Ojas Parekh

Ojas Parekh, Kevin Thompson

We resolve the approximability of the maximum energy of the Quantum Max Cut (QMC) problem using product states. A classical 0.498-approximation, using a basic semidefinite programming relaxation, is known for QMC, paralleling the celebrated 0.878-approximation for classical Max Cut. For Max Cut, improving the 0.878-approximation is Unique-Games-hard (UG-hard), and one might expect that improving the 0.498-approximation is UG-hard for QMC. In contrast, we give a classical 1/2-approximation for QMC that is unconditionally optimal, since simple examples exhibit a gap of 1/2 between the energies of an optimal product state and general quantum state. Our result relies on a new nonlinear monogamy of entanglement inequality on a triangle that is derived from the second level of the quantum Lasserre hierarchy. This inequality also applies to the quantum Heisenberg model, and our results generalize to instances of Max 2-Local Hamiltonian where each term is positive and has no 1-local parts. Finally, we give further evidence that product states are essential for approximations of 2-Local Hamiltonian.

Anuj Apte, Ojas Parekh, James Sud

We explain how the maximum energy of the Quantum MaxCut, XY, and EPR Hamiltonians on a graph $G$ are related to the spectral radii of the token graphs of $G$. From numerical study, we conjecture new bounds for these spectral radii based on properties of $G$. We show how these conjectures tighten the analysis of existing algorithms, implying state-of-the-art approximation ratios for all three Hamiltonians. Our conjectures also provide simple combinatorial bounds on the ground state energy of the antiferromagnetic Heisenberg model, which we prove for bipartite graphs.

Ojas Parekh, Cynthia A. Phillips, Conrad D. James et al.

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 $O(N^ω)$ arithmetic operations for a positive constant $ω< 3$. Our approach converts such a fast matrix multiplication algorithm into a constant-depth threshold circuit with approximately $O(N^ω)$ 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.

J. Darby Smith, William Severa, Aaron J. Hill et al.

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.

William Severa, Rich Lehoucq, Ojas Parekh et al.

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.