Noise Modeling and Mitigation

Noise Modeling and Mitigation

This guide covers noise models, noisy simulation, characterization, and error mitigation in Cirq.

Noise Channels

Depolarizing Noise

import cirq
import numpy as np

# Single-qubit depolarizing channel
depol_channel = cirq.depolarize(p=0.01)

# Apply to qubit
q = cirq.LineQubit(0)
noisy_op = depol_channel(q)

# Add to circuit
circuit = cirq.Circuit(
    cirq.H(q),
    depol_channel(q),
    cirq.measure(q, key='m')
)

Amplitude Damping

# Amplitude damping (T1 decay)
gamma = 0.1
amp_damp = cirq.amplitude_damp(gamma)

# Apply after gate
circuit = cirq.Circuit(
    cirq.X(q),
    amp_damp(q)
)

Phase Damping

# Phase damping (T2 dephasing)
gamma = 0.1
phase_damp = cirq.phase_damp(gamma)

circuit = cirq.Circuit(
    cirq.H(q),
    phase_damp(q)
)

Bit Flip Noise

# Bit flip channel
bit_flip_prob = 0.01
bit_flip = cirq.bit_flip(bit_flip_prob)

circuit = cirq.Circuit(
    cirq.H(q),
    bit_flip(q)
)

Phase Flip Noise

# Phase flip channel
phase_flip_prob = 0.01
phase_flip = cirq.phase_flip(phase_flip_prob)

circuit = cirq.Circuit(
    cirq.H(q),
    phase_flip(q)
)

Generalized Amplitude Damping

# Generalized amplitude damping
p = 0.1  # Damping probability
gamma = 0.2  # Excitation probability
gen_amp_damp = cirq.generalized_amplitude_damp(p=p, gamma=gamma)

Reset Channel

# Reset to |0⟩ or |1⟩
reset_to_zero = cirq.reset(q)

# Reset appears as measurement followed by conditional flip
circuit = cirq.Circuit(
    cirq.H(q),
    reset_to_zero
)

Noise Models

Constant Noise Model

# Apply same noise to all qubits
noise = cirq.ConstantQubitNoiseModel(
    qubit_noise_gate=cirq.depolarize(0.01)
)

# Simulate with noise
simulator = cirq.DensityMatrixSimulator(noise=noise)
result = simulator.run(circuit, repetitions=1000)

Gate-Specific Noise

class CustomNoiseModel(cirq.NoiseModel):
    """Apply different noise to different gate types."""

    def noisy_operation(self, op):
        # Single-qubit gates: depolarizing noise
        if len(op.qubits) == 1:
            return [op, cirq.depolarize(0.001)(op.qubits[0])]

        # Two-qubit gates: higher depolarizing noise
        elif len(op.qubits) == 2:
            return [
                op,
                cirq.depolarize(0.01)(op.qubits[0]),
                cirq.depolarize(0.01)(op.qubits[1])
            ]

        return op

# Use custom noise model
noise_model = CustomNoiseModel()
simulator = cirq.DensityMatrixSimulator(noise=noise_model)

Qubit-Specific Noise

class QubitSpecificNoise(cirq.NoiseModel):
    """Different noise for different qubits."""

    def __init__(self, qubit_noise_map):
        self.qubit_noise_map = qubit_noise_map

    def noisy_operation(self, op):
        noise_ops = [op]
        for qubit in op.qubits:
            if qubit in self.qubit_noise_map:
                noise = self.qubit_noise_map[qubit]
                noise_ops.append(noise(qubit))
        return noise_ops

# Define per-qubit noise
q0, q1, q2 = cirq.LineQubit.range(3)
noise_map = {
    q0: cirq.depolarize(0.001),
    q1: cirq.depolarize(0.005),
    q2: cirq.depolarize(0.002)
}

noise_model = QubitSpecificNoise(noise_map)

Thermal Noise

class ThermalNoise(cirq.NoiseModel):
    """Thermal relaxation noise."""

    def __init__(self, T1, T2, gate_time):
        self.T1 = T1  # Amplitude damping time
        self.T2 = T2  # Dephasing time
        self.gate_time = gate_time

    def noisy_operation(self, op):
        # Calculate probabilities
        p_amp = 1 - np.exp(-self.gate_time / self.T1)
        p_phase = 1 - np.exp(-self.gate_time / self.T2)

        noise_ops = [op]
        for qubit in op.qubits:
            noise_ops.append(cirq.amplitude_damp(p_amp)(qubit))
            noise_ops.append(cirq.phase_damp(p_phase)(qubit))

        return noise_ops

# Typical superconducting qubit parameters
T1 = 50e-6  # 50 μs
T2 = 30e-6  # 30 μs
gate_time = 25e-9  # 25 ns

noise_model = ThermalNoise(T1, T2, gate_time)

