We give an efficient 0.8395-approximation algorithm for the EPR Hamiltonian. Our improvement comes from a new nonlinear monogamy-of-entanglement bound on star graphs and a refined parameterization of a shallow quantum circuit from previous works. We also prove limitations showing that current methods cannot achieve substantially better approximation ratios, indicating that further progress will require fundamentally new techniques.

Finding a high (or low) energy state of a given quantum Hamiltonian is a potential area to gain a provable and practical quantum advantage. A line of recent studies focuses on Quantum Max Cut, where one is asked to find a high energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this work, we present a classical approximation algorithm for Quantum Max Cut that achieves an approximation ratio of 0.595, outperforming the previous best algorithms of Lee [Eunou Lee, 2022] (0.562, generic input graph) and King [King, 2023] (0.582, triangle-free input graph). The algorithm is based on finding the maximum weighted matching of an input graph and outputs a product of at most 2-qubit states, which is simpler than the fully entangled output states of the previous best algorithms.

While the search for quantum advantage typically focuses on speed­ups in execution time, quantum algorithms also offer the potential for advantage in space complexity. Previous work has shown such advantages for data stream problems, in which elements arrive and must be processed sequentially without random access, but these have been restricted to specially-constructed problems Le Gall, SPAA ‘06 or polynomial advantage Kallaugher, FOCS ‘21. We show an exponential quantum space advantage for the maximum directed cut problem. This is the first known exponential quantum space advantage for any natural streaming problem. This also constitutes the first unconditional exponential quantum resource advantage for approximating a discrete optimization problem in any setting.

Finding the maximum cut of a graph (MAXCUT) is a classic optimization problem that has motivated parallel algorithm development. While approximate algorithms to MAXCUT offer attractive theoretical guarantees and demonstrate compelling empirical performance, such approximation approaches can shift the dominant computational cost to the stochastic sampling operations. Neuromorphic computing, which uses the organizing principles of the nervous system to inspire new parallel computing architectures, offers a possible solution. One ubiquitous feature of natural brains is stochasticity: the individual elements of biological neural networks possess an intrinsic randomness that serves as a resource enabling their unique computational capacities. By designing circuits and algorithms that make use of randomness similarly to natural brains, we hypothesize that the intrinsic randomness in microelectronics devices could be turned into a valuable component of a neuromorphic architecture enabling more efficient computations. Here, we present neuromorphic circuits that transform the stochastic behavior of a pool of random devices into useful correlations that drive stochastic solutions to MAXCUT. We show that these circuits perform favorably in comparison to software solvers and argue that this neuromorphic hardware implementation provides a path for scaling advantages. This work demonstrates the utility of combining neuromorphic principles with intrinsic randomness as a computational resource for new computational architectures.

The Gaussian noise stability of a function f: ℝn → {-1,1} is the expected value of f (x) · f (y) over ρ-correlated Gaussian random variables x and y. Borell's inequality states that for —1 ≤ ρ ≤ 0, this is minimized by the mean-zero halfspace f (x) = sign(x1). In this work, we conjecture that a natural generalization of this result holds for functions f: ℝn → Sk-1 which output k-dimensional unit vectors. Our main conjecture, which we call the vector-valued Borell's inequality, asserts that the expectation Ex~ρy 〈f(x), f(y)〉 is minimized by the function f (x) = x≤k/||x≤k||, where x≤k = (x1,…, xk). We give several pieces of evidence in favor of this conjecture, including a proof that it does indeed hold in the special case of n = k.

This article highlights synergies between quantum information science (QIS) and operations research for QIS-curious operations researchers (and vice versa), and the author challenges the community to discover more. History: This “Challenge” paper was invited by the Editor in Chief and based on the topics raised by the author at his plenary address at the 2022 INFORMS Computing Society Conference in Tampa, Florida. Funding: This material is based on work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum Computing, Fundamental Algorithmic Research for Quantum Computing. This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC, under Contract DE-NA0003525 with the U.S. Department of Energy (DOE).

