Solving constrained quadratic binary problems via quantum adiabatic evolution

Abstract

Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.

A subgradient approach for constrained binary optimization via quantum adiabatic evolution

“…It is worth mentioning that the performance of the outer approximation method of [18] for the GQSS problem is similar to the penalty methods. In the outer approximation method, a linear programming (LP) problem is initialized with a set of sufficiently large bounds (box constraints) on the Lagrangian multipliers; iteratively, the solution of the LP problem is employed to form (L λ,µ ), and, based on the solution of (L λ,µ ), a linear constraint (cut) is added to the LP problem; this procedure is terminated when no new cuts are generated.…”

“…The quantum adiabatic approach to solving UBQP problems. We refer the reader to [18] for a short introduction on quantum adiabatic computation. For a more extensive study, we refer the reader to [4] and [5] for the proposal of a quantum adiabatic algorithm by Farhi et al, and to [20] for an exposition on its computational aspects.…”

“…In [18], the authors proposed a method for solving CBQP problems using a branch-and-bound framework in which the bounding strategy is to solve the Lagrangian dual of the primal problem by successive application of quantum adiabatic evolution. The method described in [18] shows a fast rate of convergence to solution of the dual problem in every node of the branch-and-bound tree and provides a tight bound that drastically reduces the number of nodes traversed by the algorithm. However, it is important to mention that quantum annealers are coupled to an environment, and this significantly affects their performance.…”

“…In [18], Ronagh et al explored application of outer approximation method for solving constrained problems using quantum adiabatic computation. In the present paper, we explore an alternative approach to solving a constrained binary polynomial programming (CBPP) problem stated formally as (CBPP) max f (x), s.t.…”

“…

An earlier work [18] proposes a method for solving the Lagrangian dual of a constrained binary quadratic programming problem via quantum adiabatic evolution using an outer approximation method. This should be an efficient prescription for solving the Lagrangian dual problem in the presence of an ideally noise-free quantum adiabatic system.

…”