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Wikipedia / Quadratic unconstrained binary optimization

Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem with a wide range of applications from finance and economics to machine learning. QUBO is an NP hard problem, and for many classical problems from theoretical computer science, like maximum cut, graph coloring and the partition problem, embeddings into QUBO have been formulated. Embeddings for machine learning models include support-vector machines, clustering and probabilistic graphical models. Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing.

Definition

the set of binary digits (or bits. Given a symmetric or upper triangular matrix through

Alternatively, the linear and quadratic parts can be separated as

using the diag.

.

.

Sometimes, QUBO is defined as the problem of maximizing.

Properties

unchanged:

  1. .

In its general form, QUBO is NP-hard and cannot be solved efficiently by any known polynomial-time algorithm. has certain properties, for example:

  • .

  • is diagonal otherwise.

  • are non-positive, the corresponding QUBO problem is solvable in polynomial time.

QUBO can be solved using integer linear programming solvers like CPLEX or Gurobi Optimizer can also be relaxed to continuous variables within the bounds zero and one.

Applications

QUBO is a structurally simple, yet computationally hard optimization problem. It can be used to encode a wide range of optimization problems from various scientific areas.

Maximum cut

, the maximum cut is maximized.

The more general weighted max-cut, and asks for a partition, i.e.,

this becomes equivalent to the original max-cut problem above, which is why we focus on this more general form in the following.

with the interpretation and into two subsets.

are in different subsets, equivalent to logical XOR. By extending above expression to matrix-vector form we find that

. As this is a quadratic function, it is a QUBO problem whose parameter matrix we can read from above expression as

after flipping the sign to make it a minimization problem.

Cluster analysis

Next, we consider the problem of cluster analysis-dimensional space and want to assign each point to one of two classes or clusters, where each row contains two cartesian coordinates that corresponds to a cluster assignment of all points (see figure).

denote the Euclidean distance, i.e.,

  1. ,

.

, and subtract it when they are in different clusters. This way, an optimal solution tends to place points which are far apart into different clusters, and points that are close into the same cluster.

, such a cost function is given by

.

From the second line we can see that this expression can be re-arranged to a QUBO problem by defining

and ignoring the constant term will correspond to an optimal cluster assignment w.r.t. above cost function.

Connection to Ising models

QUBO is very closely related and computationally equivalent to the Ising model, whose Hamiltonian function is defined as

. The spin variables are typically Pauli operators are not neighbors.

yields an equivalent QUBO problem

is given by

can be converted to an equivalent Ising model using the same technique, yielding

.

References