Defining Models

BQM, CQM, and DQM — choosing and building the right model

Defining Models

Dynex SDK supports three model types, each suited to a different class of problems. All models wrap a dimod object and pass through to DynexSampler.

Model Selection Guide

Problem type	Model	Example
Binary variables, no constraints	BQM	QUBO, Ising, MaxCut, graph problems
Binary/integer variables with constraints	CQM	Portfolio optimization, scheduling with budget
Multi-valued discrete variables	DQM	Multi-class assignment, routing

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BQM — Binary Quadratic Model

The most common model type. Represents an objective function over binary variables (0/1 or ±1).

import dynex
import dimod

# Define BQM with dimod
bqm = dimod.BinaryQuadraticModel(
    {0: 1.0, 1: -1.0},    # Linear terms: h_i * x_i
    {(0, 1): 0.5},         # Quadratic terms: J_ij * x_i * x_j
    0.0,                   # Constant offset
    'BINARY'               # Variable type: 'BINARY' (0/1) or 'SPIN' (-1/+1)
)

model = dynex.BQM(bqm)

Using PyQUBO

from pyqubo import Binary

x0, x1, x2 = Binary('x0'), Binary('x1'), Binary('x2')
H = x0 + x1 - 2*x0*x1 + x2
qubo, offset = H.compile().to_qubo()

bqm = dimod.BinaryQuadraticModel.from_qubo(qubo)
model = dynex.BQM(bqm)

Working with named variables

bqm = dimod.BinaryQuadraticModel(
    {'a': -1.0, 'b': -1.0, 'c': 1.0},
    {('a', 'b'): 2.0, ('b', 'c'): -0.5},
    0.0,
    'BINARY'
)
model = dynex.BQM(bqm)

---

CQM — Constrained Quadratic Model

For problems where constraints are first-class: equality and inequality constraints are encoded directly without penalty terms.

import dynex
import dimod

cqm = dimod.ConstrainedQuadraticModel()

# Define binary variables
x = dimod.Binary('x')
y = dimod.Binary('y')
z = dimod.Binary('z')

# Objective: maximize x + 2y + 3z
cqm.set_objective(-x - 2*y - 3*z)

# Constraint: x + y + z <= 2 (budget constraint)
cqm.add_constraint(x + y + z <= 2, label='budget')

# Constraint: x + z >= 1 (minimum requirement)
cqm.add_constraint(x + z >= 1, label='min_req')

model = dynex.CQM(cqm)

CQM is ideal when constraints cannot be easily penalized. The SDK handles constraint-to-QUBO conversion internally.

---

DQM — Discrete Quadratic Model

For problems with multi-valued variables — where each variable can take one of several discrete values.

import dynex
import dimod

dqm = dimod.DiscreteQuadraticModel()

# Add variable 'color' with 3 possible values (0, 1, 2)
dqm.add_variable(3, label='color')

# Add variable 'size' with 4 possible values (0, 1, 2, 3)
dqm.add_variable(4, label='size')

# Set linear biases: prefer color=1 and size=2
dqm.set_linear('color', [1.0, -1.0, 0.5])
dqm.set_linear('size', [0.0, 0.5, -1.0, 0.5])

# Set quadratic interaction between variables
dqm.set_quadratic('color', 'size', {(0, 0): 1.0, (1, 2): -0.5})

model = dynex.DQM(dqm)

---

Preprocessing

For QPU backends, preprocessing can improve solution quality by scaling and normalizing coefficients:

from dynex import DynexConfig, ComputeBackend, QPUModel

config = DynexConfig(
    compute_backend=ComputeBackend.QPU,
    qpu_model=QPUModel.APOLLO_RC1
)

sampler = dynex.DynexSampler(model, config=config)
sampleset = sampler.sample(
    num_reads=50,       # QPU: 1–100
    annealing_time=200, # QPU: 10–1000
    preprocess=True   # Enable automatic preprocessing
)