Quantum Algorithms and Applications

Quantum Algorithms and Applications

Qiskit supports a wide range of quantum algorithms for optimization, chemistry, machine learning, and physics simulations.

Table of Contents

1. Optimization Algorithms
2. Chemistry and Materials Science
3. Machine Learning
4. Algorithm Libraries

Optimization Algorithms

Variational Quantum Eigensolver (VQE)

VQE finds the minimum eigenvalue of a Hamiltonian using a hybrid quantum-classical approach.

Use Cases:

• Molecular ground state energy
• Combinatorial optimization
• Materials simulation

Implementation:

from qiskit import QuantumCircuit, transpile
from qiskit_ibm_runtime import QiskitRuntimeService, EstimatorV2 as Estimator, Session
from qiskit.quantum_info import SparsePauliOp
from scipy.optimize import minimize
import numpy as np

def vqe_algorithm(hamiltonian, ansatz, backend, initial_params):
    """
    Run VQE algorithm

    Args:
        hamiltonian: Observable (SparsePauliOp)
        ansatz: Parameterized quantum circuit
        backend: Quantum backend
        initial_params: Initial parameter values
    """

    with Session(backend=backend) as session:
        estimator = Estimator(session=session)

        def cost_function(params):
            # Bind parameters to circuit
            bound_circuit = ansatz.assign_parameters(params)

            # Transpile for hardware
            qc_isa = transpile(bound_circuit, backend=backend, optimization_level=3)

            # Compute expectation value
            job = estimator.run([(qc_isa, hamiltonian)])
            result = job.result()
            energy = result[0].data.evs

            return energy

        # Classical optimization
        result = minimize(
            cost_function,
            initial_params,
            method='COBYLA',
            options={'maxiter': 100}
        )

    return result.fun, result.x

# Example: H2 molecule Hamiltonian
hamiltonian = SparsePauliOp(
    ["IIII", "ZZII", "IIZZ", "ZZZI", "IZZI"],
    coeffs=[-0.8, 0.17, 0.17, -0.24, 0.17]
)

# Create ansatz
qc = QuantumCircuit(4)
# ... define ansatz structure ...

service = QiskitRuntimeService()
backend = service.backend("ibm_brisbane")

energy, params = vqe_algorithm(hamiltonian, qc, backend, np.random.rand(10))
print(f"Ground state energy: {energy}")

Quantum Approximate Optimization Algorithm (QAOA)

QAOA solves combinatorial optimization problems like MaxCut, TSP, and graph coloring.

Use Cases:

• MaxCut problems
• Portfolio optimization
• Vehicle routing
• Scheduling problems

Implementation:

from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
import networkx as nx

def qaoa_maxcut(graph, p, backend):
    """
    QAOA for MaxCut problem

    Args:
        graph: NetworkX graph
        p: Number of QAOA layers
        backend: Quantum backend
    """
    num_qubits = len(graph.nodes())
    qc = QuantumCircuit(num_qubits)

    # Initial superposition
    qc.h(range(num_qubits))

    # Alternating layers
    betas = [Parameter(f'β_{i}') for i in range(p)]
    gammas = [Parameter(f'γ_{i}') for i in range(p)]

    for i in range(p):
        # Problem Hamiltonian (MaxCut)
        for edge in graph.edges():
            u, v = edge
            qc.cx(u, v)
            qc.rz(2 * gammas[i], v)
            qc.cx(u, v)

        # Mixer Hamiltonian
        for qubit in range(num_qubits):
            qc.rx(2 * betas[i], qubit)

    qc.measure_all()
    return qc, betas + gammas

# Example: MaxCut on 4-node graph
G = nx.Graph()
G.add_edges_from([(0, 1), (1, 2), (2, 3), (3, 0)])

qaoa_circuit, params = qaoa_maxcut(G, p=2, backend=backend)

# Run with Sampler and optimize parameters
# ... (similar to VQE optimization loop)

Grover's Algorithm

Quantum search algorithm providing quadratic speedup for unstructured search.

Use Cases:

• Database search
• SAT solving
• Finding marked items

Implementation:

from qiskit import QuantumCircuit

def grover_oracle(marked_states):
    """Create oracle that marks target states"""
    num_qubits = len(marked_states[0])
    qc = QuantumCircuit(num_qubits)

    for target in marked_states:
        # Flip phase of target state
        for i, bit in enumerate(target):
            if bit == '0':
                qc.x(i)

        # Multi-controlled Z
        qc.h(num_qubits - 1)
        qc.mcx(list(range(num_qubits - 1)), num_qubits - 1)
        qc.h(num_qubits - 1)

        for i, bit in enumerate(target):
            if bit == '0':
                qc.x(i)

    return qc

def grover_diffusion(num_qubits):
    """Create Grover diffusion operator"""
    qc = QuantumCircuit(num_qubits)

    qc.h(range(num_qubits))
    qc.x(range(num_qubits))

    qc.h(num_qubits - 1)
    qc.mcx(list(range(num_qubits - 1)), num_qubits - 1)
    qc.h(num_qubits - 1)

    qc.x(range(num_qubits))
    qc.h(range(num_qubits))

    return qc

def grover_algorithm(marked_states, num_iterations):
    """Complete Grover's algorithm"""
    num_qubits = len(marked_states[0])
    qc = QuantumCircuit(num_qubits)

