arXiv subjects

Jeffrey Larson

Publications and source records attributed to Jeffrey Larson.

54 records · Page 3Linked to original sources

arXiv2022

Characterizing Error Mitigation by Symmetry Verification in QAOA

Hardware errors are a major obstacle to demonstrating quantum advantage with the quantum approximate optimization algorithm (QAOA). Recently, symmetry verification has been proposed and empirically demonstrated to boost the quantum state fidelity, the expected solution quality, and the success probability of QAOA on a superconducting quantum processor. Symmetry verification uses parity checks that leverage the symmetries of the objective function to be optimized. We develop a theoretical framework for analyzing this approach under local noise and derive explicit formulas for fidelity improvements on problems with global $\mathbb{Z}_2$ symmetry. We numerically investigate the symmetry verification on the MaxCut problem and identify the error regimes in which this approach improves the QAOA objective. We observe that these regimes correspond to the error rates present in near-term hardware. We further demonstrate the efficacy of symmetry verification on an IonQ trapped ion quantum processor where an improvement in the QAOA objective of up to 19.2\% is observed.

quant-ph↗

arXiv2022

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

We propose a manifold sampling algorithm for minimizing a nonsmooth composition $f= h\circ F$, where we assume $h$ is nonsmooth and may be inexpensively computed in closed form and $F$ is smooth but its Jacobian may not be available. We additionally assume that the composition $h\circ F$ defines a continuous selection. Manifold sampling algorithms can be classified as model-based derivative-free methods, in that models of $F$ are combined with particularly sampled information about $h$ to yield local models for use within a trust-region framework. We demonstrate that cluster points of the sequence of iterates generated by the manifold sampling algorithm are Clarke stationary. We consider the tractability of three particular subproblems generated by the manifold sampling algorithm and the extent to which inexact solutions to these subproblems may be tolerated. Numerical results demonstrate that manifold sampling as a derivative-free algorithm is competitive with state-of-the-art algorithms for nonsmooth optimization that utilize first-order information about $f$.

math.OC↗

arXiv2022

Multistart Algorithm for Identifying All Optima of Nonconvex Stochastic Functions

We propose a multistart algorithm to identify all local minima of a constrained, nonconvex stochastic optimization problem. The algorithm uniformly samples points in the domain and then starts a local stochastic optimization run from any point that is the "probabilistically best" point in its neighborhood. Under certain conditions, our algorithm is shown to asymptotically identify all local optima with high probability; this holds even though our algorithm is shown to almost surely start only finitely many local stochastic optimization runs. We demonstrate the performance of an implementation of our algorithm on nonconvex stochastic optimization problems, including identifying optimal variational parameters for the quantum approximate optimization algorithm.

math.OC↗

arXiv2022

Joint Routing of Conventional and Range-Extended Electric Vehicles in a Large Metropolitan Network

Range-extended electric vehicles combine the higher efficiency and environmental benefits of battery-powered electric motors with the longer mileage and autonomy of conventional internal combustion engines. This combination is particularly advantageous for time-constrained delivery routing in dense urban areas, where battery recharging along routes can be too time-consuming to economically justify the use of all-electric vehicles. However, switching from electric to conventional fossil fuel modes also results in higher costs and emissions and lower efficiency. This paper analyzes this heterogeneous vehicle routing problem and describes two solution methods: an exact branch-price-and-cut algorithm and an iterated tabu search metaheuristic. From a methodological perspective, we find that the exact algorithm consistently obtains tight lower bounds that also serve to certify the metaheuristic solutions as near-optimal. From a policy standpoint, we examine a large-scale real-world case study concerning parcel deliveries in the Chicago metropolitan area and quantify various operational metrics including energy costs and vehicle miles traveled. We find that by deploying roughly 20% of range-extended vehicles with a modest all-electric range of 33 miles, parcel distributors can save energy costs by up to 17% while incurring less than 0.5% increase in vehicle miles traveled. Increasing the range to 60 miles further reduces costs by only 4%, which can alternatively be achieved by decreasing the average service time by 1 minute or increasing driver working time by 1 hour. Our study reveals several key areas of improvement on which vehicle manufacturers, distributors, and policy makers can focus their attention.

math.OC↗

arXiv2021

LEAP: Scaling Numerical Optimization Based Synthesis Using an Incremental Approach

