Models: BQM, CQM, QM, Others#

Constrained Versus Unconstrained#

Many real-world problems include constraints. For example, a routing problem might limit the number of airplanes on the ground at an airport and a scheduling problem might require a minimum interval between shifts.

Constrained models such as ConstrainedQuadraticModel can support constraints by encoding both an objective and its set of constraints, as models or in symbolic form.

Unconstrained quadratic models are used to submit problems to samplers such as D-Wave quantum computers[1] and some hybrid quantum-classical samplers[2].

When using such samplers to handle problems with constraints, you typically formulate the constraints as penalties: see Getting Started with D-Wave Solvers. (Constrained models, such as the ConstrainedQuadraticModel, can support constraints natively.)

Supported Models#

  • Higher-Order Models

    dimod provides some Higher-Order Composites and functionality such as reducing higher-order polynomials to BQMs.

Model Construction#

dimod provides a variety of model generators. These are especially useful for testing code and learning.

See examples of using QPU solvers and Leap hybrid solvers on these models in Ocean documentation’s Getting Started examples and the dwave-examples GitHub repository.

Typically you construct a model when reformulating your problem, using such techniques as those presented in D-Wave’s system documentation’s Problem-Solving Handbook.

CQM Example: Using a dimod Generator#

This example creates a CQM representing a knapsack problem of ten items.

>>> cqm = dimod.generators.random_knapsack(10)

CQM Example: Symbolic Formulation#

This example constructs a CQM from symbolic math, which is especially useful for learning and testing with small CQMs.

>>> x = dimod.Binary('x')
>>> y = dimod.Integer('y')
>>> cqm = dimod.CQM()
>>> objective = cqm.set_objective(x+y)
>>> cqm.add_constraint(y <= 3) 

For very large models, you might read the data from a file or construct from a NumPy array.

BQM Example: Using a dimod Generator#

This example generates a BQM from a fully-connected graph (a clique) where all linear biases are zero and quadratic values are uniformly selected -1 or +1 values.

>>> bqm = dimod.generators.random.ran_r(1, 7)

BQM Example: Python Formulation#

For learning and testing with small models, construction in Python is convenient.

The maximum cut problem is to find a subset of a graph’s vertices such that the number of edges between it and the complementary subset is as large as possible.

Four-node star graph

Star graph with four nodes.#

The dwave-examples Maximum Cut example demonstrates how such problems can be formulated as QUBOs:

\[\begin{split}Q = \begin{bmatrix} -3 & 2 & 2 & 2\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{bmatrix}\end{split}\]
>>> qubo = {(0, 0): -3, (1, 1): -1, (0, 1): 2, (2, 2): -1,
...         (0, 2): 2, (3, 3): -1, (0, 3): 2}
>>> bqm = dimod.BQM.from_qubo(qubo)

BQM Example: Construction from NumPy Arrays#

For performance, especially with very large BQMs, you might read the data from a file using methods, such as from_file() or from NumPy arrays.

This example creates a BQM representing a long ferromagnetic loop with two opposite non-zero biases.

>>> import numpy as np
>>> linear = np.zeros(1000)
>>> quadratic = (np.arange(0, 1000), np.arange(1, 1001), -np.ones(1000))
>>> bqm = dimod.BinaryQuadraticModel.from_numpy_vectors(linear, quadratic, 0, "SPIN")
>>> bqm.add_quadratic(0, 10, -1)
>>> bqm.set_linear(0, -1)
>>> bqm.set_linear(500, 1)
>>> bqm.num_variables

QM Example: Interaction Between Integer Variables#

This example constructs a QM with an interaction between two integer variables.

>>> qm = dimod.QuadraticModel()
>>> qm.add_variables_from('INTEGER', ['i', 'j'])
>>> qm.add_quadratic('i', 'j', 1.5)

Data Structure#

Quadratic models are implemented with an adjacency structure in which each variable tracks its own linear bias and its neighborhood. The figure below shows the graph and adjacency representations for an example BQM,

\[E(x) = .5 x_0 - 3 x_1 - x_0 x_1 + x_0 x_2 + 2 x_0 x_3 + x_2 x_3\]
Adjacency Structure

Adjacency structure of a 4-variable binary quadratic model.#