Number Partition Problem
Introduction
In the Number Partitioning Problem [1] we need to find how to partition a set of integers into two subsets of equal sums. In case such a partition does not exist, we can ask for a partition where the difference between the sums is minimal.
Mathematical formulation
Given a set of numbers \(S=\{s_1,s_2,...,s_n\}\), a partition is defined as \(P_1,P_2 \subset \{1,...,n\}\), with \(P_1\cup P_2=\{1,...,n\}\) and \(P_1\cap P_2=\emptyset\). In the Number Partitioning Problem we need to determine a partition such that \(|\sum_{j\in P_1}s_j-\sum_{j\in P_2}s_j|\) is minimal. A partition can be represented by a binary vector \(x\) of size \(n\), where we assign 0 or 1 for being in \(P_1\) or \(P_2\), respectively. The quantity we ask to minimize is \(|\vec{x}\cdot \vec{s}-(1-\vec{x})\cdot\vec{s}|=|(2\vec{x}-1)\cdot\vec{s}|\). In practice we will minimize the square of this expression.
Solving with the Classiq platform
We go through the steps of solving the problem with the Classiq platform, using QAOA algorithm [2]. The solution is based on defining a pyomo model for the optimization problem we would like to solve.
from typing import cast
import networkx as nx
import numpy as np
import pyomo.core as pyo
from IPython.display import Markdown, display
from matplotlib import pyplot as plt
Building the Pyomo model from a graph input
We proceed by defining the pyomo model that will be used on the Classiq platform, using the mathematical formulation defined above:
# we define a matrix which gets a set of integers s and returns a pyomo model for the partitioning problem
def partite(s) -> pyo.ConcreteModel:
model = pyo.ConcreteModel()
SetSize = len(s) # the set size
model.x = pyo.Var(
range(SetSize), domain=pyo.Binary
) # our variable is a binary vector
# we define a cost function
model.cost = pyo.Objective(
expr=sum(((2 * model.x[i] - 1) * s[i]) for i in range(SetSize)) ** 2,
sense=pyo.minimize,
)
return model
Myset = np.random.randint(1, 12, 10)
mylist = [int(x) for x in Myset]
print("This is my list: ", mylist)
set_partition_model = partite(mylist)
This is my list: [4, 8, 11, 7, 1, 8, 8, 5, 7, 11]
set_partition_model.pprint()
1 Set Declarations
x_index : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 10 : {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
1 Var Declarations
x : Size=10, Index=x_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : None : 1 : False : True : Binary
1 : 0 : None : 1 : False : True : Binary
2 : 0 : None : 1 : False : True : Binary
3 : 0 : None : 1 : False : True : Binary
4 : 0 : None : 1 : False : True : Binary
5 : 0 : None : 1 : False : True : Binary
6 : 0 : None : 1 : False : True : Binary
7 : 0 : None : 1 : False : True : Binary
8 : 0 : None : 1 : False : True : Binary
9 : 0 : None : 1 : False : True : Binary
1 Objective Declarations
cost : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : minimize : ((2*x[0] - 1)*4 + (2*x[1] - 1)*8 + (2*x[2] - 1)*11 + (2*x[3] - 1)*7 + 2*x[4] - 1 + (2*x[5] - 1)*8 + (2*x[6] - 1)*8 + (2*x[7] - 1)*5 + (2*x[8] - 1)*7 + (2*x[9] - 1)*11)**2
3 Declarations: x_index x cost
Setting Up the Classiq Problem Instance
In order to solve the Pyomo model defined above, we use the Classiq combinatorial optimization engine. For the quantum part of the QAOA algorithm (QAOAConfig) - define the number of repetitions (num_layers):
from classiq import construct_combinatorial_optimization_model
from classiq.applications.combinatorial_optimization import OptimizerConfig, QAOAConfig
qaoa_config = QAOAConfig(num_layers=3)
For the classical optimization part of the QAOA algorithm we define the maximum number of classical iterations (max_iteration) and the \(\alpha\)-parameter (alpha_cvar) for running CVaR-QAOA, an improved variation of the QAOA algorithm [3]:
optimizer_config = OptimizerConfig(max_iteration=60, alpha_cvar=0.7)
Lastly, we load the model, based on the problem and algorithm parameters, which we can use to solve the problem:
qmod = construct_combinatorial_optimization_model(
pyo_model=set_partition_model,