While showing great promise, circuit synthesis techniques that combine numerical optimization with search over circuit structures face scalability challenges due to a large number of parameters, exponential search spaces, and complex objective functions. The LEAP algorithm improves scaling across these dimensions using iterative circuit synthesis, incremental re-optimization, dimensionality reduction, and improved numerical optimization. LEAP draws on the design of the optimal synthesis algorithm QSearch by extending it with an incremental approach to determine constant prefix solutions for a circuit. By narrowing the search space, LEAP improves scalability from four to six qubit circuits. LEAP was evaluated with known quantum circuits such as QFT and physical simulation circuits like the VQE, TFIM, and QITE. LEAP can compile four qubit unitaries up to $59\times$ faster than QSearch and five and six qubit unitaries with up to $1.2\times$ fewer CNOTs compared to the QFAST package. LEAP can reduce the CNOT count by up to $36\times$, or $7\times$ on average, compared to the CQC Tket compiler. Despite its heuristics, LEAP has generated optimal circuits for many test cases with a priori known solutions. The techniques introduced by LEAP are applicable to other numerical-optimization-based synthesis approaches.

quant-ph↗

arXiv2021

Sequential Linearization Method for Bound-Constrained Mathematical Programs with Complementarity Constraints

We propose an algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables. Each iteration of the algorithm involves solving a linear program with complementarity constraints in order to obtain an estimate of the active set. The algorithm enforces descent on the objective function to promote global convergence to B-stationary points. We provide a convergence analysis and preliminary numerical results on a range of test problems. We also study the effect of fixing the active constraints in a bound-constrained quadratic program that can be solved on each iteration in order to obtain fast convergence.

math.OC↗

arXiv2021

Derivative-Free Optimization of a Rapid-Cycling Synchrotron

We develop and solve a constrained optimization model to identify an integrable optics rapid-cycling synchrotron lattice design that performs well in several capacities. Our model encodes the design criteria into 78 linear and nonlinear constraints, as well as a single nonsmooth objective, where the objective and some constraints are defined from the output of Synergia, an accelerator simulator. We detail the difficulties of the 23-dimensional simulation-constrained decision space and establish that the space is nonempty. We use a derivative-free manifold sampling algorithm to account for structured nondifferentiability in the objective function. Our numerical results quantify the dependence of solutions on constraint parameters and the effect of the form of objective function.

math.OC↗

arXiv2021

Lookahead Acquisition Functions for Finite-Horizon Time-Dependent Bayesian Optimization and Application to Quantum Optimal Control

We propose a novel Bayesian method to solve the maximization of a time-dependent expensive-to-evaluate stochastic oracle. We are interested in the decision that maximizes the oracle at a finite time horizon, given a limited budget of noisy evaluations of the oracle that can be performed before the horizon. Our recursive two-step lookahead acquisition function for Bayesian optimization makes nonmyopic decisions at every stage by maximizing the expected utility at the specified time horizon. Specifically, we propose a generalized two-step lookahead framework with a customizable \emph{value} function that allows users to define the utility. We illustrate how lookahead versions of classic acquisition functions such as the expected improvement, probability of improvement, and upper confidence bound can be obtained with this framework. We demonstrate the utility of our proposed approach on several carefully constructed synthetic cases and a real-world quantum optimal control problem.

stat.CO↗

arXiv2021

libEnsemble: A Library to Coordinate the Concurrent Evaluation of Dynamic Ensembles of Calculations

Almost all applications stop scaling at some point; those that don't are seldom performant when considering time to solution on anything but aspirational/unicorn resources. Recognizing these tradeoffs as well as greater user functionality in a near-term exascale computing era, we present libEnsemble, a library aimed at particular scalability- and capability-stretching uses. libEnsemble enables running concurrent instances of an application in dynamically allocated ensembles through an extensible Python library. We highlight the structure, execution, and capabilities of the library on leading pre-exascale environments as well as advanced capabilities for exascale environments and beyond.

