Quantum Computing

The quantum module provides quantum computing capabilities, including quantum circuit construction, quantum algorithms, and integration with leading quantum frameworks like Qiskit and Cirq. This enables hybrid classical-quantum machine learning and quantum-enhanced optimization.

Overview

Quantum computing leverages quantum mechanical phenomena like superposition and entanglement to solve certain problems exponentially faster than classical computers. Neurenix provides:

Quantum circuit construction and simulation
Integration with Qiskit and Cirq
Variational quantum algorithms
Hybrid classical-quantum neural networks

Quantum Circuits

Basic Circuit Construction

from neurenix.quantum import QuantumCircuit
import neurenix as nx

# Create a quantum circuit with 3 qubits
circuit = QuantumCircuit(num_qubits=3, name="my_circuit")

# Add gates
circuit.h(0)           # Hadamard gate on qubit 0
circuit.cx(0, 1)       # CNOT gate from qubit 0 to 1
circuit.x(2)           # Pauli-X gate on qubit 2
circuit.ry(1, 0.5)     # RY rotation on qubit 1

# Add measurement
circuit.measure(0, 0)
circuit.measure(1, 1)
circuit.measure(2, 2)

print(circuit)

Quantum Gates

from neurenix.quantum import QuantumCircuit
import numpy as np

circuit = QuantumCircuit(2)

# Single-qubit gates
circuit.h(0)           # Hadamard
circuit.x(0)           # Pauli-X (NOT)
circuit.y(0)           # Pauli-Y
circuit.z(0)           # Pauli-Z
circuit.rx(0, np.pi/4) # Rotation around X
circuit.ry(0, np.pi/4) # Rotation around Y
circuit.rz(0, np.pi/4) # Rotation around Z

# Two-qubit gates
circuit.cx(0, 1)       # CNOT (controlled-X)
circuit.cz(0, 1)       # Controlled-Z

Running Circuits

# Run the circuit
results = circuit.run(shots=1024)
print(results)  # {'000': 512, '111': 512}

# Get unitary matrix
unitary = circuit.to_matrix()
print(unitary.shape)  # (8, 8) for 3 qubits

# Convert to tensor
tensor = circuit.to_tensor()

Parameterized Circuits

Parameterized circuits are essential for variational quantum algorithms.

from neurenix.quantum import ParameterizedCircuit

# Create parameterized circuit
circuit = ParameterizedCircuit(
    num_qubits=2,
    parameters=['theta1', 'theta2', 'theta3']
)

# Add parameterized gates
circuit.rx_param(0, 'theta1')
circuit.ry_param(1, 'theta2')
circuit.cx(0, 1)
circuit.rz_param(1, 'theta3')

# Bind parameters to values
bound_circuit = circuit.bind_parameters({
    'theta1': 0.5,
    'theta2': 1.2,
    'theta3': 0.8
})

# Run the bound circuit
results = bound_circuit.run(shots=1024)

Circuit Templates

Pre-built circuits for common quantum states and operations.

from neurenix.quantum import CircuitTemplate

# Bell pair (entangled state)
bell_circuit = CircuitTemplate.bell_pair()

# GHZ state
ghz_circuit = CircuitTemplate.ghz_state(num_qubits=5)

# Quantum Fourier Transform
qft_circuit = CircuitTemplate.qft(num_qubits=4)

# W state
w_circuit = CircuitTemplate.w_state(num_qubits=3)

Backend Integration

Qiskit Backend

from neurenix.quantum import QiskitBackend, QuantumCircuit

# Create circuit
circuit = QuantumCircuit(2)
circuit.h(0).cx(0, 1)
circuit.measure(0, 0).measure(1, 1)

# Use Qiskit backend
backend = QiskitBackend()
results = backend.run(circuit, shots=1024)
print(results)  # Bell state measurement results

# Get unitary
unitary = backend.to_matrix(circuit)

Cirq Backend

from neurenix.quantum import CirqBackend

# Use Cirq backend
backend = CirqBackend()
results = backend.run(circuit, shots=1024)
unitary = backend.to_matrix(circuit)

Variational Quantum Algorithms

VQE (Variational Quantum Eigensolver)

Find ground state energy of molecular Hamiltonians.

from neurenix.quantum import VQE, ParameterizedCircuit
from neurenix.optim import Adam
import neurenix as nx

# Define ansatz (variational form)
ansatz = ParameterizedCircuit(num_qubits=4)
for i in range(4):
    ansatz.ry_param(i, f'theta_{i}')
for i in range(3):
    ansatz.cx(i, i+1)