Adding Noise to Circuits

with_noise Method

# Add noise to all operations
noisy_circuit = circuit.with_noise(cirq.depolarize(p=0.01))

# Simulate noisy circuit
simulator = cirq.DensityMatrixSimulator()
result = simulator.run(noisy_circuit, repetitions=1000)

insert_into_circuit Method

# Manual noise insertion
def add_noise_to_circuit(circuit, noise_model):
    noisy_moments = []
    for moment in circuit:
        ops = []
        for op in moment:
            ops.extend(noise_model.noisy_operation(op))
        noisy_moments.append(cirq.Moment(ops))
    return cirq.Circuit(noisy_moments)

Readout Noise

Measurement Error Model

class ReadoutNoiseModel(cirq.NoiseModel):
    """Model readout/measurement errors."""

    def __init__(self, p0_given_1, p1_given_0):
        # p0_given_1: Probability of measuring 0 when state is 1
        # p1_given_0: Probability of measuring 1 when state is 0
        self.p0_given_1 = p0_given_1
        self.p1_given_0 = p1_given_0

    def noisy_operation(self, op):
        if isinstance(op.gate, cirq.MeasurementGate):
            # Apply bit flip before measurement
            noise_ops = []
            for qubit in op.qubits:
                # Average readout error
                p_error = (self.p0_given_1 + self.p1_given_0) / 2
                noise_ops.append(cirq.bit_flip(p_error)(qubit))
            noise_ops.append(op)
            return noise_ops
        return op

# Typical readout errors
readout_noise = ReadoutNoiseModel(p0_given_1=0.02, p1_given_0=0.01)

Noise Characterization

Randomized Benchmarking

import cirq

def generate_rb_circuit(qubits, depth):
    """Generate randomized benchmarking circuit."""
    # Random Clifford gates
    clifford_gates = [cirq.X, cirq.Y, cirq.Z, cirq.H, cirq.S]

    circuit = cirq.Circuit()
    for _ in range(depth):
        for qubit in qubits:
            gate = np.random.choice(clifford_gates)
            circuit.append(gate(qubit))

    # Add inverse to return to initial state (ideally)
    # (simplified - proper RB requires tracking full sequence)

    circuit.append(cirq.measure(*qubits, key='result'))
    return circuit

# Run RB experiment
def run_rb_experiment(qubits, depths, repetitions=1000):
    """Run randomized benchmarking at various depths."""
    simulator = cirq.DensityMatrixSimulator(
        noise=cirq.ConstantQubitNoiseModel(cirq.depolarize(0.01))
    )

    survival_probs = []
    for depth in depths:
        circuits = [generate_rb_circuit(qubits, depth) for _ in range(20)]

        total_survival = 0
        for circuit in circuits:
            result = simulator.run(circuit, repetitions=repetitions)
            # Calculate survival probability (returned to |0⟩)
            counts = result.histogram(key='result')
            survival = counts.get(0, 0) / repetitions
            total_survival += survival

        avg_survival = total_survival / len(circuits)
        survival_probs.append(avg_survival)

    return survival_probs

# Fit to extract error rate
# p_survival = A * p^depth + B
# Error per gate ≈ (1 - p) / 2

Cross-Entropy Benchmarking (XEB)

def xeb_fidelity(circuit, simulator, ideal_probs, repetitions=10000):
    """Calculate XEB fidelity."""

    # Run noisy simulation
    result = simulator.run(circuit, repetitions=repetitions)
    measured_probs = result.histogram(key='result')

    # Normalize
    for key in measured_probs:
        measured_probs[key] /= repetitions

    # Calculate cross-entropy
    cross_entropy = 0
    for bitstring, prob in measured_probs.items():
        if bitstring in ideal_probs:
            cross_entropy += prob * np.log2(ideal_probs[bitstring])

    # Convert to fidelity
    n_qubits = len(circuit.all_qubits())
    fidelity = (2**n_qubits * cross_entropy + 1) / (2**n_qubits - 1)

    return fidelity

Noise Visualization

Heatmap Visualization

import matplotlib.pyplot as plt

def plot_noise_heatmap(device, noise_metric):
    """Plot noise characteristics across 2D grid device."""

