Kidney Exchange QAOA Example
Author: Bill Wisotsky
What is the Problem?
Currently there are more than 100,000 patients on the waitling list in the United States for a kidney transplant from a deceased donor. This is addressed by the a program called the Kidney Exchange Program. This program won the Nobel Prize in Economics for Alvin E. Roth and Lloyd S. Shapley's contributions to the theory of stable matchings and the design of markets on 2012.
In summary, in a donor pair there is a recipient who needs a kidney transplant and a donor who is willing to give their kidney to the recipient. About \(\frac{1}{3}\) of those pairs are not compatible for a direct exchange. This is tackled by considering two incompatible pairs together: donor 1 may be compatible with recpient 2 and donor 2 may be compatible with recpient 1. In this example a two-way swap becomes feasible. This is the core of the kideny exchange program.
This is consdered an NP-Hard combinatorial optimization problem that becomes exponetially more difficult as the size of the pool increases. The longest chain in history involved 35 tranplants in the United States in 2015.
import warnings
from itertools import product
from typing import List, Tuple, cast # noqa
import networkx as nx # noqa
import numpy as np
from classiq import *
warnings.filterwarnings("ignore")
Create the pyomo model for a simple kidney exhange problem
In this very simple example, patients and donors represent sets of patients that receive a kidney from a donor. Compatibility is a dictionary mapping of patient-donor paris to their compatibilty scores. Binary decision variables are defined for each patient-donor pair x[donor,patient]. The objective is to maximize the total compatibility score. \(Maximize \sum_{d,p\in A}^{} \sum_{m\in M}c_{dp}x_{dpm}\) where d=donors, p=patients and c=compatability score. The contraints are added to ensure that each donor donates only once \(\sum_{d,p\in A}^{}x_{dpm} = y_{dm}\) and each patient receives once \(\sum_{d,p\in A}^{}x_{dpm} = y_{pm}\). We are creating a PYOMO model that gets fed into Classiq, as illustrated in Classiq documentation. We also solve initially with a classical solver to get inital results which can be compared to the QAOA results in the end.
from pyomo.environ import *
# Sample data: patient-donor pairs and compatibility scores
donors = ["donor1", "donor2", "donor3"]
patients = ["patient1", "patient2", "patient3"]
N = len(patients)
M = len(donors)
# Parameters
compatibility_scores = {
("donor1", "patient1"): 0.9,
("donor1", "patient2"): 0.7,
("donor1", "patient3"): 0.6,
("donor2", "patient1"): 0.8,
("donor2", "patient2"): 0.75,
("donor2", "patient3"): 0.65,
("donor3", "patient1"): 0.85,
("donor3", "patient2"): 0.8,
("donor3", "patient3"): 0.7,
}
# Create Pyomo model
model = ConcreteModel()
# Variables
model.x = Var(donors, patients, within=Binary)
# Objective
model.obj = Objective(
expr=sum(
compatibility_scores[donor, patient] * model.x[donor, patient]
for donor in donors
for patient in patients
),
sense=maximize,
)
# Constraints
model.donor_constraint = ConstraintList()
for donor in donors:
model.donor_constraint.add(
sum(model.x[donor, patient] for patient in patients) <= 1
)
model.patient_constraint = ConstraintList()
for patient in patients:
model.patient_constraint.add(sum(model.x[donor, patient] for donor in donors) <= 1)
# Install "glpk" and unommente for runing this part
# Solve
# solver = SolverFactory("glpk")
# solver.solve(model)
# Output
print("\033[1m\033[4mOptimal solution:\033[0m")
for donor in donors:
for patient in patients:
if model.x[donor, patient].value == 1:
print(f"{donor} donates kidney to {patient}")
print("\n\033[1m\033[4mModel Details\033[0m")
model.pprint()
[1m[4mOptimal solution:[0m
[1m[4mModel Details[0m
5 Set Declarations
donor_constraint_index : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 3 : {1, 2, 3}
patient_constraint_index : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 3 : {1, 2, 3}
x_index : Size=1, Index=None, Ordered=True
Key : Dimen : Domain : Size : Members
