11
votes
Accepted
Inverting the depolarizing channel
The existence of the inverse of a linear map is independent of the way the map affects the trace. Moreover, if an invertible map preserves a property then its inverse necessarily also preserves the ...
- 24.2k
10
votes
Why does the twirl of a quantum channel give a depolarizing channel?
I hope you do not mind if I zoom out a bit and talk about representation theory. I think a more general approach helps understanding the essential bits and will be helpful if you encounter similar ...
- 5,317
8
votes
Accepted
Why does the twirl of a quantum channel give a depolarizing channel?
Nielsen's paper cited in the question simplifies the arguments originally laid out in two papers by Horodecki family. This answer sketches the original arguments and is meant to complement the nice ...
- 24.2k
8
votes
Accepted
How can the depolarizing channel, defined as $\mathcal E(\rho) = (1-p)\rho + p\frac{I}{2}$, be a linear quantum operation?
The correct linear form of the depolarizing channel is
$$
\varepsilon(\rho) = (1-p)\rho + p\frac{I}{2}{\rm Tr}(\rho).
$$
For density matrices ${\rm Tr}(\rho)=1$, so you can usually see the form ...
- 7,612
7
votes
Accepted
What is the adjoint of the depolarizing channel?
TL;DR: $\Phi^*=\Phi$.
If quantum channel $\Psi:\mathcal{X}\to\mathcal{Y}$ has a Kraus representation $\Psi(X)=\sum_iK_iXK_i^\dagger$ then its adjoint $\Psi^*:\mathcal{Y}\to\mathcal{X}$ has a Kraus ...
- 24.2k
6
votes
Accepted
Decoding the Steane Code
The error correction protocol for any CSS code (such as the Steane code) is described in e.g. Section 10.4.2 of Nielsen & Chuang, although it's much easier to understand in the stabilizer picture. ...
- 316
4
votes
Accepted
Depolarization of density operator with zeros in diagonal
Quantum channels are foremost, linear operators. So given a basis for the Hilbert-Schmidt operator space (for example the states $\{|0\rangle\langle 0|,|0\rangle\langle 1|,|1\rangle\langle 0|,|1\...
- 1,853
4
votes
Accepted
Is the "unitary twirling operation" physically realizable?
Question 1: I guess it depends what your understanding of "physical" is. In my understanding, everything you can do in the lab is physical. Thus, twirling is perfectly physical. Note that ...
- 5,317
4
votes
Accepted
What is the operator-sum representation of the two-qubit depolarizing channel?
As pointed out in the comments, you cannot use the one-qubit formula because something like $X\rho X$ does not make sense if $\rho$ is a 2-qubit state. In fact, for this reason the answer you based ...
- 3,003
3
votes
Deriving the choi matrix definition of the quantum depolarizing channel
For $\mathcal E(X)=p\mathrm{tr}(X)\,\frac{\mathbb I}{2}+(1-p)X$, then
$$\begin{align}
\sigma &= (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle \Omega|)\\
&= \sum_{ij} \mathcal E(|i\...
- 171
3
votes
Accepted
How do I apply depolarization error on 2 qubits using Qiskit?
If you like to add a 2-qubit error to CNOT gates, you have to create a depolarizing error with err = depolarizing_error(param, num_qubits=2) which you may add to ...
- 161
3
votes
Accepted
Depolarizing channel for $n$ qubits: why is there a trace term?
There are situations where it is profitable or convenient to think about quantum channels as defined on the space of all linear operators on a given Hilbert space, not just density matrices. In this ...
- 24.2k
3
votes
Accepted
Two qubit state + Depolarizing channel = Bell diagonal state?
Firstly, note that every Bell state $|\psi_{ij}\rangle=(|0i\rangle+(-1)^j|1\bar i\rangle)/\sqrt{2}$ is an eigenstate of $E_i\otimes E_i$ for all $i$ (the eigenvalues are either $\pm 1$). Hence, a Bell-...
- 60.8k
3
votes
Accepted
How to find the operator-sum representation of the depolarizing channel?
Consider the map $\mathcal{E}: L_H \to L_H$ given by
$$
\mathcal{E}(\rho) = \frac{1}{4} (\rho + X\rho X + Y\rho Y + Z\rho Z)
$$
where $L_H$ denotes the four-dimensional real vector space of $2\times 2$...
- 24.2k
3
votes
Accepted
What is an algorithm to generate random error in depolarizing channel
I will assume you want a symmetric depolarizing channel that acts globally on an $n$-qubit system and applies one of $4^n-1$ nontrivial Paulis with equal probability. Note that we can generalize and ...
