Ising formulations of many NP problems

Abstract

We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering, and satisfiability. In each case, the required number of spins is at most cubic in the size of the problem. This work may be useful in designing adiabatic quantum optimization algorithms.

Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis

“…Many NP-complete problems such as graph and number partitioning, covering and set packing, satisfiability, matching, spanning tree as well as others can be converted into the Ising form as shown in [14]. Ising problems replace x ∈ {0, 1} n by x ∈ {−1, 1} n and can be put in the form of (1) by defining x ' j = (x j + 1)/2 and then redefining x j to be x j '.…”

Performance of two different quantum annealing correction codes

“…[42] to significantly improve the performance of quantum annealing in a minor embedding setting, a technique enabling the embedding of a given graph of interactions G into one of a smaller degree, by using several physical qubits to represent a single logical qubit. This is crucial for applications, where one often starts from a logical problem defined on a high-degree (even complete) graph [43]. In particular, the [4, 1, 4] 0 code can be viewed as a minor embedding on the Chimera graph of two interconnected square graphs, as shown in Figure 2(b).…”

Coherent Ising machines—optical neural networks operating at the quantum limit

“…The Ising coupling J ij and the local field h i take continuous (real) values, the magnitude of which are determined by mapping a given combinatorial optimization problem on the Ising model. 13,21 The three-dimensional Ising model and the two-dimensional Ising model with local fields belong to the NP-hard class in complexity theory. 18 Therefore, you can imagine that many hard problems in the real world can be solved through the Ising model.…”

“…13 Networks of degenerate optical parametric oscillators (DOPOs) are an alternative physical system, with an unconventional operating mechanism, [14][15][16][17] for solving the Ising problem, [18][19][20] and by extension many other combinatorial optimization problems. 21 Formally, the N-spin Ising problem is to find the configuration of spins σ i 2 À1; þ1 f g i ¼ 1; ; N ð Þthat minimizes the energy function H ¼ À P 1 i<j N J ij σ i σ j À P 1 i N h i σ i , where the particular problem instance being solved is specified by the N × N matrix J (with elements J ij ) and the N-dimensional vector h (with elements h i ).…”