Solving MaxCut with QAOA

This tutorial shows how to solve the maximum cut (MaxCut) combinatorial optimization problem on a graph using the Quantum Approximate Optimization Algorithm (QAOA), first introduced by Farhi et al. in 2014 ¹.

Given an arbitrary graph, the MaxCut problem consists in finding a graph cut which partitions the nodes into two disjoint sets, such that the number of edges in the cut is maximized. This is a very common combinatorial optimization problem known to be computationally hard (NP-hard).

The graph used for this tutorial is an unweighted graph randomly generated using the networkx library with a certain probability \(p\) of having an edge between two arbitrary nodes (known as Erdős–Rényi graph).

import numpy as np
import networkx as nx
import matplotlib.pyplot as plt
import random

# ensure reproducibility
seed = 42
np.random.seed(seed)
random.seed(seed)

# Create random graph
n_nodes = 4
edge_prob = 0.8
graph = nx.gnp_random_graph(n_nodes, edge_prob)

nx.draw(graph)

The goal of the MaxCut algorithm is to maximize the following cost function:

\[\mathcal{C}(p) = \sum_{\alpha}^m \mathcal{C}_{\alpha}(p)\]

where \(p\) is a given cut of the graph, \(\alpha\) is an index over the edges and \(\mathcal{C}_{\alpha}(p)\) is written such that if the nodes connected by the \(\alpha\) edge are in the same set, it returns \(0\), otherwise it returns \(1\). We will represent a cut \(p\) as a bitstring of length \(N\), where \(N\) is the number of nodes, and where the bit in position \(i\) shows to which partition node \(i\) belongs. We assign value 0 to one of the partitions defined by the cut and 1 to the other. Since this choice is arbitrary, every cut is represented by two bitstrings, e.g. "0011" and "1100" are equivalent.

Since in this tutorial we are only dealing with small graphs, we can find the maximum cut by brute force to make sure QAOA works as intended.

# Function to calculate the cost associated with a cut
def calculate_cost(cut: str, graph: nx.graph) -> float:
    """Returns the cost of a given cut (represented by a bitstring)"""
    cost = 0
    for edge in graph.edges():
        (i, j) = edge
        if cut[i] != cut[j]:
            cost += 1
    return cost


# Function to get a binary representation of an int
get_binary = lambda x, n: format(x, "b").zfill(n)

# List of all possible cuts
all_possible_cuts = [bin(k)[2:].rjust(n_nodes, "0") for k in range(2**n_nodes)]

# List with the costs associated to each cut
all_costs = [calculate_cost(cut, graph) for cut in all_possible_cuts]

# Get the maximum cost
maxcost = max(all_costs)

# Get all cuts that correspond to the maximum cost
maxcuts = [get_binary(i, n_nodes) for i, j in enumerate(all_costs) if j == maxcost]
print(f"The maximum cut is represented by the bitstrings {maxcuts}, with a cost of {maxcost}")

The maximum cut is represented by the bitstrings ['0011', '0101', '0110', '1001', '1010', '1100'], with a cost of 4

The QAOA quantum circuit

The Max-Cut problem can be solved by using the QAOA algorithm. QAOA belongs to the class of Variational Quantum Algorithms (VQAs), which means that its quantum circuit contains a certain number of parametrized quantum gates that need to be optimized with a classical optimizer. The QAOA circuit is composed of two operators:

The cost operator \(U_c\): a circuit generated by the cost Hamiltonian which encodes the cost function described above into a quantum circuit. The solution to the optimization problem is encoded in the ground state of the cost Hamiltonian \(H_c\). The cost operator is simply the evolution of the cost Hamiltonian parametrized by a variational parameter \(\gamma\) so that \(U_c = e^{i\gamma H_c}.\)
The mixing operator \(U_b\): a simple set of single-qubit rotations with adjustable angles which are tuned during the classical optimization loop to minimize the cost

The cost Hamiltonian of the MaxCut problem can be written as:

\[H_c = \frac12 \sum_{\langle i,j\rangle} (\mathbb{1} - Z_iZ_j)\]

where \(\langle i,j\rangle\) represents the edge between nodes \(i\) and \(j\). The solution of the MaxCut problem is encoded in the ground state of the above Hamiltonian.