cs.DC↗

arXiv2021

CutQC: Using Small Quantum Computers for Large Quantum Circuit Evaluations

Quantum computing (QC) is a new paradigm offering the potential of exponential speedups over classical computing for certain computational problems. Each additional qubit doubles the size of the computational state space available to a QC algorithm. This exponential scaling underlies QC's power, but today's Noisy Intermediate-Scale Quantum (NISQ) devices face significant engineering challenges in scalability. The set of quantum circuits that can be reliably run on NISQ devices is limited by their noisy operations and low qubit counts. This paper introduces CutQC, a scalable hybrid computing approach that combines classical computers and quantum computers to enable evaluation of quantum circuits that cannot be run on classical or quantum computers alone. CutQC cuts large quantum circuits into smaller subcircuits, allowing them to be executed on smaller quantum devices. Classical postprocessing can then reconstruct the output of the original circuit. This approach offers significant runtime speedup compared with the only viable current alternative--purely classical simulations--and demonstrates evaluation of quantum circuits that are larger than the limit of QC or classical simulation. Furthermore, in real-system runs, CutQC achieves much higher quantum circuit evaluation fidelity using small prototype quantum computers than the state-of-the-art large NISQ devices achieve. Overall, this hybrid approach allows users to leverage classical and quantum computing resources to evaluate quantum programs far beyond the reach of either one alone.

quant-ph↗

arXiv2020

Recursive Two-Step Lookahead Expected Payoff for Time-Dependent Bayesian Optimization

We propose a novel Bayesian method to solve the maximization of a time-dependent expensive-to-evaluate oracle. We are interested in the decision that maximizes the oracle at a finite time horizon, when relatively few noisy evaluations can be performed before the horizon. Our recursive, two-step lookahead expected payoff ($\texttt{r2LEY}$) acquisition function makes nonmyopic decisions at every stage by maximizing the estimated expected value of the oracle at the horizon. $\texttt{r2LEY}$ circumvents the evaluation of the expensive multistep (more than two steps) lookahead acquisition function by recursively optimizing a two-step lookahead acquisition function at every stage; unbiased estimators of this latter function and its gradient are utilized for efficient optimization. $\texttt{r2LEY}$ is shown to exhibit natural exploration properties far from the time horizon, enabling accurate emulation of the oracle, which is exploited in the final decision made at the horizon. To demonstrate the utility of $\texttt{r2LEY}$, we compare it with time-dependent extensions of popular myopic acquisition functions via both synthetic and real-world datasets.

math.OC↗

arXiv2020

A Repeated Route-then-Schedule Approach to Coordinated Vehicle Platooning: Algorithms, Valid Inequalities and Computation

Platooning of vehicles is a promising approach for reducing fuel consumption, increasing vehicle safety, and using road space more efficiently. We consider the important but difficult problem of assigning optimal routes and departure schedules to a collection of vehicles. We propose an iterative route-then-schedule heuristic for centralized planning that quickly converges to high-quality solutions. We also propose and analyze a collection of valid inequalities for the individual problems of assigning vehicles to routes and scheduling the times that vehicles traverse their routes. These inequalities are shown to reduce the computational time or optimality gap of solving the routing and scheduling problem instances. Our approach uses the valid inequalities in both the routing and scheduling portions of each iteration; numerical experiments highlight the speed of the approach for routing vehicles on a real-world road network.

math.OC↗

arXiv2020

A Method for Convex Black-Box Integer Global Optimization

We study the problem of minimizing a convex function on a nonempty, finite subset of the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to (sub)gradients of the objective but, rather, uses secant linear functions that interpolate the objective function at previously evaluated points. These linear mappings are shown to underestimate the objective in disconnected portions of the domain. Therefore, the union of these conditional cuts provides a nonconvex underestimator of the objective. We propose an algorithm that alternates between updating the underestimator and evaluating the objective function. We prove that the algorithm converges to a global minimum of the objective function on the feasible set. We present two approaches for representing the underestimator and compare their computational effectiveness. We also compare implementations of our algorithm with existing methods for minimizing functions on a subset of the integer lattice. We discuss the difficulty of this problem class and provide insights into why a computational proof of optimality is challenging even for moderate problem sizes.

math.OC↗

arXiv2019

Derivative-free optimization methods

In many optimization problems arising from scientific, engineering and artificial intelligence applications, objective and constraint functions are available only as the output of a black-box or simulation oracle that does not provide derivative information. Such settings necessitate the use of methods for derivative-free, or zeroth-order, optimization. We provide a review and perspectives on developments in these methods, with an emphasis on highlighting recent developments and on unifying treatment of such problems in the non-linear optimization and machine learning literature. We categorize methods based on assumed properties of the black-box functions, as well as features of the methods. We first overview the primary setting of deterministic methods applied to unconstrained, non-convex optimization problems where the objective function is defined by a deterministic black-box oracle. We then discuss developments in randomized methods, methods that assume some additional structure about the objective (including convexity, separability and general non-smooth compositions), methods for problems where the output of the black-box oracle is stochastic, and methods for handling different types of constraints.

