Neuromorphic computing, which aims to replicate the computational structure and architecture of the brain in synthetic hardware, has typically focused on artificial intelligence applications. What is less explored is whether such brain-inspired hardware can provide value beyond cognitive tasks. Here we show that the high degree of parallelism and configurability of spiking neuromorphic architectures makes them well suited to implement random walks via discrete-time Markov chains. These random walks are useful in Monte Carlo methods, which represent a fundamental computational tool for solving a wide range of numerical computing tasks. Using IBM’s TrueNorth and Intel’s Loihi neuromorphic computing platforms, we show that our neuromorphic computing algorithm for generating random walk approximations of diffusion offers advantages in energy-efficient computation compared with conventional approaches. We also show that our neuromorphic computing algorithm can be extended to more sophisticated jump-diffusion processes that are useful in a range of applications, including financial economics, particle physics and machine learning.

Graph algorithms enable myriad large-scale applications including cybersecurity, social network analysis, resource allocation, and routing. The scalability of current graph algorithm implementations on conventional computing architectures are hampered by the demise of Moore’s law. We present a theoretical framework for designing and assessing the performance of graph algorithms executing in networks of spiking artificial neurons. Although spiking neural networks (SNNs) are capable of general-purpose computation, few algorithmic results with rigorous asymptotic performance analysis are known. SNNs are exceptionally well-motivated practically, as neuromorphic computing systems with 100 million spiking neurons are available, and systems with a billion neurons are anticipated in the next few years. Beyond massive parallelism and scalability, neuromorphic computing systems offer energy consumption orders of magnitude lower than conventional high-performance computing systems. We employ our framework to design and analyze new spiking algorithms for shortest path and dynamic programming problems. Our neuromorphic algorithms are message-passing algorithms relying critically on data movement for computation. For fair and rigorous comparison with conventional algorithms and architectures, which is challenging but paramount, we develop new models of data-movement in conventional computing architectures. This allows us to prove polynomial-factor advantages, even when we assume a SNN consisting of a simple grid-like network of neurons. To the best of our knowledge, this is one of the first examples of a rigorous asymptotic computational advantage for neuromorphic computing.

The quantum k-Local Hamiltonian problem is a natural generalization of classical constraint satisfaction problems (k-CSP) and is complete for QMA, a quantum analog of NP. Although the complexity of k-Local Hamiltonian problems has been well studied, only a handful of approximation results are known. For Max 2-Local Hamiltonian where each term is a rank 3 projector, a natural quantum generalization of classical Max 2-SAT, the best known approximation algorithm was the trivial random assignment, yielding a 0.75-approximation. We present the first approximation algorithm beating this bound, a classical polynomial-time 0.764-approximation. For strictly quadratic instances, which are maximally entangled instances, we provide a 0.801 approximation algorithm, and numerically demonstrate that our algorithm is likely a 0.821-approximation. We conjecture these are the hardest instances to approximate. We also give improved approximations for quantum generalizations of other related classical 2-CSPs. Finally, we exploit quantum connections to a generalization of the Grothendieck problem to obtain a classical constant-factor approximation for the physically relevant special case of strictly quadratic traceless 2-Local Hamiltonians on bipartite interaction graphs, where a inverse logarithmic approximation was the best previously known (for general interaction graphs). Our work employs recently developed techniques for analyzing classical approximations of CSPs and is intended to be accessible to both quantum information scientists and classical computer scientists.

We present a theoretical framework for designing and assessing the performance of algorithms executing in networks consisting of spiking artificial neurons. Although spiking neural networks (SNNs) are capable of general-purpose computation, few algorithmic results with rigorous asymptotic performance analysis are known. SNNs are exceptionally well-motivated practically, as neuromorphic computing systems with 100 million spiking neurons are available, and systems with a billion neurons are anticipated in the next few years. Beyond massive parallelism and scalability, neuromorphic computing systems offer energy consumption orders of magnitude lower than conventional high-performance computing systems. We employ our framework to design and analyze neuromorphic graph algorithms, focusing on shortest path problems. Our neuromorphic algorithms are message-passing algorithms relying critically on data movement for computation, and we develop data-movement lower bounds for conventional algorithms. A fair and rigorous comparison with conventional algorithms and architectures is challenging but paramount. We prove a polynomial-factor advantage even when we assume an SNN consisting of a simple grid-like network of neurons. To the best of our knowledge, this is one of the first examples of a provable asymptotic computational advantage for neuromorphic computing.

