arXiv subjects

Shravan Veerapaneni

Publications and source records attributed to Shravan Veerapaneni.

At least 37 records · Page 2Linked to original sources

arXiv2022

Digital quantum simulation of Schrödinger dynamics using adaptive approximations of potential functions

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.

quant-ph↗

arXiv2022

Fast integral equation methods for linear and semilinear heat equations in moving domains

We present a family of integral equation-based solvers for the linear or semilinear heat equation in complicated moving (or stationary) geometries. This approach has significant advantages over more standard finite element or finite difference methods in terms of accuracy, stability and space-time adaptivity. In order to be practical, however, a number of technical capabilites are required: fast algorithms for the evaluation of heat potentials, high-order accurate quadratures for singular and weakly integrals over space-time domains, and robust automatic mesh refinement and coarsening capabilities. We describe all of these components and illustrate the performance of the approach with numerical examples in two space dimensions.

math.NA↗

arXiv2022

Toward Neural Network Simulation of Variational Quantum Algorithms

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.

quant-ph↗

arXiv2022

Tensor-Train Compression of Discrete Element Method Simulation Data

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.

math.NA↗

arXiv2022

Scalable neural quantum states architecture for quantum chemistry

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.

physics.chem-ph↗

arXiv2022

Quantum-inspired variational algorithms for partial differential equations: Application to financial derivative pricing

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 Schrödinger 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.

math.NA↗

arXiv2022

Numerical and geometrical aspects of flow-based variational quantum Monte Carlo

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.

quant-ph↗

arXiv2022

A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings

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 Poincaré 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.

math.NA↗

arXiv2022

Continuous-variable optimization with neural network quantum states

Inspired by proposals for continuous-variable quantum approximate optimization (CV-QAOA), we investigate the utility of continuous-variable neural network quantum states (CV-NQS) for performing continuous optimization, focusing on the ground state optimization of the classical antiferromagnetic rotor model. Numerical experiments conducted using variational Monte Carlo with CV-NQS indicate that although the non-local algorithm succeeds in finding ground states competitive with the local gradient search methods, the proposal suffers from unfavorable scaling. A number of proposed extensions are put forward which may help alleviate the scaling difficulty.

quant-ph↗

arXiv2022

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula. Numerical results illustrate the efficiency of the solver in accelerating particulate Stokes simulations.

math.NA↗

arXiv2021

Shape optimization of peristaltic pumps transporting rigid particles in Stokes flow

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.

math.OC↗

arXiv2021

Optimal Ciliary Locomotion of Axisymmetric Microswimmers

Many biological microswimmers locomote by periodically beating the densely-packed cilia on their cell surface in a wave-like fashion. While the swimming mechanisms of ciliated microswimmers have been extensively studied both from the analytical and the numerical point of view, the optimization of the ciliary motion of microswimmers has received limited attention, especially for non-spherical shapes. In this paper, using an envelope model for the microswimmer, we numerically optimize the ciliary motion of a ciliate with an arbitrary axisymmetric shape. The forward solutions are found using a fast boundary integral method, and the efficiency sensitivities are derived using an adjoint-based method. Our results show that a prolate microswimmer with a 2:1 aspect ratio shares similar optimal ciliary motion as the spherical microswimmer, yet the swimming efficiency can increase two-fold. More interestingly, the optimal ciliary motion of a concave microswimmer can be qualitatively different from that of the spherical microswimmer, and adding a constraint to the ciliary length is found to improve, on average, the efficiency for such swimmers.

cond-mat.soft↗

arXiv2021

Continuous-variable neural-network quantum states and the quantum rotor model

We initiate the study of neural-network quantum state algorithms for analyzing continuous-variable lattice quantum systems in first quantization. A simple family of continuous-variable trial wavefunctons is introduced which naturally generalizes the restricted Boltzmann machine (RBM) wavefunction introduced for analyzing quantum spin systems. By virtue of its simplicity, the same variational Monte Carlo training algorithms that have been developed for ground state determination and time evolution of spin systems have natural analogues in the continuum. We offer a proof of principle demonstration in the context of ground state determination of a stoquastic quantum rotor Hamiltonian. Results are compared against those obtained from partial differential equation (PDE) based scalable eigensolvers. This study serves as a benchmark against which future investigation of continuous-variable neural quantum states can be compared, and points to the need to consider deep network architectures and more sophisticated training algorithms.