None : 2 : x_index_0*x_index_1 : 9 : {('donor1', 'patient1'), ('donor1', 'patient2'), ('donor1', 'patient3'), ('donor2', 'patient1'), ('donor2', 'patient2'), ('donor2', 'patient3'), ('donor3', 'patient1'), ('donor3', 'patient2'), ('donor3', 'patient3')}
x_index_0 : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 3 : {'donor1', 'donor2', 'donor3'}
x_index_1 : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 3 : {'patient1', 'patient2', 'patient3'}
1 Var Declarations
x : Size=9, Index=x_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
('donor1', 'patient1') : 0 : None : 1 : False : True : Binary
('donor1', 'patient2') : 0 : None : 1 : False : True : Binary
('donor1', 'patient3') : 0 : None : 1 : False : True : Binary
('donor2', 'patient1') : 0 : None : 1 : False : True : Binary
('donor2', 'patient2') : 0 : None : 1 : False : True : Binary
('donor2', 'patient3') : 0 : None : 1 : False : True : Binary
('donor3', 'patient1') : 0 : None : 1 : False : True : Binary
('donor3', 'patient2') : 0 : None : 1 : False : True : Binary
('donor3', 'patient3') : 0 : None : 1 : False : True : Binary
1 Objective Declarations
obj : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : maximize : 0.9*x[donor1,patient1] + 0.7*x[donor1,patient2] + 0.6*x[donor1,patient3] + 0.8*x[donor2,patient1] + 0.75*x[donor2,patient2] + 0.65*x[donor2,patient3] + 0.85*x[donor3,patient1] + 0.8*x[donor3,patient2] + 0.7*x[donor3,patient3]
2 Constraint Declarations
donor_constraint : Size=3, Index=donor_constraint_index, Active=True
Key : Lower : Body : Upper : Active
1 : -Inf : x[donor1,patient1] + x[donor1,patient2] + x[donor1,patient3] : 1.0 : True
2 : -Inf : x[donor2,patient1] + x[donor2,patient2] + x[donor2,patient3] : 1.0 : True
3 : -Inf : x[donor3,patient1] + x[donor3,patient2] + x[donor3,patient3] : 1.0 : True
patient_constraint : Size=3, Index=patient_constraint_index, Active=True
Key : Lower : Body : Upper : Active
1 : -Inf : x[donor1,patient1] + x[donor2,patient1] + x[donor3,patient1] : 1.0 : True
2 : -Inf : x[donor1,patient2] + x[donor2,patient2] + x[donor3,patient2] : 1.0 : True
3 : -Inf : x[donor1,patient3] + x[donor2,patient3] + x[donor3,patient3] : 1.0 : True
9 Declarations: x_index_0 x_index_1 x_index x obj donor_constraint_index donor_constraint patient_constraint_index patient_constraint
Start Generatng the QAOA Process
Create the inital parameters for the quantum circuit. These can me modified as needed.
-
Defining the number of layers (num_layers) of the QAOA Ansatz.
-
Define the penalty_energy for invalid solutions, which influences the convergence rate. Smaller positive values are preferred, but shoudl be tweaked.
from classiq import (
Preferences,
construct_combinatorial_optimization_model,
set_preferences,
)
from classiq.applications.combinatorial_optimization import OptimizerConfig, QAOAConfig
qaoa_config = QAOAConfig(num_layers=5, penalty_energy=2)
Create the classical optimizer part of the QAOA. These parameters can be modified.
-
opt_type is the classical optimizer type. Choices include, COBYLA, SPSA, ADAM, L_BFGS_B, and NELDER_MEAD
-
The max_iterations is the maximum number of optimzer iterations and is set to 100.
-
The alpha_cvar is a parameter that describes the quantile considered in the CVAR expectation value. See https://arxiv.org/abs/1907.04769 for more information.
optimizer_config = OptimizerConfig(
# opt_type='COBYLA',
max_iteration=200,
alpha_cvar=1,
)
Combine everthing together to form the entire QAOA model as a QMOD.
-
PYOMO Model
-
QAOA quantum circuit
-
Clasical optimizer
qmod = construct_combinatorial_optimization_model(
pyo_model=model,
qaoa_config=qaoa_config,
optimizer_config=optimizer_config,
)
# defining cosntraint such as computer and parameters for a quicker and more optimized circuit.
preferences = Preferences(transpilation_option="none", timeout_seconds=300)
qmod = set_preferences(qmod, preferences)
-
Wite out the QMOD and preferences to a JSON file
-
Synthesize the model in Classiq interface
write_qmod(qmod, "kidney_exchange_problem") # optional
-
Sythesize the quantum model
-
Show the quantm model in the Classiq platform
qmod = set_constraints(qmod, Constraints(optimization_parameter="width"))
qprog = synthesize(qmod)
# show(qprog)
Execute the quantum model and store the result.