- 7,448
3
votes
Why is the Choi matrix I get for the depolarizing channel $\mathcal{E}(\rho) = (1 - p) \rho + p \frac{I}{2}$ different from what it shold be?
Your definition of the depolarizing channel is slightly off, it should read
$$
\mathcal E(\rho)=(1-p)\rho+p\,\boxed{{\rm tr}(\rho)}\frac I2\,.
$$
This extra term ${\rm tr}(\rho)$ is necessary for $\...
- 3,003
3
votes
Accepted
Can a quantum operation inflate the Bloch sphere?
I'll try to answer questions 1. and 2.
Your construction is indeed not a quantum channel, because the projective measurements that are being executed are dependent on the dominant eigenvector (...
- 482
2
votes
Qiskit noise model question (from textbook)
As CX gate is a two-qubit gate, therefore, the size of the matrix which represents the two-qubit gate is 4x4. That's the reason why you have to use a tensor product with two error_gate1 (2x2) to ...
- 189
2
votes
Inverting the depolarizing channel
Ah, the channel is trace preserving so its straightforward to invert in this case. Let $Y = \mathcal{D}_p (X)$ so that
\begin{align}
\text{Tr}(Y) &= p\text{Tr}(X) + (1-p) \frac{\text{Tr}(X)}{2^n} ...
- 7,448
2
votes
Why does the twirl of a quantum channel give a depolarizing channel?
I will give an extended explanation of Nielsen's proof, i.e. your first ref link.
The idea is that, $\rho=\sum_ip_i|i\rangle\langle i|$, we can prove it's depolarizing channel for each $|i\rangle\...
- 3,019
2
votes
Accepted
How to calculate the action of a channel on part of a quantum state?
you can absolutely do the calculation in this comment. That is, you can just compute
$$(I\otimes \mathcal E)\rho = \sum_{ij} (I\otimes \mathcal E)(|i\rangle\!\langle j|\otimes \sigma_{ij})
= \sum_{ij} ...
glS♦
- 26.4k
2
votes
Given a list of stabilizers (or parity check matrix), find an encoding circuit
The first thing you'll want to do is to convert the raw matrix bits into objects Stim understands and can operate efficiently upon. Stim has a method ...
- 40.8k
2
votes
Does applying a random Pauli matrix to a density matrix result in the identity?
You can generalize the question as follows: given a set of unitaries $X\subset\mathbf U(d)$ and weights $p_U$ normalized as $\sum_{U\in X}p_U=1$, when do we have
$$\sum_{U\in X} p_U U\rho U^\dagger=\...
glS♦
- 26.4k
2
votes
What is an instruction when we `add_all_qubit_quantum_error`?
It depends if you want to apply the same error to any two-qubits gate you use in your circuit or not.
This answer explains it very well
How to selectively apply noise in Qiskit simulations?
If you ...
2
votes
What is an instruction when we `add_all_qubit_quantum_error`?
This depends on your circuit. If you have only 1-qubit gates, you can construct a noise model from single-qubit errors. But if you want to do something interesting, you need to entangle your qubits ...
- 161
2
votes
What do quantum noise models have in common?
Operator sum representation
Quantum channels may be represented in an operator-sum form: For a channel $\Lambda$ acting on a $d$-dimensional state $\rho$, there exists some set of $d$-dimensional ...
- 7,448
2
votes
Accepted
Why does depolarising channel shrinks Bloch vectors?
It doesn't really make much sense to do this calculation just with an input state $|0\rangle$. So, instead, I'll replace $|0\rangle$ with $|\psi\rangle$.
If you calculate $\text{Tr}_\Omega\rho$, you'...
- 60.8k
1
vote
Accepted
How to see the effects of the depolarizing channel in qiskit?
You should start noticing the effect of depolarizing error with the current circuit. However, the issue with your code is that you are using X in uppercase in this line:
...
- 11.5k
1
vote
Accepted
Diamond norm distances between some channel and the identity
You can find some useful results in these links: https://quantum-journal.org/papers/q-2021-08-09-522/, https://arxiv.org/abs/1004.4110v1, https://doi.org/10.1088/1367-2630/ab8e5c, https://arxiv.org/...
- 734
1
vote
Recover the noisy POVMs of Bell basis measurement
To set up the problem and the notations, let's first analyze why your figure corresponds to projective measurement on four Bell basis. Let's use $U_C$ stands for CNOT gate and $H$ stands for Hadamard ...
- 3,019
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