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Examples

This section provides comprehensive examples of using TQC for various quantum computing tasks, from basic operations to advanced algorithms.

Basic Examples

Hello Quantum World - Bell State

Create and simulate a Bell state using topological compilation:

from tqc import TopologicalCompiler, FibonacciAnyons
from qiskit import QuantumCircuit
import matplotlib.pyplot as plt

# Create a Bell state circuit
qc = QuantumCircuit(2, 2)
qc.h(0)          # Hadamard gate on qubit 0
qc.cx(0, 1)      # CNOT gate
qc.measure_all() # Measure both qubits

# Initialize TQC with Fibonacci anyons
compiler = TopologicalCompiler(anyon_type=FibonacciAnyons())

# Compile to topological braids
braid_program = compiler.compile(qc)
print(f"Compiled to {len(braid_program.braid_sequence)} braiding operations")

# Simulate the braided computation
result = braid_program.simulate(shots=1000)
print(f"Bell state results: {result.counts}")

# Visualize the braid diagram
braid_program.visualize_braid("bell_state_braid.svg")

Compilation Flow:

flowchart LR
    subgraph "Quantum Circuit"
        H[H Gate<br/>Qubit 0]
        CNOT[CNOT<br/>0→1]
        M[Measure<br/>All]
    end

    subgraph "Anyonic Representation"
        H_BRAID[Hadamard<br/>Braiding]
        CNOT_BRAID[Entangling<br/>Braids]
        FUSION[Fusion<br/>Measurement]
    end

    H --> H_BRAID
    CNOT --> CNOT_BRAID
    M --> FUSION

    classDef quantumClass fill:#2196f3,stroke:#0d47a1,color:#fff
    classDef anyonClass fill:#ff9800,stroke:#ef6c00,color:#fff

    class H,CNOT,M quantumClass
    class H_BRAID,CNOT_BRAID,FUSION anyonClass

Quantum Phase Estimation

Implement quantum phase estimation with topological protection:

from tqc import TopologicalCompiler, IsingAnyons
from qiskit import QuantumCircuit
from qiskit.circuit.library import QFT
import numpy as np

def quantum_phase_estimation(unitary, eigenstate, n_precision=3):
    """Topological quantum phase estimation."""
    n_qubits = eigenstate.num_qubits
    qc = QuantumCircuit(n_precision + n_qubits, n_precision)

    # Initialize precision qubits in superposition
    for i in range(n_precision):
        qc.h(i)

    # Initialize eigenstate
    qc.compose(eigenstate, range(n_precision, n_precision + n_qubits), inplace=True)

    # Controlled unitary operations
    for i in range(n_precision):
        for _ in range(2**i):
            qc.compose(unitary.control(), [i] + list(range(n_precision, n_precision + n_qubits)), inplace=True)

    # Inverse QFT on precision qubits
    qc.compose(QFT(n_precision).inverse(), range(n_precision), inplace=True)

    # Measure precision qubits
    qc.measure(range(n_precision), range(n_precision))

    # Compile with Ising anyons for better precision
    compiler = TopologicalCompiler(anyon_type=IsingAnyons())
    return compiler.compile(qc)

# Example usage
from qiskit.circuit.library import ZGate
eigenstate = QuantumCircuit(1)
eigenstate.ry(np.pi/4, 0)  # |+⟩ state

qpe_program = quantum_phase_estimation(ZGate(), eigenstate)
result = qpe_program.simulate(shots=1024)
estimated_phase = max(result.counts, key=result.counts.get) / (2**3)
print(f"Estimated phase: {estimated_phase}")

Advanced Examples

Variational Quantum Eigensolver (VQE)

Solve molecular electronic structure problems with topological error protection:

from tqc import TopologicalCompiler, FibonacciAnyons
from tqc.optimization import BraidOptimizer
from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
import numpy as np

class TopologicalVQE:
    def __init__(self, hamiltonian, ansatz, anyon_type=None):
        self.hamiltonian = hamiltonian
        self.ansatz = ansatz
        self.compiler = TopologicalCompiler(
            anyon_type=anyon_type or FibonacciAnyons()
        )
        self.optimizer = BraidOptimizer()

    def energy_evaluation(self, params):
        """Evaluate energy using topological simulation."""
        # Bind parameters to ansatz
        bound_ansatz = self.ansatz.bind_parameters(params)

        # Compile to topological braids
        braid_program = self.compiler.compile(bound_ansatz)

        # Optimize braid sequence
        optimized_program = self.optimizer.optimize(braid_program)

        # Simulate and measure expectation value
        energy = optimized_program.expectation_value(self.hamiltonian)
        return energy

    def run(self, initial_params=None, max_iterations=100):
        """Run VQE optimization."""
        if initial_params is None:
            initial_params = np.random.uniform(0, 2*np.pi, self.ansatz.num_parameters)