qaoa_config=qaoa_config,
optimizer_config=optimizer_config,
)
We also set the quantum backend we want to execute on:
from classiq import set_execution_preferences
from classiq.execution import ClassiqBackendPreferences, ExecutionPreferences
backend_preferences = ExecutionPreferences(
backend_preferences=ClassiqBackendPreferences(backend_name="simulator")
)
qmod = set_execution_preferences(qmod, backend_preferences)
from classiq import write_qmod
write_qmod(qmod, "set_partition")
Synthesizing the QAOA Circuit and Solving the Problem
We can now synthesize and view the QAOA circuit (ansatz) used to solve the optimization problem:
from classiq import show, synthesize
qprog = synthesize(qmod)
show(qprog)
Opening: https://platform.classiq.io/circuit/0ff868cf-2ccd-47e7-b5f7-4500845af8c3?version=0.41.0.dev39%2B79c8fd0855
We now solve the problem by calling the execute function on the quantum program we have generated:
from classiq import execute
res = execute(qprog).result()
We can check the convergence of the run:
from classiq.execution import VQESolverResult
vqe_result = res[0].value
vqe_result.convergence_graph
Optimization Results
We can also examine the statistics of the algorithm:
import pandas as pd
from classiq.applications.combinatorial_optimization import (
get_optimization_solution_from_pyo,
)
solution = get_optimization_solution_from_pyo(
set_partition_model,
vqe_result=vqe_result,
penalty_energy=qaoa_config.penalty_energy,
)
optimization_result = pd.DataFrame.from_records(solution)
optimization_result.sort_values(by="cost", ascending=True).head(5)
| probability | cost | solution | count | |
|---|---|---|---|---|
| 525 | 0.001 | 0.0 | [0, 1, 1, 1, 1, 0, 1, 0, 0, 0] | 1 |
| 29 | 0.004 | 0.0 | [1, 1, 1, 0, 0, 0, 0, 1, 1, 0] | 4 |
| 511 | 0.001 | 0.0 | [1, 1, 0, 0, 0, 1, 1, 0, 1, 0] | 1 |
| 494 | 0.001 | 0.0 | [0, 1, 0, 0, 0, 1, 1, 0, 0, 1] | 1 |
| 424 | 0.001 | 0.0 | [0, 0, 0, 1, 1, 1, 1, 0, 0, 1] | 1 |
And the histogram:
optimization_result.hist("cost", weights=optimization_result["probability"])
array([[<Axes: title={'center': 'cost'}>]], dtype=object)
Let us plot the solution:
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]
p1 = [mylist[i] for i in range(len(mylist)) if best_solution[i] == 0]
p2 = [mylist[i] for i in range(len(mylist)) if best_solution[i] == 1]
print("P1=", p1, ", total sum: ", sum(p1))
print("P2=", p2, ", total sum: ", sum(p2))
print("difference= ", abs(sum(p1) - sum(p2)))
P1= [7, 1, 8, 8, 11] , total sum: 35
P2= [4, 8, 11, 5, 7] , total sum: 35
difference= 0
Lastly, we can compare to the classical solution of the problem:
from pyomo.opt import SolverFactory
solver = SolverFactory("couenne")
solver.solve(set_partition_model)
set_partition_model.display()
Model unknown
Variables:
x : Size=10, Index=x_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : 1.0 : 1 : False : False : Binary
1 : 0 : 1.0 : 1 : False : False : Binary
2 : 0 : 1.0 : 1 : False : False : Binary
3 : 0 : 0.0 : 1 : False : False : Binary
4 : 0 : 1.0 : 1 : False : False : Binary
5 : 0 : 0.0 : 1 : False : False : Binary
6 : 0 : 0.0 : 1 : False : False : Binary
7 : 0 : 0.0 : 1 : False : False : Binary
8 : 0 : 0.0 : 1 : False : False : Binary
9 : 0 : 1.0 : 1 : False : False : Binary
Objectives:
cost : Size=1, Index=None, Active=True
Key : Active : Value
None : True : 0.0
Constraints:
None
classical_solution = [pyo.value(set_partition_model.x[i]) for i in range(len(mylist))]
p1 = [mylist[i] for i in range(len(mylist)) if classical_solution[i] == 0]
p2 = [mylist[i] for i in range(len(mylist)) if classical_solution[i] == 1]
print("P1=", p1, ", total sum: ", sum(p1))
print("P2=", p2, ", total sum: ", sum(p2))
print("difference= ", abs(sum(p1) - sum(p2)))
P1= [7, 8, 8, 5, 7] , total sum: 35
P2= [4, 8, 11, 1, 11] , total sum: 35
difference= 0
References
[1]: Number Partitioning Problem (Wikipedia)
[2]: Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028 (2014).
[3]: Barkoutsos, Panagiotis Kl, et al. "Improving variational quantum optimization using CVaR." Quantum 4 (2020): 256.