Solving Quadratic Unconstrained Binary Optimization with divide-and-conquer and quantum algorithms

Abstract

Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several approximate methods have been devised to reduce such cost. With the growing maturity of quantum computing, quantum algorithms have been proposed to speed up the solution by using either quantum annealers or universal quantum computers. Here we apply the divide-and-conquer approach to reduce the original problem to a collection of smaller problems whose solutions can be assembled to form a single Polynomial Binary Unconstrained Optimization instance with fewer variables. This technique can be applied to any QUBO instance and leads to either an all-classical or a hybrid quantum-classical approach. When quantum heuristics like the Quantum Approximate Optimization Algorithm (QAOA) are used, our proposal leads to a double advantage: a substantial reduction of quantum resources, specifically an average of ∼ 42% fewer qubits to solve MaxCut on random 3-regular graphs, together with an improvement in the quality of the approximate solutions reached.

Unsupervised strategies for identifying optimal parameters in Quantum Approximate Optimization Algorithm

“…From an application perspective, we envision these techniques to be employed in algorithms were QAOA is run on a small part of the problem to solve such as divide-and-conquer [9,10] and iterative algorithms [11,12,13]. Restricting to a few number of circuit calls will help decrease the runtime of quantum-featured or quantum-enhanced algorithms, making them closer to compete with classical heuristics.…”

“…Hence, our approaches allow balancing between circuit calls of small quantum computers and performances. Such settings for instance naturally occur in divide-and-conquer-type schemes to enable smaller quantum computers to improve optimization [9,10,11,12], or in Recursive-QAOA [13] as we demonstrate later.…”

“…It exhibits a few properties that makes it interesting for combinatorial optimization [5,6,7,8]. These can be useful when using it as a subroutine [9,10,11,12,13], especially the concentration of parameters [6]. This concentration property suggests that optimal parameters found for one instance can be reused on another.…”

“…On the one hand, the «reasonable» aspect may not be straightforward to characterize [18]. On the other hand, in many QAOAfeatured algorithms [9,10,11,12,13], many instances are generated and could give rise to many distributions. Finally, within a distribution, several areas of concentration may rise.…”

Globally optimizing QAOA circuit depth for constrained optimization problems

“…QUBOs are implementable on current hardware [10,11]. Recent work has examined how QAOA can be used on CO problems that can be written as polynomial unconstrained binary optimization problems [12,13]. When solving a CO problem using QAOA, each monomial in k vertices in the problem formulation corresponds to a k qubit gate.…”