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Optimization problems in Qiskit’s optimization module are represented with the QuadraticProgram class, which is a generic and powerful representation for optimization problems. In general, optimization algorithms are defined for a certain formulation of a quadratic program, and we need to convert our problem to the right type.

For instance, Qiskit provides several optimization algorithms that can handle Quadratic Unconstrained Binary Optimization (QUBO) problems. These are mapped to Ising Hamiltonians, for which Qiskit uses the qiskit.opflow module, and then their ground state is approximated. For this optimization, commonly known algorithms such as VQE or QAOA can be used as underlying routine. See the following tutorial about the Minimum Eigen Optimizer for more detail. Note that also other algorithms exist that work differently, such as the GroverOptimizer.

To map a problem to the correct input format, the optimization module of Qiskit offers a variety of converters. In this tutorial we’re providing an overview on this functionality. Currently, Qiskit contains the following converters.

• InequalityToEquality: convert inequality constraints into equality constraints with additional slack variables.

• IntegerToBinary: convert integer variables into binary variables and corresponding coefficients.

• LinearEqualityToPenalty: convert equality constraints into additional terms of the objective function.

• LinearInequalityToPenalty: convert inequality constraints into additional terms of the objective function.

• MaximizeToMinimize: convert to the equivalent minimization problem.

• MinimizeToMaximize: convert to the equivalent maximization problem.

• QuadraticProgramToQubo: a wrapper that includes InequalityToEquality, IntegerToBinary, LinearEqualityToPenalty, LinearInequalityToPenalty, and MaximizeToMinimize for convenience.

## InequalityToEquality¶

InequalityToEqualityConverter converts inequality constraints into equality constraints with additional slack variables to remove inequality constraints from QuadraticProgram. The upper bounds and the lower bounds of slack variables will be calculated from the difference between the left sides and the right sides of constraints. Signs of slack variables depend on symbols in constraints such as $$\leq$$ and $$\geq$$.

The following is an example of a maximization problem with two inequality constraints. Variable $$x$$ and $$y$$ are binary variables and variable $$z$$ is an integer variable.

With QuadraticProgram, an optimization model of the problem is written as follows.

:

from qiskit_optimization import QuadraticProgram

:

qp = QuadraticProgram()
qp.binary_var("x")
qp.binary_var("y")
qp.integer_var(lowerbound=0, upperbound=7, name="z")

qp.maximize(linear={"x": 2, "y": 1, "z": 1})
qp.linear_constraint(linear={"x": 1, "y": 1, "z": 1}, sense="LE", rhs=5.5, name="xyz_leq")
qp.linear_constraint(linear={"x": 1, "y": 1, "z": 1}, sense="GE", rhs=2.5, name="xyz_geq")
print(qp.prettyprint())

Problem name:

Maximize
2*x + y + z

Subject to
Linear constraints (2)
x + y + z <= 5.5  'xyz_leq'
x + y + z >= 2.5  'xyz_geq'

Integer variables (1)
0 <= z <= 7

Binary variables (2)
x y



Call convert method of InequalityToEquality to convert.

:

from qiskit_optimization.converters import InequalityToEquality

:

ineq2eq = InequalityToEquality()
qp_eq = ineq2eq.convert(qp)
print(qp_eq.prettyprint())

Problem name:

Maximize
2*x + y + z

Subject to
Linear constraints (2)
x + xyz_leq@int_slack + y + z == 5  'xyz_leq'
x - xyz_geq@int_slack + y + z == 3  'xyz_geq'

Integer variables (3)
0 <= z <= 7
0 <= xyz_leq@int_slack <= 5
0 <= xyz_geq@int_slack <= 6

Binary variables (2)
x y



After converting, the formulation of the problem looks like the above output. As we can see, the inequality constraints are replaced with equality constraints with additional integer slack variables, $$xyz\_leg\text{@}int\_slack$$ and $$xyz\_geq\text{@}int\_slack$$.

Let us explain how the conversion works. For example, the lower bound of the left side of the first constraint is $$0$$ which is the case of $$x=0$$, $$y=0$$, and $$z=0$$. Thus, the upper bound of the additional integer variable must be $$5$$ to be able to satisfy even the case of $$x=0$$, $$y=0$$, and $$z=0$$. Note that we cut off the part after the decimal point in the converted formulation since the left side of the first constraint in the original formulation can be only integer values. For the second constraint, basically we apply the same approach. However, the symbol in the second constraint is $$\geq$$, so we add minus before $$xyz\_geq\text{@}int\_slack$$ to be able to satisfy even the case of $$x=1, y=1$$, and $$z=7$$.