from classiq import execute
res = execute(qprog).result()
View the convergence graph
Important to remember that this is a maximization problem when looking at the graph
# from classiq.execution import VQESolverResult
vqe_result = res[0].value
vqe_result.convergence_graph
Retrieve and Display the Solutions
-
Print them out
-
Graph using a histogram
-
Show Donor - Recipients in Network Graph
Print out the top 10 solutions with the highest cost or objective
import pandas as pd
from classiq.applications.combinatorial_optimization import (
get_optimization_solution_from_pyo,
)
solution = get_optimization_solution_from_pyo(
model, vqe_result=vqe_result, penalty_energy=qaoa_config.penalty_energy
)
optimization_result = pd.DataFrame.from_records(solution)
print("\n\033[1m\033[4mTop 10 Solutions\033[0m")
optimization_result.sort_values(by="cost", ascending=False).head(10)
[1m[4mTop 10 Solutions[0m
| probability | cost | solution | count | |
|---|---|---|---|---|
| 4 | 0.021484 | 2.35 | [1, 0, 0, 0, 0, 1, 0, 1, 0] | 44 |
| 1 | 0.027344 | 2.35 | [1, 0, 0, 0, 1, 0, 0, 0, 1] | 56 |
| 0 | 0.028320 | 2.20 | [0, 1, 0, 1, 0, 0, 0, 0, 1] | 58 |
| 3 | 0.021973 | 2.20 | [0, 0, 1, 1, 0, 0, 0, 1, 0] | 45 |
| 2 | 0.024902 | 2.20 | [0, 1, 0, 0, 0, 1, 1, 0, 0] | 51 |
| 5 | 0.020996 | 2.20 | [0, 0, 1, 0, 1, 0, 1, 0, 0] | 43 |
| 53 | 0.004395 | 1.70 | [1, 0, 0, 0, 0, 0, 0, 1, 0] | 9 |
| 54 | 0.004395 | 1.65 | [1, 0, 0, 0, 1, 0, 0, 0, 0] | 9 |
| 39 | 0.005859 | 1.60 | [0, 0, 0, 1, 0, 0, 0, 1, 0] | 12 |
| 35 | 0.005859 | 1.60 | [1, 0, 0, 0, 0, 0, 0, 0, 1] | 12 |
Histogram of Cost and Weighted by Probability
import matplotlib.pyplot as plt
optimization_result["cost"].plot(
kind="hist", bins=30, edgecolor="black", weights=optimization_result["probability"]
)
plt.ylabel("Probability", fontsize=12)
plt.xlabel("Cost", fontsize=12)
plt.tick_params(axis="both", labelsize=12)
plt.title("Histogram of Cost Weighted by Probability", fontsize=16)
plt.show()
Create a Network Graph for the Best Solution Found
\(\star\) Very important to remember that this is a mximization problem and the classical solver of the QAOA process returns all possible results. We need to filter out the solution with the highest cost which would represent the the highest compatability score.
# This function plots the solution in a table and a graph
def plotting_sol(x_sol, cost):
x_sol_to_mat = np.reshape(np.array(x_sol), [N, M]) # vector to matrix
print("\033[1m\033[4m** QAOA SOLUTION **\033[0m")
print("\033[4mHighest Compatibility Score\033[0m = ", cost)
# plotting in a table
df = pd.DataFrame(x_sol_to_mat)
df.columns = patients
df.index = donors
print(df)
# plotting in a graph
graph_sol = nx.DiGraph()
graph_sol.add_nodes_from(donors + patients)
for n, m in product(range(N), range(M)):
if x_sol_to_mat[n, m] > 0:
graph_sol.add_edges_from(
[(donors[m], patients[n])],
weight=compatibility_scores[(donors[m], patients[n])],
)
plt.figure(figsize=(10, 6))
left = nx.bipartite.sets(graph_sol, top_nodes=patients)[0]
pos = nx.bipartite_layout(graph_sol, left)
nx.draw_networkx(
graph_sol, pos=pos, nodelist=patients, font_size=22, font_color="None"
)
nx.draw_networkx_nodes(
graph_sol, pos, nodelist=patients, node_color="#119DA4", node_size=500
)
for d in donors:
x, y = pos[d]
plt.text(
x,
y,
s=d,
bbox=dict(facecolor="#F43764", alpha=1),
horizontalalignment="center",
fontsize=12,
)
nx.draw_networkx_edges(graph_sol, pos, width=2)
labels = nx.get_edge_attributes(graph_sol, "weight")
nx.draw_networkx_edge_labels(
graph_sol, pos, edge_labels=labels, font_size=12, label_pos=0.6
)
nx.draw_networkx_labels(
graph_sol,
pos,
labels={co: co for co in patients},
font_size=12,
# font_color="#F4F9E9",
)
plt.title("Network Graph of the Best Solution", fontsize=16)
plt.axis("off")
plt.show()
# best_solution = optimization_result.loc[optimization_result.probability.idxmax()]
# plotting_sol(best_solution.solution, best_solution.probability)
best_solution = optimization_result.loc[optimization_result.cost.idxmax()]
plotting_sol(best_solution.solution, best_solution.cost)
[1m[4m** QAOA SOLUTION **[0m
[4mHighest Compatibility Score[0m = 2.3499999999999996
patient1 patient2 patient3
donor1 1 0 0
donor2 0 1 0
donor3 0 0 1
