Converters for Quadratic Programs

Optimization problems in Qiskit 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 optimization provides several optimization algorithms that can handle Quadratic Unconstrained Binary Optimization (QUBO) problems. These are mapped to Ising Hamiltonians, for which Qiskit optimization uses the qiskit.quantum_info.SparsePauliOp object, 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 optimization offers a variety of converters. In this tutorial we're providing an overview on this functionality. Currently, Qiskit optimization 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 and .

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

\begin{aligned} & \text{maximize} & 3x + 2y + z\ & \text{subject to:} & x+y+z \leq 5.5\ & & x+y+z \geq 2.5\ & & x, y \in {0,1}\ & & z \in {0,1,2,3,4,5,6,7} \ \end{aligned}

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

Call convert method of InequalityToEquality to convert.

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, and .

Let us explain how the conversion works. For example, the lower bound of the left side of the first constraint is which is the case of , , and . Thus, the upper bound of the additional integer variable must be to be able to satisfy even the case of , , and . 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 , so we add minus before to be able to satisfy even the case of , and .

\begin{aligned} & \text{maximize} & 3x + 2y + z\ & \text{subject to:} & x+y+z+ xyz_leg\text{@}int_slack= 5\ & & x+y+z+xyz_geq\text{@}int_slack= 3\ & & x, y \in {0,1}\ & & z \in {0,1,2,3,4,5,6,7} \ & & xyz_leg\text{@}int_slack \in {0,1,2,3,4,5} \ & & xyz_geq\text{@}int_slack \in {0,1,2,3,4,5,6} \ \end{aligned}

Let us see how the interpret method works. The purpose of this method is to convert the solution of the converted problem back to that of the original problem. To use this method, we would first require to solve the problem. For the purpose of this tutorial, we will use docplex to solve. We will first translate the quadratic problem into a docplex.mp model.

The result result_eq of qp_eq has 5 variable values (x=1.0, y=1.0, z=3.0, xyz_leq@int_slack=0.0, xyz_geq@int_slack=2.0) while result result_orig of the original qp has three values (x=1.0, y=1.0, z=3.0). We can call InequalityToEquality.interpret method by passing a list or an array to the method that has values of qp_eq.variables as follows. result_eq.x has the list of values that each variable takes in the solution in correspondence to their position in the variable list qp_eq.variables.

We notice that are the values taken in the solution of the converted problem for the common variables between converted and original problems (variables: , , ). The interpret method shows the same values are the solution of the original problem. This is because the objective function for the converted and original problems is exactly the same. The slack variables are just ensuring equality in the constraints of the converted problem, where the constraints are also exactly same between the original and converted problems, except that the original problem has inequality constraints.

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 and are binary variables, while the variable and the slack variables and are integer variables. We print the problem again for reference.

Call convert method of IntegerToBinary to convert.

After converting, the integer variable is replaced with three binary variables , and with coefficients 1, 2 and 4, respectively as the above. The slack variables and 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 , , and to represent that acceptable values , , and , 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. Let us see how the interpret method works.

result_eq_bin has more binary variables due to the conversion by IntegerToBinary.convert. IntegerToBinary.interpret translates them back to the integer values by aggregating binary variables values associated with the original integer variables of qp_eq.

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. where and are numbers and is a variable, will be added to the objective function in the form of where is a large number as penalty factor. By default . 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.

Call convert method of LinearEqualityToPenalty to convert.

After converting, the equality constraints are added to the objective function as additional terms with the default penalty factor provided by Qiskit optimization. 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.

Like we did for the other converters, let us see how interpret method works for this case.

We can see that the result of the interpret method implies that both the original and converted problems have exactly the same solution. This is expected because the converted problem has exactly the same variables as the original problem, the objective has been modified in such a way that we do not have the constraints anymore in the converted problem.

Finally, let us see how we interpret the result of QUBO back to the solution of the original problem qp. The following code shows that the interpreted values are equivalent to the result of the original problem qp.