Noisy simulation
Digital Noise
In the description of closed quantum systems, the complete quantum state is a pure quantum state represented by a state vector $|\psi \rangle $.
However, this description is not sufficient to describe open quantum systems. When the system interacts with its surrounding environment, it transitions into a mixed state where quantum information is no longer entirely contained in a single state vector but is distributed probabilistically.
To address these more general cases, we consider a probabilistic combination \(p_i\) of possible pure states \(|\psi_i \rangle\). Thus, the system is described by a density matrix \(\rho\) defined as follows:
The transformations of the density operator of an open quantum system interacting with its noisy environment are represented by the super-operator \(S: \rho \rightarrow S(\rho)\), often referred to as a quantum channel. Quantum channels, due to the conservation of the probability distribution, must be CPTP (Completely Positive and Trace Preserving). Any CPTP super-operator can be written in the following form:
Where \(K_i\) are Kraus operators satisfying the closure property \(\sum_i K_i K^{\dagger}_i = \mathbb{I}\). As noise is the result of system interactions with its environment, it is therefore possible to simulate noisy quantum circuit with noise type gates.
Thus, horqrux implements a large selection of single qubit noise gates such as:
- The bit flip channel defined as: \(\textbf{BitFlip}(\rho) =(1-p) \rho + p X \rho X^{\dagger}\)
- The phase flip channel defined as: \(\textbf{PhaseFlip}(\rho) = (1-p) \rho + p Z \rho Z^{\dagger}\)
- The depolarizing channel defined as: \(\textbf{Depolarizing}(\rho) = (1-p) \rho + \frac{p}{3} (X \rho X^{\dagger} + Y \rho Y^{\dagger} + Z \rho Z^{\dagger})\)
- The pauli channel defined as: \(\textbf{PauliChannel}(\rho) = (1-p_x-p_y-p_z) \rho + p_x X \rho X^{\dagger} + p_y Y \rho Y^{\dagger} + p_z Z \rho Z^{\dagger}\)
- The amplitude damping channel defined as: \(\textbf{AmplitudeDamping}(\rho) = K_0 \rho K_0^{\dagger} + K_1 \rho K_1^{\dagger}\) with: \(\begin{equation*} K_{0} \ =\begin{pmatrix} 1 & 0\\ 0 & \sqrt{1-\ \gamma } \end{pmatrix} ,\ K_{1} \ =\begin{pmatrix} 0 & \sqrt{\ \gamma }\\ 0 & 0 \end{pmatrix} \end{equation*}\)
- The phase damping channel defined as: \(\textbf{PhaseDamping}(\rho) = K_0 \rho K_0^{\dagger} + K_1 \rho K_1^{\dagger}\) with: \(\begin{equation*} K_{0} \ =\begin{pmatrix} 1 & 0\\ 0 & \sqrt{1-\ \gamma } \end{pmatrix}, \ K_{1} \ =\begin{pmatrix} 0 & 0\\ 0 & \sqrt{\ \gamma } \end{pmatrix} \end{equation*}\)
- The generalize amplitude damping channel is defined as: \(\textbf{GeneralizedAmplitudeDamping}(\rho) = K_0 \rho K_0^{\dagger} + K_1 \rho K_1^{\dagger} + K_2 \rho K_2^{\dagger} + K_3 \rho K_3^{\dagger}\) with: \(\begin{cases} K_{0} \ =\sqrt{p} \ \begin{pmatrix} 1 & 0\\ 0 & \sqrt{1-\ \gamma } \end{pmatrix} ,\ K_{1} \ =\sqrt{p} \ \begin{pmatrix} 0 & 0\\ 0 & \sqrt{\ \gamma } \end{pmatrix} \\ K_{2} \ =\sqrt{1\ -p} \ \begin{pmatrix} \sqrt{1-\ \gamma } & 0\\ 0 & 1 \end{pmatrix} ,\ K_{3} \ =\sqrt{1-p} \ \begin{pmatrix} 0 & 0\\ \sqrt{\ \gamma } & 0 \end{pmatrix} \end{cases}\)
Noise protocols can be added to gates by instantiating DigitalNoiseInstance providing the DigitalNoiseType and the error_probability (either float or tuple of float):
from horqrux.noise import DigitalNoiseInstance, DigitalNoiseType
noise_prob = 0.3
AmpD = DigitalNoiseInstance(DigitalNoiseType.AMPLITUDE_DAMPING, error_probability=noise_prob)
Then a gate can be instantiated by providing a tuple of DigitalNoiseInstance instances. Let’s show this through the simulation of a realistic \(X\) gate.
For instance, an \(X\) gate flips the state of the qubit: \(X|0\rangle = |1\rangle\). In practice, it's common for the target qubit to stay in its original state after applying \(X\) due to the interactions between it and its environment. The possibility of failure can be represented by the BitFlip subclass of DigitalNoiseInstance, which flips the state again after the application of the \(X\) gate, returning it to its original state with a probability 1 - gate_fidelity.
from horqrux.api import sample
from horqrux.noise import DigitalNoiseInstance, DigitalNoiseType
from horqrux.utils import density_mat, product_state
from horqrux.primitive import X
noise = (DigitalNoiseInstance(DigitalNoiseType.BITFLIP, 0.1),)
ops = [X(0)]
noisy_ops = [X(0, noise=noise)]
state = product_state("0")
noiseless_samples = sample(state, ops)
noisy_samples = sample(density_mat(state), noisy_ops)