The Lasserre Hierarchy, [18, 19], is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a higher level corresponding to a more accurate relaxation. High level programs have proven to be invaluable components of approximation algorithms for many NP-hard optimization problems [3, 7, 26]. There is a natural analogous quantum hierarchy [5, 8, 24], which is also parameterized by level and provides a relaxation of many (QMA-hard) quantum problems of interest [5, 6, 9]. In contrast to the classical case, however, there is only one approximation algorithm which makes use of higher levels of the hierarchy [5]. Here we provide the first ever use of the level-2 hierarchy in an approximation algorithm for a particular QMA-complete problem, so-called Quantum Max Cut [2, 9]. We obtain modest improvements on state-of-the-art approximation factors for this problem, as well as demonstrate that the level-2 hierarchy satisfies many physically-motivated constraints that the level-1 does not satisfy. Indeed, this observation is at the heart of our analysis and indicates that higher levels of the quantum Lasserre Hierarchy may be very useful tools in the design of approximation algorithms for QMA-complete problems.

This report summarizes the work performed under the project "Linear Programming in Strongly Polynomial Time." Linear programming (LP) is a classic combinatorial optimization problem heavily used directly and as an enabling subroutine in integer programming (IP). Specifically IP is the same as LP except that some solution variables must take integer values (e.g. to represent yes/no decisions). Together LP and IP have many applications in resource allocation including general logistics, and infrastructure design and vulnerability analysis. The project was motivated by the PI's recent success developing methods to efficiently sample Voronoi vertices (essentially finding nearest neighbors in high-dimensional point sets) in arbitrary dimension. His method seems applicable to exploring the high-dimensional convex feasible space of an LP problem. Although the project did not provably find a strongly-polynomial algorithm, it explored multiple algorithm classes. The new medial simplex algorithms may still lead to solvers with improved provable complexity. We describe medial simplex algorithms and some relevant structural/complexity results. We also designed a novel parallel LP algorithm based on our geometric insights and implemented it in the Spoke-LP code. A major part of the computational step is many independent vector dot products. Our parallel algorithm distributes the problem constraints across processors. Current commercial and high-quality free LP solvers require all problem details to fit onto a single processor or multicore. Our new algorithm might enable the solution of problems too large for any current LP solvers. We describe our new algorithm, give preliminary proof-of-concept experiments, and describe a new generator for arbitrarily large LP instances.

We present a classical approximation algorithm for the MAX-2-Local Hamiltonian problem. This is a maximization version of the QMA-complete 2-Local Hamiltonian problem in quantum computing, with the additional assumption that each local term is positive semidefinite. The MAX-2-Local Hamiltonian problem generalizes NP-hard constraint satisfaction problems, and our results may be viewed as generalizations of approximation approaches for the MAX-2-CSP problem. We work in the product state space and extend the framework of Goemans and Williamson for approximating MAX-2-CSPs. The key difference is that in the product state setting, a solution consists of a set of normalized 3-dimensional vectors rather than boolean numbers, and we leverage approximation results for rank-constrained Grothendieck inequalities. For MAX-2-Local Hamiltonian we achieve an approximation ratio of 0.328. This is the first example of an approximation algorithm beating the random quantum assignment ratio of 0.25 by a constant factor.

The widely parallel, spiking neural networks of neuromorphic processors can enable computationally powerful formulations. While recent interest has focused on primarily machine learning tasks, the space of appropriate applications is wide and continually expanding. Here, we leverage the parallel and event-driven structure to solve a steady state heat equation using a random walk method. The random walk can be executed fully within a spiking neural network using stochastic neuron behavior, and we provide results from both IBM TrueNorth and Intel Loihi implementations. Additionally, we position this algorithm as a potential scalable benchmark for neuromorphic systems.

We study a trajectory analysis problem we call the Trajectory Capture Problem (TCP), in which, for a given input set T of trajectories in the plane, and an integer k-2, we seek to compute a set of k points ("portals") to maximize the total weight of all subtrajectories of T between pairs of portals. This problem naturally arises in trajectory analysis and summarization. We show that the TCP is NP-hard (even in very special cases) and give some first approximation results. Our main focus is on attacking the TCP with practical algorithm-engineering approaches, including integer linear programming (to solve instances to provable optimality) and local search methods. We study the integrality gap arising from such approaches. We analyze our methods on different classes of data, including benchmark instances that we generate. Our goal is to understand the best performing heuristics, based on both solution time and solution quality. We demonstrate that we are able to compute provably optimal solutions for real-world instances. 2012 ACM Subject Classification Theory of computation ! Design and analysis of algorithms.

Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively.

With the advent of large-scale neuromorphic platforms, we seek to better understand the applications of neuromorphic computing to more general-purpose computing domains. Graph analysis problems have grown increasingly relevant in the wake of readily available massive data. We demonstrate that a broad class of combinatorial and graph problems known as dynamic programs enjoy simple and efficient neuromorphic implementations, by developing a general technique to convert dynamic programs to spiking neuromorphic algorithms. Dynamic programs have been studied for over 50 years and have dozens of applications across many fields.