Quantum Error Mitigation

Introduction to Quantum Error Mitigation

Quantum error mitigation is a set of techniques used to reduce the impact of errors in quantum computations without the need for full quantum error correction. These techniques are essential for improving the reliability of quantum computations on near-term quantum devices, which are prone to various types of noise and errors.

Techniques for Error Mitigation

Zero-Noise Extrapolation (ZNE)

Zero-Noise Extrapolation is a technique that involves running the same quantum circuit at different noise levels and extrapolating the results to estimate the outcome at zero noise.

Probabilistic Error Cancellation (PEC)

Probabilistic Error Cancellation is a technique that involves applying a series of operations to cancel out the effects of noise, effectively reducing the overall error in the computation.

Other Techniques

• Richardson Extrapolation: A specific form of zero-noise extrapolation that uses a polynomial fit to estimate the zero-noise result.
• Symmetry Verification: A technique that leverages the symmetry properties of quantum states to detect and correct errors.

Example Implementation of Error Mitigation Techniques using Qiskit

Zero-Noise Extrapolation Implementation

from qiskit import QuantumCircuit, Aer, transpile, assemble
from qiskit.providers.aer.noise import NoiseModel, depolarizing_error
from qiskit.visualization import plot_histogram
import numpy as np

# Define a simple quantum circuit
qc = QuantumCircuit(1)
qc.h(0)
qc.measure_all()

# Create a noise model with depolarizing error
noise_model = NoiseModel()
error = depolarizing_error(0.1, 1)
noise_model.add_all_qubit_quantum_error(error, ['h'])

# Simulate the circuit with noise
simulator = Aer.get_backend('qasm_simulator')
compiled_circuit = transpile(qc, simulator)
qobj = assemble(compiled_circuit, shots=1000)
result = simulator.run(qobj, noise_model=noise_model).result()
counts = result.get_counts()

# Plot the noisy results
plot_histogram(counts, title="Noisy Results")

# Perform Zero-Noise Extrapolation
# (This is a simplified example; in practice, multiple noise levels would be used)
noisy_results = [counts]
noise_levels = [0.1]

# Extrapolate to zero noise
zero_noise_result = np.mean([counts for counts in noisy_results], axis=0)

# Plot the extrapolated results
plot_histogram(zero_noise_result, title="Zero-Noise Extrapolated Results")

Probabilistic Error Cancellation Implementation

from qiskit import QuantumCircuit, Aer, transpile, assemble
from qiskit.providers.aer.noise import NoiseModel, depolarizing_error
from qiskit.visualization import plot_histogram
import numpy as np

# Define a simple quantum circuit
qc = QuantumCircuit(1)
qc.h(0)
qc.measure_all()

# Create a noise model with depolarizing error
noise_model = NoiseModel()
error = depolarizing_error(0.1, 1)
noise_model.add_all_qubit_quantum_error(error, ['h'])

# Simulate the circuit with noise
simulator = Aer.get_backend('qasm_simulator')
compiled_circuit = transpile(qc, simulator)
qobj = assemble(compiled_circuit, shots=1000)
result = simulator.run(qobj, noise_model=noise_model).result()
counts = result.get_counts()

# Plot the noisy results
plot_histogram(counts, title="Noisy Results")

# Perform Probabilistic Error Cancellation
# (This is a simplified example; in practice, more sophisticated techniques would be used)
corrected_counts = {key: val * (1 / (1 - 2 * 0.1)) for key, val in counts.items()}

# Plot the corrected results
plot_histogram(corrected_counts, title="Probabilistic Error Cancellation Results")

Importance of Error Mitigation in Near-Term Quantum Devices

Quantum error mitigation is crucial for enhancing the reliability of quantum computations on near-term quantum devices. These devices are prone to various types of noise and errors, which can significantly impact the accuracy of quantum computations. By applying error mitigation techniques, we can improve the fidelity of quantum computations, enabling practical applications with current quantum hardware.

Conclusion

In this notebook, we have explored the fundamental concepts of quantum error mitigation, including various techniques for mitigating errors in quantum computations and their implementation using Qiskit. Understanding quantum error mitigation is crucial for improving the reliability of quantum computations on near-term quantum devices. As quantum computing technology continues to advance, these techniques will play a key role in ensuring the accuracy and reliability of quantum computations.