The QAOA quantum circuit consists of a number of layers, each layer containing a cost and a mixing operator. Below, the QAOA quantum circuit is defined using qadence operations. First, a layer of Hadamard gates is applied to all qubits to prepare the initial state \(|+\rangle ^{\otimes n}\). The cost operator of each layer can be built "manually", implementing the \(e^{iZZ\gamma}\) terms with CNOTs and a \(\rm{RZ}(2\gamma)\) rotation, or it can also be automatically decomposed into digital single and two-qubits operations via the .digital_decomposition() method. The decomposition is exact since the Hamiltonian generator is diagonal.

from qadence import tag, kron, chain, RX, RZ, Z, H, CNOT, I, add
from qadence import HamEvo, QuantumCircuit, Parameter

n_qubits = graph.number_of_nodes()
n_edges = graph.number_of_edges()
n_layers = 6

# Generate the cost Hamiltonian
zz_ops = add(Z(edge[0]) @ Z(edge[1]) for edge in graph.edges)
cost_ham = 0.5 * (n_edges * kron(I(i) for i in range(n_qubits)) - zz_ops)


# QAOA circuit
def build_qaoa_circuit(n_qubits, n_layers, graph):
    layers = []
    # Layer of Hadamards
    initial_layer = kron(H(i) for i in range(n_qubits))
    layers.append(initial_layer)
    for layer in range(n_layers):

        # cost layer with digital decomposition
        # cost_layer = HamEvo(cost_ham, f"g{layer}").digital_decomposition(approximation="basic")
        cost_layer = []
        for edge in graph.edges():
            (q0, q1) = edge
            zz_term = chain(
                CNOT(q0, q1),
                RZ(q1, Parameter(f"g{layer}")),
                CNOT(q0, q1),
            )
            cost_layer.append(zz_term)
        cost_layer = chain(*cost_layer)
        cost_layer = tag(cost_layer, "cost")

        # mixing layer with single qubit rotations
        mixing_layer = kron(RX(i, f"b{layer}") for i in range(n_qubits))
        mixing_layer = tag(mixing_layer, "mixing")

        # putting all together in a single ChainBlock
        layers.append(chain(cost_layer, mixing_layer))

    final_b = chain(*layers)
    return QuantumCircuit(n_qubits, final_b)


circuit = build_qaoa_circuit(n_qubits, n_layers, graph)

# Print a single layer of the circuit

Train the QAOA circuit to solve MaxCut

Given the QAOA circuit above, one can construct the associated Qadence QuantumModel and train it using standard gradient based optimization.

The loss function to be minimized reads:

\[\mathcal{L} =-\langle \psi | H_c| \psi \rangle= -\frac12 \sum_{\langle i,j\rangle} \left(1 - \langle \psi | Z_i Z_j | \psi \rangle \right)\]

where \(|\psi\rangle(\beta, \gamma)\) is the wavefunction obtained by running the QAQA quantum circuit and the sum runs over the edges of the graph \(\langle i,j\rangle\).

import torch
from qadence import QuantumModel

torch.manual_seed(seed)


def loss_function(model: QuantumModel):
    # The loss corresponds to the expectation
    # value of the cost Hamiltonian
    return -1.0 * model.expectation().squeeze()


# initialize the parameters to random values
model = QuantumModel(circuit, observable=cost_ham)
model.reset_vparams(torch.rand(model.num_vparams))
initial_loss = loss_function(model)
print(f"Initial loss: {initial_loss}")

# train the model
n_epochs = 100
lr = 0.1

optimizer = torch.optim.Adam(model.parameters(), lr=lr)

for i in range(n_epochs):
    optimizer.zero_grad()
    loss = loss_function(model)
    loss.backward()
    optimizer.step()
    if (i + 1) % (n_epochs // 10) == 0:
        print(f"MaxCut cost at iteration {i+1}: {-loss.item()}")

Initial loss: -2.1782381363858794
MaxCut cost at iteration 10: 3.7470706807026417
MaxCut cost at iteration 20: 3.8378810288930216
MaxCut cost at iteration 30: 3.9424197899236133
MaxCut cost at iteration 40: 3.9981256255766002
MaxCut cost at iteration 50: 3.996470528508214
MaxCut cost at iteration 60: 3.9991374608876606
MaxCut cost at iteration 70: 3.9994678542919555
MaxCut cost at iteration 80: 3.999872558672829
MaxCut cost at iteration 90: 3.9999475834121063
MaxCut cost at iteration 100: 3.9999793311641003

Qadence offers some convenience functions to implement this training loop with advanced logging and metrics track features.

Results

Given the trained quantum model, one needs to sample the resulting quantum state to recover the bitstring with the highest probability which corresponds to the maximum cut of the graph.

samples = model.sample(n_shots=100)[0]
most_frequent = max(samples, key=samples.get)

print(f"Most frequently sampled bitstring corresponding to the maximum cut: {most_frequent}")

# let's now draw the cut obtained with the QAOA procedure
colors = []
labels = {}
for node, b in zip(graph.nodes(), most_frequent):
    colors.append("green") if int(b) == 0 else colors.append("red")
    labels[node] = "A" if int(b) == 0 else "B"

nx.draw_networkx(graph, node_color=colors, with_labels=True, labels=labels)

Most frequently sampled bitstring corresponding to the maximum cut: 1001

References

Farhi et al. - A Quantum Approximate Optimization Algorithm ↩