# Define Hamiltonian (example: H2 molecule)
hamiltonian = nx.tensor([
    [1.0, 0.0, 0.0, 0.0],
    [0.0, 0.5, 0.2, 0.0],
    [0.0, 0.2, 0.5, 0.0],
    [0.0, 0.0, 0.0, 2.0]
])

# Create VQE instance
vqe = VQE(
    ansatz=ansatz,
    hamiltonian=hamiltonian,
    optimizer=Adam(lr=0.01)
)

# Run optimization
ground_state_energy, optimal_params = vqe.run(max_iterations=100)
print(f"Ground state energy: {ground_state_energy}")

QAOA (Quantum Approximate Optimization Algorithm)

Solve combinatorial optimization problems.

from neurenix.quantum import QAOA
import networkx as netx

# Define problem (MaxCut)
graph = netx.Graph()
graph.add_edges_from([(0, 1), (1, 2), (2, 3), (3, 0), (0, 2)])

# Create QAOA instance
qaoa = QAOA(
    graph=graph,
    num_layers=3,
    optimizer=Adam(lr=0.1)
)

# Run optimization
best_solution, best_value = qaoa.run(max_iterations=100)
print(f"Best cut value: {best_value}")
print(f"Best partition: {best_solution}")

Quantum Phase Estimation

from neurenix.quantum import QuantumPhaseEstimation

# Estimate phase of unitary operator
qpe = QuantumPhaseEstimation(
    unitary=my_unitary,
    precision_qubits=5
)

phase = qpe.estimate()
print(f"Estimated phase: {phase}")

Hybrid Quantum-Classical Models

Quantum Layer in Neural Network

from neurenix.quantum import QuantumLayer, ParameterizedCircuit
from neurenix.nn import Module, Linear
import neurenix as nx

class HybridModel(Module):
    def __init__(self):
        super().__init__()
        
        # Classical layers
        self.fc1 = Linear(10, 4)
        
        # Quantum layer
        circuit = ParameterizedCircuit(num_qubits=4)
        for i in range(4):
            circuit.rx_param(i, f'input_{i}')
        for i in range(3):
            circuit.cx(i, i+1)
        for i in range(4):
            circuit.ry_param(i, f'weight_{i}')
        
        self.quantum_layer = QuantumLayer(
            circuit=circuit,
            input_params=['input_0', 'input_1', 'input_2', 'input_3'],
            weight_params=['weight_0', 'weight_1', 'weight_2', 'weight_3']
        )
        
        # Output layer
        self.fc2 = Linear(4, 2)
    
    def forward(self, x):
        x = self.fc1(x).tanh()
        x = self.quantum_layer(x)
        x = self.fc2(x)
        return x

# Create and train model
model = HybridModel()
optimizer = nx.optim.Adam(model.parameters(), lr=0.01)

# Training loop
for epoch in range(100):
    outputs = model(X_train)
    loss = criterion(outputs, y_train)
    
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

Quantum Convolutional Layer

from neurenix.quantum import QuantumConv2d

class QuantumCNN(Module):
    def __init__(self):
        super().__init__()
        self.qconv1 = QuantumConv2d(
            in_channels=1,
            out_channels=4,
            kernel_size=2,
            num_qubits=4
        )
        self.fc = Linear(4 * 13 * 13, 10)
    
    def forward(self, x):
        x = self.qconv1(x)
        x = x.flatten(1)
        x = self.fc(x)
        return x

Quantum Algorithms

Grover’s Search Algorithm

from neurenix.quantum import Grover

# Search for marked item in unsorted database
grover = Grover(
    num_qubits=4,
    marked_states=[5, 12]  # Items to search for
)

# Run algorithm
results = grover.search(shots=1000)
print(f"Found states: {results}")  # Should find 5 and 12 with high probability

Shor’s Factoring Algorithm

from neurenix.quantum import Shor

# Factor a number using quantum algorithm
shor = Shor()
factors = shor.factor(15)  # Factor N=15
print(f"Factors of 15: {factors}")  # [3, 5]

Quantum Utilities

from neurenix.quantum.utils import (
    state_fidelity,
    density_matrix,
    circuit_to_tensor,
    quantum_gradient
)

# Calculate state fidelity
fidelity = state_fidelity(state1, state2)

# Get density matrix
rho = density_matrix(circuit)

# Convert circuit to tensor
tensor = circuit_to_tensor(circuit)

# Compute quantum gradient
grad = quantum_gradient(circuit, observable)

Example: Quantum Classifier

import neurenix as nx
from neurenix.quantum import QuantumLayer, ParameterizedCircuit
from neurenix.nn import Module, Linear
from neurenix.optim import Adam

class QuantumClassifier(Module):
    def __init__(self, num_qubits=4, num_classes=2):
        super().__init__()
        