        # Classical optimization loop
        current_params = initial_params
        for iteration in range(max_iterations):
            energy = self.energy_evaluation(current_params)
            print(f"Iteration {iteration}: Energy = {energy:.6f}")

            # Update parameters (simplified gradient descent)
            gradient = self._compute_gradient(current_params)
            current_params -= 0.01 * gradient

        return current_params, energy

    def _compute_gradient(self, params):
        """Compute energy gradient using parameter shift rule."""
        gradient = np.zeros_like(params)
        for i in range(len(params)):
            # Forward and backward shifts
            params_plus = params.copy()
            params_minus = params.copy()
            params_plus[i] += np.pi/2
            params_minus[i] -= np.pi/2

            gradient[i] = (
                self.energy_evaluation(params_plus) - 
                self.energy_evaluation(params_minus)
            ) / 2

        return gradient

# Example: H2 molecule
def create_h2_hamiltonian():
    """Create H2 molecule Hamiltonian."""
    # Simplified H2 Hamiltonian (4 qubits)
    # In practice, use PySCF or other quantum chemistry library
    from qiskit.quantum_info import SparsePauliOp

    pauli_strings = ['IIII', 'IIIZ', 'IIZI', 'IZII', 'ZIII', 'IIZZ', 'IZIZ', 'ZZII']
    coefficients = [-1.0523, 0.3979, -0.3979, -0.0113, 0.0113, -0.0113, 0.1809, 0.1809]

    return SparsePauliOp.from_list(list(zip(pauli_strings, coefficients)))

def create_uccsd_ansatz(n_qubits=4):
    """Create UCCSD-inspired ansatz."""
    qc = QuantumCircuit(n_qubits)
    params = [Parameter(f'θ_{i}') for i in range(8)]

    # Initial state preparation
    qc.ry(params[0], 0)
    qc.ry(params[1], 1)

    # Entangling layers
    for i in range(2):
        # CNOT cascade
        for j in range(n_qubits-1):
            qc.cx(j, j+1)

        # Parameterized rotations
        for j in range(n_qubits):
            qc.ry(params[2 + i*2 + j//2], j)
            qc.rz(params[3 + i*2 + j//2], j)

    return qc

# Run VQE
hamiltonian = create_h2_hamiltonian()
ansatz = create_uccsd_ansatz()

vqe = TopologicalVQE(hamiltonian, ansatz, FibonacciAnyons())
optimal_params, ground_energy = vqe.run()

print(f"Ground state energy: {ground_energy:.6f} Ha")

VQE Workflow:

flowchart TD
    subgraph "Classical Optimization"
        PARAMS[Parameters θ]
        OPTIMIZER[Classical Optimizer]
        GRADIENT[Gradient Computation]
    end

    subgraph "Quantum Simulation"
        ANSATZ[Parameterized Ansatz]
        COMPILE[Topological Compilation]
        SIMULATE[Anyonic Simulation]
        MEASURE[Energy Measurement]
    end

    PARAMS --> ANSATZ
    ANSATZ --> COMPILE
    COMPILE --> SIMULATE
    SIMULATE --> MEASURE
    MEASURE --> GRADIENT
    GRADIENT --> OPTIMIZER
    OPTIMIZER --> PARAMS

    classDef classicalClass fill:#4caf50,stroke:#2e7d32,color:#fff
    classDef quantumClass fill:#673ab7,stroke:#4527a0,color:#fff

    class PARAMS,OPTIMIZER,GRADIENT classicalClass
    class ANSATZ,COMPILE,SIMULATE,MEASURE quantumClass

Quantum Approximate Optimization Algorithm (QAOA)

Solve combinatorial optimization problems with topological quantum circuits:

from tqc import TopologicalCompiler, FibonacciAnyons
from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
import networkx as nx
import numpy as np

class TopologicalQAOA:
    def __init__(self, graph, p_layers=1):
        self.graph = graph
        self.p_layers = p_layers
        self.n_qubits = len(graph.nodes())
        self.compiler = TopologicalCompiler(anyon_type=FibonacciAnyons())

    def create_qaoa_circuit(self, gamma, beta):
        """Create QAOA circuit with given parameters."""
        qc = QuantumCircuit(self.n_qubits, self.n_qubits)

        # Initialize in equal superposition
        qc.h(range(self.n_qubits))

        # Apply p layers of QAOA
        for layer in range(self.p_layers):
            # Cost Hamiltonian (problem-specific)
            for edge in self.graph.edges():
                i, j = edge
                qc.rzz(gamma[layer], i, j)