    # Get device qubits (assuming GridQubit)
    qubits = sorted(device.metadata.qubit_set)
    rows = max(q.row for q in qubits) + 1
    cols = max(q.col for q in qubits) + 1

    # Create heatmap data
    heatmap = np.full((rows, cols), np.nan)

    for qubit in qubits:
        if isinstance(qubit, cirq.GridQubit):
            value = noise_metric.get(qubit, 0)
            heatmap[qubit.row, qubit.col] = value

    # Plot
    plt.figure(figsize=(10, 8))
    plt.imshow(heatmap, cmap='RdYlGn_r', interpolation='nearest')
    plt.colorbar(label='Error Rate')
    plt.title('Qubit Error Rates')
    plt.xlabel('Column')
    plt.ylabel('Row')
    plt.show()

# Example usage
noise_metric = {q: np.random.random() * 0.01 for q in device.metadata.qubit_set}
plot_noise_heatmap(device, noise_metric)

Gate Fidelity Visualization

def plot_gate_fidelities(calibration_data):
    """Plot single- and two-qubit gate fidelities."""

    sq_fidelities = []
    tq_fidelities = []

    for qubit, metrics in calibration_data.items():
        if 'single_qubit_rb_fidelity' in metrics:
            sq_fidelities.append(metrics['single_qubit_rb_fidelity'])
        if 'two_qubit_rb_fidelity' in metrics:
            tq_fidelities.append(metrics['two_qubit_rb_fidelity'])

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

    ax1.hist(sq_fidelities, bins=20)
    ax1.set_xlabel('Single-Qubit Gate Fidelity')
    ax1.set_ylabel('Count')
    ax1.set_title('Single-Qubit Gate Fidelities')

    ax2.hist(tq_fidelities, bins=20)
    ax2.set_xlabel('Two-Qubit Gate Fidelity')
    ax2.set_ylabel('Count')
    ax2.set_title('Two-Qubit Gate Fidelities')

    plt.tight_layout()
    plt.show()

Error Mitigation Techniques

Zero-Noise Extrapolation

def zero_noise_extrapolation(circuit, noise_levels, simulator):
    """Extrapolate to zero noise limit."""

    expectation_values = []

    for noise_level in noise_levels:
        # Scale noise
        noisy_circuit = circuit.with_noise(
            cirq.depolarize(p=noise_level)
        )

        # Measure expectation
        result = simulator.simulate(noisy_circuit)
        # ... calculate expectation value
        exp_val = calculate_expectation(result)
        expectation_values.append(exp_val)

    # Extrapolate to zero noise
    from scipy.optimize import curve_fit

    def exponential_fit(x, a, b, c):
        return a * np.exp(-b * x) + c

    popt, _ = curve_fit(exponential_fit, noise_levels, expectation_values)
    zero_noise_value = popt[2]

    return zero_noise_value

Probabilistic Error Cancellation

def quasi_probability_decomposition(noisy_gate, ideal_gate, noise_model):
    """Decompose noisy gate into quasi-probability distribution."""

    # Decompose noisy gate as: N = ideal + error
    # Invert: ideal = (N - error) / (1 - error_rate)

    # This creates a quasi-probability distribution
    # (some probabilities may be negative)

    # Implementation depends on specific noise model
    pass

Readout Error Mitigation

def mitigate_readout_errors(results, confusion_matrix):
    """Apply readout error mitigation using confusion matrix."""

    # Invert confusion matrix
    inv_confusion = np.linalg.inv(confusion_matrix)

    # Get measured counts
    counts = results.histogram(key='result')

    # Convert to probability vector
    total_counts = sum(counts.values())
    measured_probs = np.array([counts.get(i, 0) / total_counts
                               for i in range(len(confusion_matrix))])

    # Apply inverse
    corrected_probs = inv_confusion @ measured_probs

    # Convert back to counts
    corrected_counts = {i: int(p * total_counts)
                       for i, p in enumerate(corrected_probs) if p > 0}

    return corrected_counts

Hardware-Based Noise Models

From Google Calibration

import cirq_google

# Get calibration data
processor = cirq_google.get_engine().get_processor('weber')
noise_props = processor.get_device_specification()

# Create noise model from calibration
noise_model = cirq_google.NoiseModelFromGoogleNoiseProperties(noise_props)

# Simulate with realistic noise
simulator = cirq.DensityMatrixSimulator(noise=noise_model)
result = simulator.run(circuit, repetitions=1000)

Best Practices

1. Use density matrix simulator for noisy simulations: State vector simulators cannot model mixed states
2. Match noise model to hardware: Use calibration data when available
3. Include all error sources: Gate errors, decoherence, readout errors
4. Characterize before mitigating: Understand noise before applying mitigation
5. Consider error propagation: Noise compounds through circuit depth
6. Use appropriate benchmarking: RB for gate errors, XEB for full circuit fidelity
7. Visualize noise patterns: Identify problematic qubits and gates
8. Apply targeted mitigation: Focus on dominant error sources
9. Validate mitigation: Verify that mitigation improves results
10. Keep circuits shallow: Minimize noise accumulation