        # Create parameterized quantum circuit
        self.circuit = ParameterizedCircuit(num_qubits=num_qubits)
        
        # Encoding layer
        for i in range(num_qubits):
            self.circuit.ry_param(i, f'input_{i}')
        
        # Variational layers
        for layer in range(3):
            # Entangling layer
            for i in range(num_qubits-1):
                self.circuit.cx(i, i+1)
            
            # Rotation layer
            for i in range(num_qubits):
                self.circuit.ry_param(i, f'w{layer}_{i}')
        
        # Quantum layer
        input_params = [f'input_{i}' for i in range(num_qubits)]
        weight_params = [f'w{l}_{i}' for l in range(3) for i in range(num_qubits)]
        
        self.quantum = QuantumLayer(
            circuit=self.circuit,
            input_params=input_params,
            weight_params=weight_params
        )
        
        # Classical output layer
        self.fc = Linear(num_qubits, num_classes)
    
    def forward(self, x):
        x = self.quantum(x)
        x = self.fc(x)
        return x

# Training
model = QuantumClassifier(num_qubits=4, num_classes=2)
optimizer = Adam(model.parameters(), lr=0.01)
criterion = nx.nn.CrossEntropyLoss()

for epoch in range(100):
    for X_batch, y_batch in dataloader:
        # Forward pass
        outputs = model(X_batch)
        loss = criterion(outputs, y_batch)
        
        # Backward pass
        optimizer.zero_grad()
        loss.backward()
        optimizer.step()
    
    print(f"Epoch {epoch}: Loss = {loss.item():.4f}")

Example: Quantum GAN

from neurenix.quantum import ParameterizedCircuit, QuantumLayer

class QuantumGenerator(Module):
    def __init__(self, latent_dim=4):
        super().__init__()
        
        circuit = ParameterizedCircuit(num_qubits=latent_dim)
        # Define generator circuit architecture
        for i in range(latent_dim):
            circuit.ry_param(i, f'z_{i}')
        
        self.quantum = QuantumLayer(circuit, ...)
    
    def forward(self, z):
        return self.quantum(z)

class QuantumDiscriminator(Module):
    def __init__(self, input_dim=4):
        super().__init__()
        
        circuit = ParameterizedCircuit(num_qubits=input_dim)
        # Define discriminator circuit
        self.quantum = QuantumLayer(circuit, ...)
        self.fc = Linear(input_dim, 1)
    
    def forward(self, x):
        x = self.quantum(x)
        return self.fc(x).sigmoid()

Best Practices

Circuit Depth: Keep circuits shallow to minimize decoherence effects
Parameterization: Use efficient parameterization schemes (hardware-efficient ansatz)
Measurement: Use sufficient shots for accurate expectation values
Classical Optimization: Choose appropriate optimizers (Adam often works well)
Hybrid Design: Combine quantum and classical layers strategically

Hardware Considerations

# Run on real quantum hardware (requires IBM Q account)
from qiskit import IBMQ

IBMQ.load_account()
provider = IBMQ.get_provider(hub='ibm-q')
backend = provider.get_backend('ibmq_manila')

# Use with Neurenix
from neurenix.quantum import QiskitBackend

qbackend = QiskitBackend(device=backend)
results = qbackend.run(circuit, shots=1024)

References

Nielsen & Chuang - “Quantum Computation and Quantum Information”
Schuld & Petruccione - “Supervised Learning with Quantum Computers”
Farhi et al. (2014) - “A Quantum Approximate Optimization Algorithm”
Peruzzo et al. (2014) - “A variational eigenvalue solver on a photonic quantum processor”

​Quantum Computing

​Overview

​Quantum Circuits

​Basic Circuit Construction

​Quantum Gates

​Running Circuits

​Parameterized Circuits

​Circuit Templates

​Backend Integration

​Qiskit Backend

​Cirq Backend

​Variational Quantum Algorithms

​VQE (Variational Quantum Eigensolver)

​QAOA (Quantum Approximate Optimization Algorithm)

​Quantum Phase Estimation

​Hybrid Quantum-Classical Models

​Quantum Layer in Neural Network

​Quantum Convolutional Layer

​Quantum Algorithms

​Grover’s Search Algorithm

​Shor’s Factoring Algorithm

​Quantum Utilities

​Example: Quantum Classifier

​Example: Quantum GAN

​Best Practices

​Hardware Considerations

​References

​See Also