# Implementing noise model in Quantum simulator

I have noticed that a few papers, such as this one, provide models two for typical noise that a qubit encounters during network transmission:

1. Λ is the dephasing channel
Λ(ρ) = Pq(ρ) = qρ + (1 − q)ZρZ
where ρ is a single qubit state, Z is the Pauli Z gate, and q ∈ [0, 1] is the noise parameter.

2. Λ is the depolarizing channel
Λ(ρ) = Dq(ρ) = qρ + (1 − q)1/2
where ρ is a single qubit state, 1/2 is a maximally mixed single-qubit state, and q ∈ [0, 1] is the noise parameter.

I would like to implement this in the SimulaQron program, but I don't really understand how to calculate the effect on the qubit. What exactly do I have to do to the qubit to apply this noise?

Nor do I understand the value of q. My interpretation is that q would occasionally be 1, and occasionally be 0, to vary the effect on the qubit. However, if I change q probabilistically, doesn't that eliminate the point of modelling the noise?

It is important to realize that the depolarizing and dephasing channel (and pretty much any other noise model for that matter) do not represent unitary operations. This means there is no unitary operation (that takes qubit states to qubit states) corresponding to these channels.

Rather, channels are more general: they map density operators to density operators. If you are not familiar with density operators (often denoted with $$\rho$$), I strongly advice you to first familiarize yourself with the concept. They are a framework that allows for the (more general than 'normal' or* pure states, denoted with the ket notation:$$|\psi\rangle$$) statistical mixtures: a statistical mixture of actual pure qubit states. The population of the various pure states determines the density matrix.

A typical noise channel creates statistical noise: therefore you need the density operator framework to describe these channels. As far as I am aware, the simulaqron software was not designed to account for density operators - it only works with pure states. This limits the possibility of simulating a quantum channel as a noise process significantly.

The only thing that would be possible tom simulate the dephasing channel is to gather statistics by repeating the same circuit over and over, but sometimes injecting a Z error (with probability $$1-q$$). (*see bottom note)

For the depolarizing channel, a different but equivalent definition of this channel is:

$$\begin{equation} \Lambda_{depo}(\rho) = (1-p)\rho + \frac{p}{3} X\rho X+ \frac{p}{3} Y\rho Y+ \frac{p}{3} Z\rho Z, \end{equation}$$ so you could apply a random $$X$$,$$Y$$ or $$Z$$ with equal probability of $$\frac{p}{3}$$, instead of only the $$Z$$ operation as with the dephasing channel.

• Note that it is customary to write the dephasing channel as $$\Lambda_{deph}(\rho) = (1-p)\rho + p Z\rho Z$$, so with the probabilities flipped compared to your definition.