Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Memristor-based hardware and algorithms for higher-order Hopfield optimization solver outperforming quadratic Ising machines

Mohammad Hizzani, Arne Heittmann, George Hutchinson, Dmitrii Dobrynin, Thomas Van Vaerenbergh, Tinish Bhattacharya, Adrien Renaudineau, Dmitri Strukov, John Paul Strachan

TL;DR

This work designs and quantify a higher-order Hopfield optimization solver, with 28nm CMOS technology and memristive couplings for lower area and energy computations, and combines algorithmic and circuit analysis to show quantitative advantages over quadratic Ising Machines (IM)s.

Abstract

Ising solvers offer a promising physics-based approach to tackle the challenging class of combinatorial optimization problems. However, typical solvers operate in a quadratic energy space, having only pair-wise coupling elements which already dominate area and energy. We show that such quadratization can cause severe problems: increased dimensionality, a rugged search landscape, and misalignment with the original objective function. Here, we design and quantify a higher-order Hopfield optimization solver, with 28nm CMOS technology and memristive couplings for lower area and energy computations. We combine algorithmic and circuit analysis to show quantitative advantages over quadratic Ising Machines (IM)s, yielding 48x and 72x reduction in time-to-solution (TTS) and energy-to-solution (ETS) respectively for Boolean satisfiability problems of 150 variables, with favorable scaling.

Memristor-based hardware and algorithms for higher-order Hopfield optimization solver outperforming quadratic Ising machines

TL;DR

This work designs and quantify a higher-order Hopfield optimization solver, with 28nm CMOS technology and memristive couplings for lower area and energy computations, and combines algorithmic and circuit analysis to show quantitative advantages over quadratic Ising Machines (IM)s.

Abstract

Ising solvers offer a promising physics-based approach to tackle the challenging class of combinatorial optimization problems. However, typical solvers operate in a quadratic energy space, having only pair-wise coupling elements which already dominate area and energy. We show that such quadratization can cause severe problems: increased dimensionality, a rugged search landscape, and misalignment with the original objective function. Here, we design and quantify a higher-order Hopfield optimization solver, with 28nm CMOS technology and memristive couplings for lower area and energy computations. We combine algorithmic and circuit analysis to show quantitative advantages over quadratic Ising Machines (IM)s, yielding 48x and 72x reduction in time-to-solution (TTS) and energy-to-solution (ETS) respectively for Boolean satisfiability problems of 150 variables, with favorable scaling.
Paper Structure (9 sections, 5 equations, 8 figures, 1 algorithm)

This paper contains 9 sections, 5 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Comparing quadratic and higher-order solvers and energy landscapes
  4. QUBO
  5. PUBO
  6. Landscape Advantages of PUBO over QUBO
  7. 28nm QUBO and PUBO Hardware Designs
  8. Conclusion
  9. Acknowledgment

Figures (8)

  • Figure 1: Left: QUBO HNN that utilizes memristors to store synapse weights in conductance to perform VMM. Right: PUBO HNN with support for cubic interactions. N is number of variables (nvar)
  • Figure 2: a. Search space vs problem size for quadratized and native space for hard 3-SAT random instances with number of clauses $4.23 \times$ number of variables, resulting in $2^N$ and $2^{5.23}$ size of search space for native and quadratized representation respectively. b. Histogram of running 3-SAT QUBO solver, showing energy reduction ($-\Delta E$) does not correlate well with satisfying more clauses $\Delta \text{SAT}$ .
  • Figure 3: The deformation of the native (PUBO) energy landscape $f(\mathbf{x})$ when reformulated as a QUBO problem $g(\mathbf{x}, \mathbf{y})$ with the penalty quadratization of Eq. \ref{['eq:general_quadratization']}. Every vertex denotes a PUBO/QUBO configuration, edges represent bit-flip neighbours. $\mathbf{x}_a$ and $\mathbf{x}_b$ have different optimal $\mathbf{y}_{a}$ and $\mathbf{y}_b$, while $\mathbf{y}_b$ is optimal for both $\mathbf{x}_b$ and $\mathbf{x}_c$. An unstable point $\mathbf{x}_c$ in PUBO can become a saddle in QUBO.
  • Figure 4: Sampled histogram of local minima valley complexity $\Sigma(s)$ vs valley entropy $s$ for the PUBO and Rosenberg QUBO landscapes, averaged over 100 SATLIB instances uf50-[901,1000]. Penalty hyperparameter for QUBO is $P = 0.5$. $T$ is temperature of QUBO connectivity method.
  • Figure 5: (Lower is better) a. TTS in algorithmic steps. b. True TTS (seconds) after circuit layout. c. ETS (Joules) for 0.99 success rate for 80 instances at each size. Curves are median values for each size.
  • ...and 3 more figures