D-Wave’s samplers mostly1 solve quadratic models of various sorts. Quadratic models are characterized by having one or two variables per term. A simple example of a quadratic model is,

\[D = Ax + By + Cxy\]

where \(A\), \(B\), and \(C\) are constants. Single variable terms—\(Ax\) and \(By\) here—are linear with the constant biasing the term’s variable. Two-variable terms—\(Cxy\) here—are quadratic with a relationship between the variables.

The variables in these models may be of the following types:

  • Binary: \(v_i \in\{-1,+1\} \text{ or } \{0,1\}\), represented by the dimod Vartype classes BINARY and SPIN.

    Typically used for applications that optimize over decisions that could either be true (or yes) or false (no); for example,

    • Should the antenna transmit or no?

    • Did a network node experience failure?

  • Discrete: for example a variable that can be assigned one of the values of the set {red, green, blue, yellow}, represented by the dimod Vartype class INTEGER.

    Typically used for applications that optimize over several distinct options; for example,

    • Which shift should employee X work?

    • Should the state be colored red, blue, green or yellow?

  • Integer: represented by the dimod Vartype class INTEGER.

    Typically used for applications that optimize the number of something; for example,

    • How many widgets should be loaded onto the truck?

  • Real: represented by the dimod Vartype class REAL.

    Typically used for applications that optimize over an uncountable set; for example,

    • Where should the sensor be built?


Ocean also provides some higher-order tools for developing and testing your code; for example, the ExactPolySolver class.

Supported Models and Hybrid Samplers

D-Wave’s quantum computers solve binary quadratic models; Leap hybrid solvers can solver models with more varied variable types.

Variable Types and Supported Models, Hybrid Samplers



Hybrid Samplers





Structural Imbalance in a Social Network

Binary, discrete



Map Coloring: Hybrid DQM Sampler

Binary, integer

QuadraticModel, ConstrainedQuadraticModel


Bin Packing, Stock-Sales Strategy in a Simplified Market

Binary, integer, real



Diet Planning

Variable Representations and Labels

Ocean enables you to represent a variable with a quadratic model, as described in dimod’s symbolic math documentation. This makes it important to distinguish between such a variable’s representation and its label.

For example, in the code below, variables a, i, j are represented by QuadraticModel objects and the ten variables in array x by BinaryQuadraticModel objects:

>>> a = dimod.Real("a")
>>> i, j = dimod.Integers(["i", "j"])
>>> x = dimod.BinaryArray([f"x{i}" for i in range(10)])

Each such variable is represented by a quadratic model that has a single linear bias of 1,

>>> x[0]
BinaryQuadraticModel({'x0': 1.0}, {}, 0.0, 'BINARY')

with its single variable having a specified label; e.g., x0 for the first model in x.

The code below adds two variables to a ConstrainedQuadraticModel. The first, using the add_variable() method, adds a variable by specifying a label, "b", and the type of required variable, "REAL". The second, using the add_constraint_from_model() method, specifies the variable i instantiated above as a QuadraticModel object.

>>> cqm = dimod.ConstrainedQuadraticModel()
>>> cqm.add_variable("b", "REAL")
>>> cqm.add_constraint_from_model(i, ">=", 2, "Min i")
'Min i'
>>> cqm.variables
Variables(['b', 'i'])