We investigate the space complexity of two graph streaming problems: MAX-CUT and its quantum analogue, QUANTUM MAX-CUT. Previous work by Kapralov and Krachun [STOC 19] resolved the classical complexity of the classical problem, showing that any (2 – ε)-approximation requires Ω(n) space (a 2-approximation is trivial with O(log n) space). We generalize both of these qualifiers, demonstrating Ω(n) space lower bounds for (2 – ε)-approximating MAX-CUT and QUANTUM MAX-CUT, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for QUANTUM MAX-CUT only gives a 4-approximation, we show tightness with an algorithm that returns a (2 + ε)-approximation to the QUANTUM MAX-CUT value of a graph in O(log n) space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using o(n) space.We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.

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.

The Lasserre Hierarchy 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. There is a natural analogous quantum hierarchy, which is also parameterized by level and provides a relaxation of many (QMA-hard) quantum problems of interest. In contrast to the classical case, however, there is only one approximation algorithm which makes use of higher levels of the hierarchy. 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. 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.

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 EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King [King, 2023]. We introduce a polynomial time (1+√5)/4≈0.809-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of 0.72 [Jorquera et al., 2024].

Neuromorphic computing offers the potential of an unprecedented level of parallelism at a local scale. Although in their infancy, current first-generation neuromorphic processing units (NPUs) deliver as many as 128K artificial neurons in a package smaller than current laptop CPUs and demanding significantly less energy. Neuromorphic systems consisting of such NPUs and offering a total of 100 million neurons are anticipated in 2020. NPUs were envisioned to accelerate machine learning, and designing neuromorphic algorithms to leverage the benefits of NPUs in other domains remains an open challenge. We design and analyze neuromorphic graph algorithms, focusing on shortest path problems. Our neuromorphic algorithms are packet-passing algorithms relying 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 paramount, and we prove a polynomial-factor advantage even when we assume an NPU with 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.

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.

The random walk is a fundamental stochastic process that underlies many numerical tasks in scientific computing applications. We consider here two neural algorithms that can be used to efficiently implement random walks on spiking neuromorphic hardware. The first method tracks the positions of individual walkers independently by using a modular code inspired by the grid cell spatial representation in the brain. The second method tracks the densities of random walkers at each spatial location directly. We analyze the scaling complexity of each of these methods and illustrate their ability to model random walkers under different probabilistic conditions.

Boolean circuits of McCulloch-Pitts threshold gates are a classic model of neural computation studied heavily in the late 20th century as a model of general computation. Recent advances in large-scale neural computing hardware has made their practical implementation a near-term possibility. We describe a theoretical approach for multiplying two N by N matrices that integrates threshold gate logic with conventional fast matrix multiplication algorithms, that perform $O(N^ømega)$ arithmetic operations for a positive constant $ømega < 3$. Our approach converts such a fast matrix multiplication algorithm into a constant-depth threshold circuit with approximately $O(N^ømega)$ gates. Prior to our work, it was not known whether the Θ(N^3)$-gate barrier for matrix multiplication was surmountable by constant-depth threshold circuits. Dense matrix multiplication is a core operation in convolutional neural network training. Performing this work on a neural architecture instead of off-loading it to a GPU may be an appealing option.

Neural-inspired spike-based computing machines often claim to achieve considerable advantages in terms of energy and time efficiency by using spikes for computation and communication. However, fundamental questions about spike-based computation remain unanswered. For instance, how much advantage do spike-based approaches have over conventional methods, and under what circumstances does spike-based computing provide a comparative advantage? Simply implementing existing algorithms using spikes as the medium of computation and communication is not guaranteed to yield an advantage. Here, we demonstrate that spike-based communication and computation within algorithms can increase throughput, and they can decrease energy cost in some cases. We present several spiking algorithms, including sorting a set of numbers in ascending/descending order, as well as finding the maximum or minimum or median of a set of numbers. We also provide an example application: a spiking median-filtering approach for image processing providing a low-energy, parallel implementation. The algorithms and analyses presented here demonstrate that spiking algorithms can provide performance advantages and offer efficient computation of fundamental operations useful in more complex algorithms.

