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 ...
Adam Zalcman's user avatar
  • 22.3k
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 ...
Markus Heinrich's user avatar
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 ...
Adam Zalcman's user avatar
  • 22.3k
8 votes
Accepted

How can the depolarizing channel be a 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 ...
Danylo Y's user avatar
  • 7,279
6 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 ...
Adam Zalcman's user avatar
  • 22.3k
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. ...
ChrisD's user avatar
  • 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\...
keisuke.akira's user avatar
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 ...
Markus Heinrich's user avatar
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\...
HerrWarum's user avatar
  • 131
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 ...
siserman's user avatar
  • 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 ...
Adam Zalcman's user avatar
  • 22.3k
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-...
DaftWullie's user avatar
  • 58.1k
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 ...
forky40's user avatar
  • 6,783
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 ...
BẢO BẠCH GIA's user avatar
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} ...
forky40's user avatar
  • 6,783
2 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 $2\times 2$...
Adam Zalcman's user avatar
  • 22.3k
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\...
narip's user avatar
  • 2,984
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's user avatar
  • 24.9k
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 ...
Craig Gidney's user avatar
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's user avatar
  • 24.9k
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 ...
Claudio Sanavio's user avatar
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 ...
siserman's user avatar
  • 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 ...
forky40's user avatar
  • 6,783
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'...
DaftWullie's user avatar
  • 58.1k
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: ...
Egretta.Thula's user avatar
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/...
Yaron Jarach's user avatar
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 ...
narip's user avatar
  • 2,984
1 vote

Prove the invariance upon change of variables in the definition of twirled channel

If $W=UV^\dagger$ then $U=WV$. This means that you can rewrite $$ U\rho U^\dagger=W(V\rho V^\dagger)W^\dagger. $$ So, let me write $\tilde\rho=V\rho V^\dagger$. Your integral becomes $$ V\mathcal{E}_T(...
DaftWullie's user avatar
  • 58.1k
1 vote

What do quantum noise models have in common?

In this paper, the authors use $$ \Lambda_1^{depol}(\rho_1)=(1-\frac{4}{3}\epsilon_1)\rho_1+\frac{1}{3}\epsilon_1\sum_{i=0}^{3}\sigma_i\rho_1\sigma_i$$ instead of $$ \Lambda_1^{depol}(\rho_1)=(1-\...
Yuchen Guo's user avatar
1 vote

Why use conjugate not transpose complex conjugate in superoperator?

This is just the standard way that you write down the superoperator. Think about the following: If I have $$ \rho\mapsto \sigma\rho \sigma^\dagger $$ then if I want to write this as a superoperator, I ...
DaftWullie's user avatar
  • 58.1k

Only top scored, non community-wiki answers of a minimum length are eligible