    # Initialize superposition
    qc.h(range(num_qubits))

    # Grover iterations
    oracle = grover_oracle(marked_states)
    diffusion = grover_diffusion(num_qubits)

    for _ in range(num_iterations):
        qc.compose(oracle, inplace=True)
        qc.compose(diffusion, inplace=True)

    qc.measure_all()
    return qc

# Search for state |101⟩ in 3-qubit space
marked = ['101']
iterations = int(np.pi/4 * np.sqrt(2**3))  # Optimal iterations
qc_grover = grover_algorithm(marked, iterations)

Chemistry and Materials Science

Molecular Ground State Energy

Install Qiskit Nature:

uv pip install qiskit-nature qiskit-nature-pyscf

Example: H2 Molecule

from qiskit_nature.second_q.drivers import PySCFDriver
from qiskit_nature.second_q.mappers import JordanWignerMapper, ParityMapper
from qiskit_nature.second_q.circuit.library import UCCSD, HartreeFock

# Define molecule
driver = PySCFDriver(
    atom="H 0 0 0; H 0 0 0.735",
    basis="sto3g",
    charge=0,
    spin=0
)

# Get electronic structure problem
problem = driver.run()

# Map fermionic operators to qubits
mapper = JordanWignerMapper()
hamiltonian = mapper.map(problem.hamiltonian.second_q_op())

# Create initial state
num_particles = problem.num_particles
num_spatial_orbitals = problem.num_spatial_orbitals

init_state = HartreeFock(
    num_spatial_orbitals,
    num_particles,
    mapper
)

# Create ansatz
ansatz = UCCSD(
    num_spatial_orbitals,
    num_particles,
    mapper,
    initial_state=init_state
)

# Run VQE
energy, params = vqe_algorithm(
    hamiltonian,
    ansatz,
    backend,
    np.zeros(ansatz.num_parameters)
)

# Add nuclear repulsion energy
total_energy = energy + problem.nuclear_repulsion_energy
print(f"Ground state energy: {total_energy} Ha")

Different Molecular Mappers

# Jordan-Wigner mapping
jw_mapper = JordanWignerMapper()
ham_jw = jw_mapper.map(problem.hamiltonian.second_q_op())

# Parity mapping (often more efficient)
parity_mapper = ParityMapper()
ham_parity = parity_mapper.map(problem.hamiltonian.second_q_op())

# Bravyi-Kitaev mapping
from qiskit_nature.second_q.mappers import BravyiKitaevMapper
bk_mapper = BravyiKitaevMapper()
ham_bk = bk_mapper.map(problem.hamiltonian.second_q_op())

Excited States

from qiskit_nature.second_q.algorithms import QEOM

# Quantum Equation of Motion for excited states
qeom = QEOM(estimator, ansatz, 'sd')  # Singles and doubles excitations
excited_states = qeom.solve(problem)

Machine Learning

Quantum Kernels

Quantum computers can compute kernel functions for machine learning.

Install Qiskit Machine Learning:

uv pip install qiskit-machine-learning

Example: Classification with Quantum Kernel

from qiskit_machine_learning.kernels import FidelityQuantumKernel
from qiskit_algorithms.state_fidelities import ComputeUncompute
from qiskit.circuit.library import ZZFeatureMap
from sklearn.svm import SVC
import numpy as np

# Create feature map
num_features = 2
feature_map = ZZFeatureMap(feature_dimension=num_features, reps=2)

# Create quantum kernel
fidelity = ComputeUncompute(sampler=sampler)
qkernel = FidelityQuantumKernel(fidelity=fidelity, feature_map=feature_map)

# Use with scikit-learn
X_train = np.random.rand(50, 2)
y_train = np.random.choice([0, 1], 50)

# Compute kernel matrix
kernel_matrix = qkernel.evaluate(X_train)

# Train SVM with quantum kernel
svc = SVC(kernel='precomputed')
svc.fit(kernel_matrix, y_train)

# Predict
X_test = np.random.rand(10, 2)
kernel_test = qkernel.evaluate(X_test, X_train)
predictions = svc.predict(kernel_test)

Variational Quantum Classifier (VQC)

from qiskit_machine_learning.algorithms import VQC
from qiskit.circuit.library import RealAmplitudes

# Create feature map and ansatz
feature_map = ZZFeatureMap(2)
ansatz = RealAmplitudes(2, reps=1)

# Create VQC
vqc = VQC(
    sampler=sampler,
    feature_map=feature_map,
    ansatz=ansatz,
    optimizer='COBYLA'
)

# Train
vqc.fit(X_train, y_train)

# Predict
predictions = vqc.predict(X_test)
accuracy = vqc.score(X_test, y_test)

Quantum Neural Networks (QNN)

from qiskit_machine_learning.neural_networks import SamplerQNN
from qiskit.circuit import QuantumCircuit, Parameter

