adaptively chosen subdomains. For a fixed precision of approximation, we show how adaptive grids can significantly reduce the total gate count at the cost of introducing a small number of ancillary qubits. We demonstrate the circuit construction with both physically motivated and artificially designed potential functions, and discuss their generalizations to higher dimensions.

We present a new algorithm for incrementally updating the tensor train decomposition of a stream of tensor data. This new algorithm, called the {\em tensor train incremental core expansion} (TT-ICE) improves upon the current state-of-the-art algorithms for compressing in tensor train format by developing a new adaptive approach that incurs significantly slower rank growth and guarantees compression accuracy. This capability is achieved by limiting the number of new vectors appended to the TT-cores of an existing accumulation tensor after each data increment. These vectors represent directions orthogonal to the span of existing cores and are limited to those needed to represent a newly arrived tensor to a target accuracy. We provide two versions of the algorithm: TT-ICE and TT-ICE accelerated with heuristics (TT-ICE$^*$). We provide a proof of correctness for TT-ICE and empirically demonstrate the performance of the algorithms in compressing large-scale video and scientific simulation datasets. Compared to existing approaches that also use rank adaptation, TT-ICE$^*$ achieves $57\times$ higher compression and up to $95\%$ reduction in computational time., Comment: 26 pages, 10 figures, for the python code of TT-ICE and TT-ICE$^*$ algorithms see https://github.com/dorukaks/TT-ICE

Variational quantum algorithms (VQAs) utilize a hybrid quantum-classical architecture to recast problems of high-dimensional linear algebra as ones of stochastic optimization. Despite the promise of leveraging near- to intermediate-term quantum resources to accelerate this task, the computational advantage of VQAs over wholly classical algorithms has not been firmly established. For instance, while the variational quantum eigensolver (VQE) has been developed to approximate low-lying eigenmodes of high-dimensional sparse linear operators, analogous classical optimization algorithms exist in the variational Monte Carlo (VMC) literature, utilizing neural networks in place of quantum circuits to represent quantum states. In this paper we ask if classical stochastic optimization algorithms can be constructed paralleling other VQAs, focusing on the example of the variational quantum linear solver (VQLS). We find that such a construction can be applied to the VQLS, yielding a paradigm that could theoretically extend to other VQAs of similar form., Comment: To appear at the workshop on AI for Science: Progress and Promises at NeurIPS 2022

We propose a framework for discrete scientific data compression based on the tensor-train (TT) decomposition. Our approach is tailored to handle unstructured output data from discrete element method (DEM) simulations, demonstrating its effectiveness in compressing both raw (e.g. particle position and velocity) and derived (e.g. stress and strain) datasets. We show that geometry-driven "tensorization" coupled with the TT decomposition (known as quantized TT) yields a hierarchical compression scheme, achieving high compression ratios for key variables in these DEM datasets.

Variational optimization of neural-network representations of quantum states has been successfully applied to solve interacting fermionic problems. Despite rapid developments, significant scalability challenges arise when considering molecules of large scale, which correspond to non-locally interacting quantum spin Hamiltonians consisting of sums of thousands or even millions of Pauli operators. In this work, we introduce scalable parallelization strategies to improve neural-network-based variational quantum Monte Carlo calculations for ab-initio quantum chemistry applications. We establish GPU-supported local energy parallelism to compute the optimization objective for Hamiltonians of potentially complex molecules. Using autoregressive sampling techniques, we demonstrate systematic improvement in wall-clock timings required to achieve CCSD baseline target energies. The performance is further enhanced by accommodating the structure of resultant spin Hamiltonians into the autoregressive sampling ordering. The algorithm achieves promising performance in comparison with the classical approximate methods and exhibits both running time and scalability advantages over existing neural-network based methods., Comment: 11 pages, 4 figures

Variational quantum Monte Carlo (VMC) combined with neural-network quantum states offers a novel angle of attack on the curse-of-dimensionality encountered in a particular class of partial differential equations (PDEs); namely, the real- and imaginary time-dependent Schr\"odinger equation. In this paper, we present a simple generalization of VMC applicable to arbitrary time-dependent PDEs, showcasing the technique in the multi-asset Black-Scholes PDE for pricing European options contingent on many correlated underlying assets.

This article aims to summarize recent and ongoing efforts to simulate continuous-variable quantum systems using flow-based variational quantum Monte Carlo techniques, focusing for pedagogical purposes on the example of bosons in the field amplitude (quadrature) basis. Particular emphasis is placed on the variational real- and imaginary-time evolution problems, carefully reviewing the stochastic estimation of the time-dependent variational principles and their relationship with information geometry. Some practical instructions are provided to guide the implementation of a PyTorch code. The review is intended to be accessible to researchers interested in machine learning and quantum information science., Comment: Review article

This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's function in complex geometries. Utilizing volumetric domain decomposition, the integral is computed over a union of regular boxes (lending the scheme compatibility with adaptive box codes) and triangular regions (which may be potentially curved near boundaries). Singular and near-singular quadrature is handled by converting integrals on volumetric regions to line integrals bounding a reference volume cell using cell mappings and elements of the Poincar\'e lemma, followed by leveraging existing one-dimensional near-singular and singular quadratures appropriate to the singular nature of the kernel. The scheme achieves compatibility with fast multipole methods (FMMs) and thereby optimal asymptotic complexity by coupling global rules for target-independent quadrature of smooth functions to local target-dependent singular quadrature corrections, and it relies on orthogonal polynomial systems on each cell for well-conditioned, high-order and efficient (with respect to number of required volume function evaluations) approximation of arbitrary volumetric sources. Our domain discretization scheme is naturally compatible with standard meshing software such as Gmsh, which are employed to discretize a narrow region surrounding the domain boundaries. We present 8th-order accurate results, demonstrate the success of the method with examples showing up to 12-digit accuracy on complex geometries, and, for static geometries, our numerical examples show well over $99\%$ of evaluation time of the particular solution is spent in the FMM step., Comment: 33 pages, 9 figures

This paper presents a computational approach for finding the optimal shapes of peristaltic pumps transporting rigid particles in Stokes flow. In particular, we consider shapes that minimize the rate of energy dissipation while pumping a prescribed volume of fluid, number of particles and/or distance traversed by the particles over a set time period. Our approach relies on a recently developed fast and accurate boundary integral solver for simulating multiphase flows through periodic geometries of arbitrary shapes. In order to fully capitalize on the dimensionality reduction feature of the boundary integral methods, shape sensitivities must ideally involve evaluating the physical variables on the particle or pump boundaries only. We show that this can indeed be accomplished owing to the linearity of Stokes flow. The forward problem solves for the particle motion in a slip-driven pipe flow while the adjoint problems in our construction solve quasi-static Dirichlet boundary value problems backwards in time, retracing the particle evolution. The shape sensitivities simply depend on the solution of one forward and one adjoint (for each shape functional) problems. We validate these analytic shape derivative formulas by comparing against finite-difference based gradients and present several examples showcasing optimal pump shapes under various constraints., Comment: 24 pages, 8 figures