math.OC↗

arXiv2019

Multistart Methods for Quantum Approximate Optimization

Hybrid quantum-classical algorithms such as the quantum approximate optimization algorithm (QAOA) are considered one of the most promising approaches for leveraging near-term quantum computers for practical applications. Such algorithms are often implemented in a variational form, combining classical optimization methods with a quantum machine to find parameters to maximize performance. The quality of the QAOA solution depends heavily on quality of the parameters produced by the classical optimizer. Moreover, the presence of multiple local optima in the space of parameters makes it harder for the classical optimizer. In this paper we study the use of a multistart optimization approach within a QAOA framework to improve the performance of quantum machines on important graph clustering problems. We also demonstrate that reusing the optimal parameters from similar problems can improve the performance of classical optimization methods, expanding on similar results for MAXCUT.

quant-ph↗

arXiv2018

Comparing optimization strategies for force field parameterization

Classical molecular dynamics (MD) simulations enable modeling of materials and examination of microscopic details that are not accessible experimentally. The predictive capability of MD relies on the force field (FF) used to describe interatomic interactions. FF parameters are typically determined to reproduce selected material properties computed from density functional theory (DFT) and/or measured experimentally. A common practice in parameterizing FFs is to use least-squares local minimization algorithms. Genetic algorithms (GAs) have also been demonstrated as a viable global optimization approach, even for complex FFs. However, an understanding of the relative effectiveness and efficiency of different optimization techniques for the determination of FF parameters is still lacking. In this work, we evaluate various FF parameter optimization schemes, using as example a training data set calculated from DFT for different polymorphs of Ir$O_2$. The Morse functional form is chosen for the pairwise interactions and the optimization of the parameters against the training data is carried out using (1) multi-start local optimization algorithms: Simplex, Levenberg-Marquardt, and POUNDERS, (2) single-objective GA, and (3) multi-objective GA. Using random search as a baseline, we compare the algorithms in terms of reaching the lowest error, and number of function evaluations. We also compare the effectiveness of different approaches for FF parameterization using a test data set with known ground truth (i.e generated from a specific Morse FF). We find that the performance of optimization approaches differs when using the Test data vs. the DFT data. Overall, this study provides insight for selecting a suitable optimization method for FF parameterization, which in turn can enable more accurate prediction of material properties and chemical phenomena.

physics.comp-ph↗

arXiv2017

Platoon formation maximization through centralized routing and departure time coordination

Platooning allows vehicles to travel with small intervehicle distance in a coordinated fashion thanks to vehicle-to-vehicle connectivity. When applied at a larger scale, platooning will create significant opportunities for energy savings due to reduced aerodynamic drag, as well as increased road capacity and congestion reduction resulting from shorter vehicle headways. However, these potential savings are maximized if platooning-capable vehicles spend most of their travel time within platoons. Ad hoc platoon formation may not ensure a high rate of platoon driving. In this paper we consider the problem of central coordination of platooning-capable vehicles. By coordinating their routes and departure times, we can maximize the fuel savings afforded by platooning vehicles. The resulting problem is a combinatorial optimization problem that considers the platoon coordination and vehicle routing problems simultaneously. We demonstrate our methodology by evaluating the benefits of a coordinated solution and comparing it with the uncoordinated case when platoons form only in an ad hoc manner. We compare the coordinated and uncoordinated scenarios on a grid network with different assumptions about demand and the time vehicles are willing to wait.

eess.SY↗

arXiv2016

Origins and optimization of entanglement in plasmonically coupled quantum dots

A system of two or more quantum dots interacting with a dissipative plasmonic nanostructure is investigated in detail by using a cavity quantum electrodynamics approach with a model Hamiltonian. We focus on determining and understanding system configurations that generate multiple bipartite quantum entanglements between the occupation states of the quantum dots. These configurations include allowing for the quantum dots to be asymmetrically coupled to the plasmonic system. Analytical solution of a simplified limit for an arbitrary number of quantum dots and numerical simulations and optimization for the two- and three-dot cases are used to develop guidelines for maximizing the bipartite entanglements. For any number of quantum dots, we show that through simple starting states and parameter guidelines, one quantum dot can be made to share a strong amount of bipartite entanglement with all other quantum dots in the system, while entangling all other pairs to a lesser degree.

quant-ph↗