Noise models · Quantum Mechanics

Intuition

Noise on qubits is described by a few standard channels. The bit flip applies $\sigma_{x}$ with probability $p$ ; the phase flip applies $\sigma_{z}$ with probability $p$ . Each keeps one axis of the Bloch ball and squeezes the other two by $1-2p$ . The depolarising channel applies $\sigma_{x}$ , $\sigma_{y}$ or $\sigma_{z}$ at random, which is the same as replacing the state by the completely mixed one, and shrinks the ball evenly. Amplitude damping pulls it towards the ground state; phase damping squeezes it onto the $z$ axis. Error-correcting codes are designed against these models, and hardware is compared by their rates.

A road can have ruts, crosswinds or ice, and each pushes a car in its own way; an engineer who knows which hazard dominates can design against it. Noise models are the typical hazards of qubits.

A cut through the Bloch ball, $x$ across and $z$ up, under the bit flip with $p=0.2$ : it keeps $r_{x}$ and squeezes by . The phase flip gives the same shape turned on its side, keeping and squeezing .

Standard noise channels of a qubit

With probability $p$ of an error:

\begin{aligned}\text{bit flip:}&\ \ \hat{\rho}\mapsto(1-p)\hat{\rho}+p\,\sigma_{x}\hat{\rho}\sigma_{x}\\ \text{phase flip:}&\ \ \hat{\rho}\mapsto(1-p)\hat{\rho}+p\,\sigma_{z}\hat{\rho}\sigma_{z}\\ \text{depolarising:}&\ \ \hat{\rho}\mapsto(1-p)\hat{\rho}+p\,\tfrac12\hat{1}\end{aligned}

Properties

Bit flip: $r_{x}$ kept, $r_{y}$ and $r_{z}$ multiplied by . Phase flip: kept, and multiplied by .

The depolarising channel from random Pauli errors

\Bigl(1-\tfrac{3p}{4}\Bigr)\hat{\rho}+\tfrac{p}{4}\sum_{i=x,y,z}\sigma_{i}\hat{\rho}\sigma_{i}=(1-p)\,\hat{\rho}+p\,\tfrac12\hat{1}

The sum of $\sigma_{i}\hat{\rho}\sigma_{i}$ over the three Pauli matrices is $2 \overset{}{1}$ for any density matrix, which follows from how each Pauli matrix flips the other two. Put it into the mixture of Pauli errors and collect terms.

Proof steps

\sigma_{i}\sigma_{j}\sigma_{i}=-\sigma_{j}\ (i\neq j),\qquad \sigma_{i}\sigma_{i}\sigma_{i}=\sigma_{i}

Pauli matrices anticommute and square to $\hat{1}$ .

\sum_{i}\sigma_{i}\hat{\rho}\sigma_{i}=2\,\hat{1}-\hat{\rho}\qquad\Bigl(\hat{\rho}=\tfrac12(\hat{1}+\mathbf{r}\cdot\vec{\sigma})\Bigr)

The identity part gains a factor 3; each $\sigma_{j}$ gets $+1-1-1=-1$ .

\Bigl(1-\tfrac{3p}{4}\Bigr)\hat{\rho}+\tfrac{p}{4}\bigl(2\,\hat{1}-\hat{\rho}\bigr)

Each Pauli error with probability $p/4$ , none with $1-3p/4$ .

=(1-p)\,\hat{\rho}+p\,\tfrac12\hat{1}

Collect the terms.

Applications

Practice

Bit Flip

The bit-flip channel applies $\sigma_{x}$ , which swaps $\ket{0}$ and $\ket{1}$ , with probability $p$ .

\hat{\rho}\mapsto(1-p)\hat{\rho}+p\,\sigma_{x}\hat{\rho}\sigma_{x}

Try it

A bit-flip channel with $p=0.1$ acts on $\ket{0}$ . What is the probability of then finding $\ket{1}$ ?