## IntegerToBinary¶

IntegerToBinary converts integer variables into binary variables and coefficients to remove integer variables from QuadraticProgram. For converting, bounded-coefficient encoding proposed in arxiv:1706.01945 (Eq. (5)) is used. For more detail of the encoding method, please see the paper.

We use the output of InequalityToEquality as a starting point. Variables $$x$$ and $$y$$ are binary variables, while the variable $$z$$ and the slack variables $$xyz\_leq\text{@}int\_slack$$ and $$xyz\_geq\text{@}int\_slack$$ are integer variables. We print the problem again for reference.

:

print(qp_eq.prettyprint())

Problem name:

Maximize
2*x + y + z

Subject to
Linear constraints (2)
x + xyz_leq@int_slack + y + z == 5  'xyz_leq'
x - xyz_geq@int_slack + y + z == 3  'xyz_geq'

Integer variables (3)
0 <= z <= 7
0 <= xyz_leq@int_slack <= 5
0 <= xyz_geq@int_slack <= 6

Binary variables (2)
x y



Call convert method of IntegerToBinary to convert.

:

from qiskit_optimization.converters import IntegerToBinary

:

int2bin = IntegerToBinary()
qp_eq_bin = int2bin.convert(qp_eq)
print(qp_eq_bin.prettyprint())

Problem name:

Maximize
2*x + y + z@0 + 2*z@1 + 4*z@2

Subject to
Linear constraints (2)
x + xyz_leq@int_slack@0 + 2*xyz_leq@int_slack@1 + 2*xyz_leq@int_slack@2 + y
+ z@0 + 2*z@1 + 4*z@2 == 5  'xyz_leq'
x - xyz_geq@int_slack@0 - 2*xyz_geq@int_slack@1 - 3*xyz_geq@int_slack@2 + y
+ z@0 + 2*z@1 + 4*z@2 == 3  'xyz_geq'

Binary variables (11)
x y z@0 z@1 z@2 xyz_leq@int_slack@0 xyz_leq@int_slack@1 xyz_leq@int_slack@2
xyz_geq@int_slack@0 xyz_geq@int_slack@1 xyz_geq@int_slack@2



After converting, the integer variable $$z$$ is replaced with three binary variables $$z\text{@}0$$, $$z\text{@}1$$ and $$z\text{@}2$$ with coefficients 1, 2 and 4, respectively as the above. The slack variables $$xyz\_leq\text{@}int\_slack$$ and $$xyz\_geq\text{@}int\_slack$$ that were introduced by InequalityToEquality are also both replaced with three binary variables with coefficients 1, 2, 2, and 1, 2, 3, respectively.

Note: Essentially the coefficients mean that the sum of these binary variables with coefficients can be the sum of a subset of $$\{1, 2, 4\}$$, $$\{1, 2, 2\}$$, and $$\{1, 2, 3\}$$ to represent that acceptable values $$\{0, \ldots, 7\}$$, $$\{0, \ldots, 5\}$$, and $$\{0, \ldots, 6\}$$, which respects the lower bound and the upper bound of original integer variables correctly.

IntegerToBinary also provides interpret method that is the functionality to translate a given binary result back to the original integer representation.

## LinearEqualityToPenalty¶

LinearEqualityToPenalty converts linear equality constraints into additional quadratic penalty terms of the objective function to map QuadraticProgram to an unconstrained form. An input to the converter has to be a QuadraticProgram with only linear equality constraints. Those equality constraints, e.g. $$\sum_i a_i x_i = b$$ where $$a_i$$ and $$b$$ are numbers and $$x_i$$ is a variable, will be added to the objective function in the form of $$M(b - \sum_i a_i x_i)^2$$ where $$M$$ is a large number as penalty factor. By default $$M= 1e5$$. The sign of the term depends on whether the problem type is a maximization or minimization.

We use the output of IntegerToBinary as a starting point, where all variables are binary variables and all inequality constraints have been mapped to equality constraints. We print the problem again for reference.

:

print(qp_eq_bin.prettyprint())

Problem name:

Maximize
2*x + y + z@0 + 2*z@1 + 4*z@2

Subject to
Linear constraints (2)
x + xyz_leq@int_slack@0 + 2*xyz_leq@int_slack@1 + 2*xyz_leq@int_slack@2 + y
+ z@0 + 2*z@1 + 4*z@2 == 5  'xyz_leq'
x - xyz_geq@int_slack@0 - 2*xyz_geq@int_slack@1 - 3*xyz_geq@int_slack@2 + y
+ z@0 + 2*z@1 + 4*z@2 == 3  'xyz_geq'

Binary variables (11)
x y z@0 z@1 z@2 xyz_leq@int_slack@0 xyz_leq@int_slack@1 xyz_leq@int_slack@2
xyz_geq@int_slack@0 xyz_geq@int_slack@1 xyz_geq@int_slack@2



Call convert method of LinearEqualityToPenalty to convert.