# Create parameterized circuit
qc = QuantumCircuit(2)
params = [Parameter(f'θ_{i}') for i in range(4)]

# Network structure
for i, param in enumerate(params[:2]):
    qc.ry(param, i)

qc.cx(0, 1)

for i, param in enumerate(params[2:]):
    qc.ry(param, i)

qc.measure_all()

# Create QNN
qnn = SamplerQNN(
    circuit=qc,
    sampler=sampler,
    input_params=[],  # No input parameters for this example
    weight_params=params
)

# Use with PyTorch or TensorFlow for training

Algorithm Libraries

Qiskit Algorithms

Standard implementations of quantum algorithms:

uv pip install qiskit-algorithms

Available Algorithms:

• Amplitude Estimation
• Phase Estimation
• Shor's Algorithm
• Quantum Fourier Transform
• HHL (Linear systems)

Example: Quantum Phase Estimation

from qiskit.circuit.library import QFT
from qiskit_algorithms import PhaseEstimation

# Create unitary operator
num_qubits = 3
unitary = QuantumCircuit(num_qubits)
# ... define unitary ...

# Phase estimation
pe = PhaseEstimation(num_evaluation_qubits=3, quantum_instance=backend)
result = pe.estimate(unitary=unitary, state_preparation=initial_state)

Qiskit Optimization

Optimization problem solvers:

uv pip install qiskit-optimization

Supported Problems:

• Quadratic programs
• Integer programming
• Linear programming
• Constraint satisfaction

Example: Portfolio Optimization

from qiskit_optimization.applications import PortfolioOptimization
from qiskit_optimization.algorithms import MinimumEigenOptimizer
from qiskit_algorithms import QAOA

# Define portfolio problem
returns = [0.1, 0.15, 0.12]  # Expected returns
covariances = [[1, 0.5, 0.3], [0.5, 1, 0.4], [0.3, 0.4, 1]]
budget = 2  # Number of assets to select

portfolio = PortfolioOptimization(
    expected_returns=returns,
    covariances=covariances,
    budget=budget
)

# Convert to quadratic program
qp = portfolio.to_quadratic_program()

# Solve with QAOA
qaoa = QAOA(sampler=sampler, optimizer='COBYLA', reps=2)
optimizer = MinimumEigenOptimizer(qaoa)

result = optimizer.solve(qp)
print(f"Optimal portfolio: {result.x}")

Physics Simulations

Time Evolution

from qiskit.synthesis import SuzukiTrotter
from qiskit.quantum_info import SparsePauliOp

# Define Hamiltonian
hamiltonian = SparsePauliOp(["XX", "YY", "ZZ"], coeffs=[1.0, 1.0, 1.0])

# Time evolution
time = 1.0
evolution_gate = SuzukiTrotter(order=2, reps=10).synthesize(
    hamiltonian,
    time
)

qc = QuantumCircuit(2)
qc.append(evolution_gate, range(2))

Partial Differential Equations

Use Case: Quantum algorithms for solving PDEs with potential exponential speedup.

# Quantum PDE solver implementation
# Requires advanced techniques like HHL algorithm
# and amplitude encoding of solution vectors

Benchmarking

Benchpress Toolkit

Benchmark quantum algorithms:

# Benchpress provides standardized benchmarks
# for comparing quantum algorithm performance

# Examples:
# - Quantum volume circuits
# - Random circuit sampling
# - Application-specific benchmarks

Best Practices

1. Start Simple

Begin with small problem instances to validate your approach:

# Test with 2-3 qubits first
# Scale up after confirming correctness

2. Use Simulators for Development

from qiskit.primitives import StatevectorSampler

# Develop with local simulator
sampler_sim = StatevectorSampler()

# Switch to hardware for production
# sampler_hw = Sampler(backend)

3. Monitor Convergence

convergence_data = []

def tracked_cost_function(params):
    energy = cost_function(params)
    convergence_data.append(energy)
    return energy

# Plot convergence after optimization
import matplotlib.pyplot as plt
plt.plot(convergence_data)
plt.xlabel('Iteration')
plt.ylabel('Energy')
plt.show()

4. Parameter Initialization

# Use problem-specific initialization when possible
# Random initialization
initial_params = np.random.uniform(0, 2*np.pi, num_params)

# Or use classical preprocessing
# initial_params = classical_solution_to_params(classical_result)

5. Save Intermediate Results

import json

checkpoint = {
    'iteration': iteration,
    'params': params.tolist(),
    'energy': energy,
    'timestamp': time.time()
}

with open(f'checkpoint_{iteration}.json', 'w') as f:
    json.dump(checkpoint, f)

Resources and Further Reading

Official Documentation:

• Qiskit Textbook
• Qiskit Nature Documentation
• Qiskit Machine Learning Documentation
• Qiskit Optimization Documentation

Research Papers:

• VQE: Peruzzo et al., Nature Communications (2014)
• QAOA: Farhi et al., arXiv:1411.4028
• Quantum Kernels: Havlíček et al., Nature (2019)