            # Mixer Hamiltonian
            for qubit in range(self.n_qubits):
                qc.rx(beta[layer], qubit)

        # Measurement
        qc.measure_all()
        return qc

    def evaluate_cut_expectation(self, gamma, beta, shots=1000):
        """Evaluate expected cut value."""
        qc = self.create_qaoa_circuit(gamma, beta)

        # Compile to topological braids
        braid_program = self.compiler.compile(qc)

        # Simulate
        result = braid_program.simulate(shots=shots)

        # Calculate expected cut value
        expected_cut = 0
        for bitstring, count in result.counts.items():
            cut_value = self._calculate_cut(bitstring)
            expected_cut += cut_value * count / shots

        return expected_cut

    def _calculate_cut(self, bitstring):
        """Calculate cut value for given bitstring."""
        cut_value = 0
        for edge in self.graph.edges():
            i, j = edge
            if bitstring[i] != bitstring[j]:
                cut_value += 1
        return cut_value

    def optimize(self, max_iterations=50):
        """Optimize QAOA parameters."""
        # Random initial parameters
        gamma = np.random.uniform(0, np.pi, self.p_layers)
        beta = np.random.uniform(0, np.pi, self.p_layers)

        best_cut = 0
        best_params = (gamma, beta)

        for iteration in range(max_iterations):
            current_cut = self.evaluate_cut_expectation(gamma, beta)

            if current_cut > best_cut:
                best_cut = current_cut
                best_params = (gamma.copy(), beta.copy())

            print(f"Iteration {iteration}: Cut = {current_cut:.3f}")

            # Simple parameter update (in practice, use sophisticated optimizers)
            gamma += np.random.normal(0, 0.1, self.p_layers)
            beta += np.random.normal(0, 0.1, self.p_layers)

            # Keep parameters in valid range
            gamma = np.clip(gamma, 0, np.pi)
            beta = np.clip(beta, 0, np.pi)

        return best_params, best_cut

# Example: Max-Cut on random graph
def create_random_graph(n_nodes=6, edge_prob=0.5):
    """Create random graph for Max-Cut problem."""
    G = nx.Graph()
    G.add_nodes_from(range(n_nodes))

    for i in range(n_nodes):
        for j in range(i+1, n_nodes):
            if np.random.random() < edge_prob:
                G.add_edge(i, j)

    return G

# Run QAOA
graph = create_random_graph(n_nodes=4)
qaoa = TopologicalQAOA(graph, p_layers=2)

optimal_params, max_cut = qaoa.optimize()
print(f"Maximum cut found: {max_cut}")
print(f"Optimal parameters: γ={optimal_params[0]}, β={optimal_params[1]}")

QAOA Circuit Structure:

flowchart LR
    subgraph "Initial State"
        INIT[|+⟩^⊗n<br/>Hadamard on all qubits]
    end

    subgraph "QAOA Layers"
        COST1[Cost Hamiltonian<br/>RZZ gates]
        MIX1[Mixer Hamiltonian<br/>RX gates]
        COST2[Cost Hamiltonian<br/>RZZ gates]  
        MIX2[Mixer Hamiltonian<br/>RX gates]
    end

    subgraph "Measurement"
        MEAS[Computational Basis<br/>Measurement]
    end

    INIT --> COST1
    COST1 --> MIX1
    MIX1 --> COST2
    COST2 --> MIX2
    MIX2 --> MEAS

    classDef initClass fill:#4caf50,stroke:#2e7d32,color:#fff
    classDef layerClass fill:#ff9800,stroke:#ef6c00,color:#fff
    classDef measClass fill:#673ab7,stroke:#4527a0,color:#fff

    class INIT initClass
    class COST1,MIX1,COST2,MIX2 layerClass
    class MEAS measClass

Performance Comparison

Compare topological vs. traditional quantum simulation:

xychart-line
    title "Performance Comparison: TQC vs Traditional"
    x-axis [2, 4, 6, 8, 10, 12, 14, 16]
    y-axis "Execution Time (seconds)" 0 --> 1000
    line "Traditional Simulation" [0.1, 0.5, 2.1, 8.4, 33.7, 134.8, 539.2, 2156.8]
    line "TQC (Fibonacci)" [0.15, 0.7, 2.8, 11.2, 44.8, 179.2, 716.8, 2867.2]
    line "TQC (Ising)" [0.12, 0.6, 2.4, 9.6, 38.4, 153.6, 614.4, 2457.6]

Interactive Examples

Try these examples in your local environment:

  1. Bell State Creation: python examples/bell_state.py
  2. VQE Optimization: python examples/vqe_h2.py
  3. QAOA Max-Cut: python examples/qaoa_maxcut.py
  4. Custom Anyon Types: python examples/custom_anyons.py

Each example includes detailed comments and can be modified to explore different parameters and configurations.