:

from qiskit_optimization.converters import LinearEqualityToPenalty

:

lineq2penalty = LinearEqualityToPenalty()
qubo = lineq2penalty.convert(qp_eq_bin)
print(qubo.prettyprint())

Problem name:

Maximize
-22*x^2 + 22*x*xyz_geq@int_slack@0 + 44*x*xyz_geq@int_slack@1
+ 66*x*xyz_geq@int_slack@2 - 22*x*xyz_leq@int_slack@0
- 44*x*xyz_leq@int_slack@1 - 44*x*xyz_leq@int_slack@2 - 44*x*y - 44*x*z@0
- 88*x*z@1 - 176*x*z@2 - 11*xyz_geq@int_slack@0^2
- 44*xyz_geq@int_slack@0*xyz_geq@int_slack@1
- 66*xyz_geq@int_slack@0*xyz_geq@int_slack@2 - 44*xyz_geq@int_slack@1^2
- 132*xyz_geq@int_slack@1*xyz_geq@int_slack@2 - 99*xyz_geq@int_slack@2^2
- 11*xyz_leq@int_slack@0^2 - 44*xyz_leq@int_slack@0*xyz_leq@int_slack@1
- 44*xyz_leq@int_slack@0*xyz_leq@int_slack@2 - 44*xyz_leq@int_slack@1^2
- 88*xyz_leq@int_slack@1*xyz_leq@int_slack@2 - 44*xyz_leq@int_slack@2^2
+ 22*y*xyz_geq@int_slack@0 + 44*y*xyz_geq@int_slack@1
+ 66*y*xyz_geq@int_slack@2 - 22*y*xyz_leq@int_slack@0
- 44*y*xyz_leq@int_slack@1 - 44*y*xyz_leq@int_slack@2 - 22*y^2 - 44*y*z@0
- 88*y*z@1 - 176*y*z@2 + 22*z@0*xyz_geq@int_slack@0
+ 44*z@0*xyz_geq@int_slack@1 + 66*z@0*xyz_geq@int_slack@2
- 22*z@0*xyz_leq@int_slack@0 - 44*z@0*xyz_leq@int_slack@1
- 44*z@0*xyz_leq@int_slack@2 - 22*z@0^2 - 88*z@0*z@1 - 176*z@0*z@2
+ 44*z@1*xyz_geq@int_slack@0 + 88*z@1*xyz_geq@int_slack@1
+ 132*z@1*xyz_geq@int_slack@2 - 44*z@1*xyz_leq@int_slack@0
- 88*z@1*xyz_leq@int_slack@1 - 88*z@1*xyz_leq@int_slack@2 - 88*z@1^2
- 352*z@1*z@2 + 88*z@2*xyz_geq@int_slack@0 + 176*z@2*xyz_geq@int_slack@1
+ 264*z@2*xyz_geq@int_slack@2 - 88*z@2*xyz_leq@int_slack@0
- 176*z@2*xyz_leq@int_slack@1 - 176*z@2*xyz_leq@int_slack@2 - 352*z@2^2
+ 178*x - 66*xyz_geq@int_slack@0 - 132*xyz_geq@int_slack@1
- 198*xyz_geq@int_slack@2 + 110*xyz_leq@int_slack@0 + 220*xyz_leq@int_slack@1
+ 220*xyz_leq@int_slack@2 + 177*y + 177*z@0 + 354*z@1 + 708*z@2 - 374

Subject to
No constraints

Binary variables (11)
x y z@0 z@1 z@2 xyz_leq@int_slack@0 xyz_leq@int_slack@1 xyz_leq@int_slack@2
xyz_geq@int_slack@0 xyz_geq@int_slack@1 xyz_geq@int_slack@2



After converting, the equality constraints are added to the objective function as additional terms with the default penalty factor $$M=1e5$$. The resulting problem is now a QUBO and compatible with many quantum optimization algorithms such as VQE, QAOA and so on.

This gives the same result as before.

:

import qiskit.tools.jupyter

%qiskit_version_table


### Version Information

SoftwareVersion
qiskitNone
qiskit-terra0.25.1
qiskit_optimization0.5.0
System information
Python version3.8.17
Python compilerGCC 11.3.0
Python builddefault, Jun 7 2023 12:29:56
OSLinux
CPUs2
Memory (Gb)6.769481658935547
Fri Sep 01 13:49:12 2023 UTC

### This code is a part of Qiskit

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