The dentate gyrus forms a critical link between the entorhinal cortex and CA3 by providing a sparse version of the signal. Concurrent with this increase in sparsity, a widely accepted theory suggests the dentate gyrus performs pattern separation-similar inputs yield decorrelated outputs. Although an active region of study and theory, few logically rigorous arguments detail the dentate gyrus's (DG) coding. We suggest a theoretically tractable, combinatorial model for this action. The model provides formal methods for a highly redundant, arbitrarily sparse, and decorrelated output signal.To explore the value of this model framework, we assess how suitable it is for two notable aspects of DG coding: how it can handle the highly structured grid cell representation in the input entorhinal cortex region and the presence of adult neurogenesis, which has been proposed to produce a heterogeneous code in the DG. We find tailoring the model to grid cell input yields expansion parameters consistent with the literature. In addition, the heterogeneous coding reflects activity gradation observed experimentally. Finally, we connect this approach with more conventional binary threshold neural circuit models via a formal embedding.

In this paper, we study two closely related problems on bipartite graphs, viz., the partial vertex cover problem and the budgeted maximum coverage problem. Both these problems arise in a number of different application domains, including, but not limited to, computer security and transportation logistics. It is well known that the vertex cover problem is solvable in polynomial time on bipartite graphs. However, the computational complexity of the partial vertex cover problem on bipartite graphs was open, thus far. In this paper, we establish that the partial vertex cover problem is \bf NP-hard, even on bipartite graphs. Our result also establishes that the closely related budgeted maximum coverage problem is \bf NP-hard on bipartite graphs. For the latter problem, we present an $\frac{8}{9}$-approximation algorithm. Our approximation guarantee matches and resolves the integrality gap of the natural linear programming relaxation for this problem and improves upon a recent $\frac{4}{5}$-approximation algorithm for the same problem.

For a graph ( V , E ), existing compact linear formulations for the minimum cut problem require Θ(| V || E |) variables and constraints and can be interpreted as a composition of | V | − 1 polyhedra for minimum s - t cuts in much the same way as early approaches to finding globally minimum cuts relied on | V | − 1 calls to a minimum s - t cut algorithm. We present the first formulation to beat this bound, one that uses O (| V | 2 ) variables and O (| V | 3 ) constraints. An immediate consequence of our result is a compact linear relaxation with O (| V | 2 ) constraints and O (| V | 3 ) variables for enforcing global connectivity constraints. This relaxation is as strong as standard cut-based relaxations and has applications in solving traveling salesman problems by integer programming as well as finding approximate solutions for survivable network design problems using Jain's iterative rounding method. Another application is a polynomial-time verifiable certificate of size n for for the NP-complete problem of l 1 -embeddability of a rational metric on an n -set (as opposed to a certificate of size n 2 known previously).

In 1995, Gremban, Miller, and Zagha introduced support-tree preconditioners and a parallel algorithm called support-tree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n × n Laplacian matrix. A Laplacian is a symmetric matrix in which the off-diagonal entries are non-positive, and the row and column sums are zero. A Laplacian A with 2m non-zeros can be interpreted as an undirected positively-weighted graph G with n vertices and m edges, where there is an edge between two nodes i and j with weight c((i, j)) = −Ai,j = −Aj,i if Ai,j = Aj,i < 0. Gremban et al. showed experimentally that STCG performs well on several classes of graphs commonly used in scientific computations. In his thesis, Gremban also proved upper bounds on the number of iterations required for STCG to converge for certain classes of graphs. In this paper, we present an algorithm for finding a preconditioner for an arbitrary graph G = (V, E) with n nodes, m edges, and a weight function c> 0 on the edges, where w.l.o.g., mine∈E c(e) = 1. Equipped with this preconditioner, STCG requires O(log 4 n · � ∆/α) iterations, where α = min U⊂V,|U|≤|V |/2 c(U, V \\U)/|U | is the minimum edge expansion of the graph, and ∆ = maxv∈V c(v) is the maximum incident weight on any vertex. Each iteration requires O(m) work and can be implemented in O(log n) steps in parallel, using only O(m) space. Our results generalize to matrices that are symmetric and diagonally-dominant (SDD). 1