quant-ph↗

arXiv2021

Overcoming barriers to scalability in variational quantum Monte Carlo

The variational quantum Monte Carlo (VQMC) method received significant attention in the recent past because of its ability to overcome the curse of dimensionality inherent in many-body quantum systems. Close parallels exist between VQMC and the emerging hybrid quantum-classical computational paradigm of variational quantum algorithms. VQMC overcomes the curse of dimensionality by performing alternating steps of Monte Carlo sampling from a parametrized quantum state followed by gradient-based optimization. While VQMC has been applied to solve high-dimensional problems, it is known to be difficult to parallelize, primarily owing to the Markov Chain Monte Carlo (MCMC) sampling step. In this work, we explore the scalability of VQMC when autoregressive models, with exact sampling, are used in place of MCMC. This approach can exploit distributed-memory, shared-memory and/or GPU parallelism in the sampling task without any bottlenecks. In particular, we demonstrate the GPU-scalability of VQMC for solving up to ten-thousand dimensional combinatorial optimization problems.

cs.DC↗

arXiv2021

High-order close evaluation of Laplace layer potentials: A differential geometric approach

This paper presents a new approach for solving the close evaluation problem in three dimensions, commonly encountered while solving linear elliptic partial differential equations via potential theory. The goal is to evaluate layer potentials close to the boundary over which they are defined. The approach introduced here converts these nearly-singular integrals on a patch of the boundary to a set of non-singular line integrals on the patch boundary using the Stokes theorem on manifolds. A function approximation scheme based on harmonic polynomials is designed to express the integrand in a form that is suitable for applying the Stokes theorem. As long as the data -- the boundary and the density function -- is given in a high-order format, the double-layer potential and its derivatives can be evaluated with high-order accuracy using this scheme both on and off the boundary. In particular, we present numerical results demonstrating seventh-order convergence on a smooth, warped torus example achieving 10-digit accuracy in evaluating double layer potential at targets that are arbitrarily close to the boundary.

math.NA↗

arXiv2021

Fast and accurate solvers for simulating Janus particle suspensions in Stokes flow

We present a novel computational framework for simulating suspensions of rigid spherical Janus particles in Stokes flow. We show that long-range Janus particle interactions for a wide array of applications may be resolved using fast, spectrally accurate boundary integral methods tailored to polydisperse suspensions of spherical particles. These are incorporated into our rigid body Stokes platform. Our approach features the use of spherical harmonic expansions for spectrally accurate integral operator evaluation, complementarity-based collision resolution, and optimal O(n) scaling with the number of particles when accelerated via fast summation techniques. We demonstrate the flexibility of our platform through three key examples of Janus particle systems prominent in biomedical applications: amphiphilic, bipolar electric and phoretic particles. We formulate Janus particle interactions in boundary integral form and showcase characteristic self-assembly and complex collective behavior for each particle type.

physics.flu-dyn↗

arXiv2020

Meta Variational Monte Carlo

An identification is found between meta-learning and the problem of determining the ground state of a randomly generated Hamiltonian drawn from a known ensemble. A model-agnostic meta-learning approach is proposed to solve the associated learning problem and a preliminary experimental study of random Max-Cut problems indicates that the resulting Meta Variational Monte Carlo accelerates training and improves convergence.

quant-ph↗

arXiv2020

Natural evolution strategies and variational Monte Carlo

A notion of quantum natural evolution strategies is introduced, which provides a geometric synthesis of a number of known quantum/classical algorithms for performing classical black-box optimization. Recent work of Gomes et al. [2019] on heuristic combinatorial optimization using neural quantum states is pedagogically reviewed in this context, emphasizing the connection with natural evolution strategies. The algorithmic framework is illustrated for approximate combinatorial optimization problems, and a systematic strategy is found for improving the approximation ratios. In particular it is found that natural evolution strategies can achieve approximation ratios competitive with widely used heuristic algorithms for Max-Cut, at the expense of increased computation time.

quant-ph↗