Noise

Table of Contents

• Brief
  • Introduction to Noise Models
  • Noise Error

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Brief

In practical quantum computers, computational errors are inevitable due to the physical characteristics of qubits. To better simulate these errors in a quantum virtual machine, QPanda3 introduces various noise models and error models, making the simulation process closer to real quantum computers. These noise and error models allow for fine-tuned adjustments of qubits, quantum logic gates, and error parameters, satisfying a variety of complex noise scenarios.

Introduction to Noise Models

• amplitude_damping_error

The amplitude_damping_error represents the relaxation noise model of a qubit. Its Kraus operators are defined as follows:

\[K_1 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1 - p} \end{bmatrix}, \quad K_2 = \begin{bmatrix} 0 & \sqrt{p} \\ 0 & 0 \end{bmatrix} \]

This model requires a single noise parameter.

• pauli_z_error

The pauli_z_error represents the dephasing noise model of a qubit. Its Kraus operators are defined as follows:

\[K_1 = \begin{bmatrix} \sqrt{1 - p} & 0 \\ 0 & \sqrt{1 - p} \end{bmatrix}, \quad K_2 = \begin{bmatrix} \sqrt{p} & 0 \\ 0 & -\sqrt{p} \end{bmatrix} \]

This model requires a single noise parameter.

• decoherence_error

The decoherence_error combines the above two models (relaxation and dephasing noise). Their relationship is described as follows:

\[P_{\text{damping}} = 1 - e^{-\frac{t_{\text{gate}}}{T_1}}, \quad P_{\text{dephasing}} = 0.5 \times \big(1 - e^{-(\frac{t_{\text{gate}}}{T_2} - \frac{t_{\text{gate}}}{2T_1})}\big) \]

The combined Kraus operators are:

\[K_1 = K_{1_{\text{damping}}} K_{1_{\text{dephasing}}}, \quad K_2 = K_{1_{\text{damping}}} K_{2_{\text{dephasing}}}, \quad K_3 = K_{2_{\text{damping}}} K_{1_{\text{dephasing}}}, \quad K_4 = K_{2_{\text{damping}}} K_{2_{\text{dephasing}}} \]

This model requires three noise parameters.

• depolarizing_error

The depolarizing_error represents depolarizing noise, where a single qubit may collapse into a completely mixed state (I/2) with a certain probability. The Kraus operators are:

\[K_1 = \sqrt{1 - 3p/4} \cdot I, \quad K_2 = \sqrt{p}/2 \cdot X, \quad K_3 = \sqrt{p}/2 \cdot Y, \quad K_4 = \sqrt{p}/2 \cdot Z \]

Here, \(I, X, Y, Z \) represent the quantum logic gate matrices.

This model requires a single noise parameter.

• pauli_x_error

The pauli_x_error represents pauli_x_error noise. Its Kraus operators are defined as follows:

\[K_1 = \begin{bmatrix} \sqrt{1 - p} & 0 \\ 0 & \sqrt{1 - p} \end{bmatrix}, \quad K_2 = \begin{bmatrix} 0 & \sqrt{p} \\ \sqrt{p} & 0 \end{bmatrix} \]

This model requires a single noise parameter.

• pauli_y_error

The pauli_y_error represents bit-phase flip noise. Its Kraus operators are:

\[K_1 = \begin{bmatrix} \sqrt{1 - p} & 0 \\ 0 & \sqrt{1 - p} \end{bmatrix}, \quad K_2 = \begin{bmatrix} 0 & -i \times \sqrt{p} \\ i \times \sqrt{p} & 0 \end{bmatrix} \]

This model requires a single noise parameter.

• phase_damping_error

The phase_damping_error represents phase damping noise. Its Kraus operators are:

\[K_1 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1 - p} \end{bmatrix}, \quad K_2 = \begin{bmatrix} 0 & 0 \\ 0 & \sqrt{p} \end{bmatrix} \]

This model requires a single noise parameter.

Noise Error

The noise errors and noise models in pyqpanda3 are separated. Firstly, we need to construct the noise errors, and then set them through the noise model. In the morning, the model can be set to support which quantum bits and specific quantum logic gates

import os
import sys
 
from pyqpanda3.core import *
 
# pauli x error , There is a probability of 0.5 that flipping qubit
x_error = pauli_x_error(0.5)
 
# pauli x error , There is a probability of 0.5 that an additional Y gate will be applied
y_error = pauli_y_error(0.5)
 
# pauli z error , There is a probability of 0.5 that an additional Z gate will be applied
z_error = pauli_z_error(0.5)
 
# kraus Error
amplitude_damping_error = amplitude_damping_error(0.5)
decoherence_error = decoherence_error(5, 5, 0.2)
depolarizing_error = depolarizing_error(0.5)

The NoiseModel can be used to set noise errors, including specific applications to quantum bits and quantum logic gates

from pyqpanda3.core import QCircuit,H,RX,CNOT,QProg,CPUQVM,measure
from pyqpanda3.core import NoiseModel,pauli_x_error,depolarizing_error,GateType
 
circuit = QCircuit(2)
circuit << H(0)
circuit << CNOT(0, 1)
circuit << RX(0, 3.6)
 
prog = QProg()
prog << circuit
prog << measure([0,1,2],[0,1,2])
 
model = NoiseModel()
 
# Set noise to apply to all qubits
model.add_all_qubit_quantum_error(pauli_x_error(0.5),GateType.RX)
model.add_all_qubit_quantum_error(pauli_x_error(0.9),GateType.H)
 
# Set noise to apply to select qubits
model.add_quantum_error(depolarizing_error(0.5),GateType.H,[0,1,2])
 
machine = CPUQVM()
machine.run(prog, 1000, model)
 
print(machine.result().get_counts())

The output is as follows.

{'001': 468, '010': 479, '011': 24, '000': 29}