Skip to main content
11 votes
Accepted

What is the relationship between Choi and Chi matrix in Qiskit?

( I copied some text from a previous answer of mine) Defining the Choi and $\chi$ matrix The Choi matrix is a direct result of the Choi-Jamiolkowski isomorphism. Some intuition on what this is can be ...
JSdJ's user avatar
  • 5,539
10 votes
Accepted

How to perform Quantum Process Tomography for three qubit gates?

I am sure that since you are asking this question you probably already understand this, but for future & other's reference let me give a quick recap of what we are trying to achieve. Quantum ...
JSdJ's user avatar
  • 5,539
5 votes
Accepted

How can I fit an unknown quantum channel?

The 2-norm difference typically isn't particularly physical. So no, this is most likely not the right distance. What you want from a physical point of view is a distance measure which measures the ...
Norbert Schuch's user avatar
4 votes
Accepted

How to find the Kraus operators from the process matrix?

There might be a better way to do this (directly converting to Kraus operators from process matrix), but my suggestion would be to convert from the $\chi$ process matrix to the Choi matrix $\mathcal{C}...
Bebotron's user avatar
  • 425
4 votes

How to compute the unitary from the $\chi$ matrix obtained from QPT

Not all quantum operations are unitary. A more general type of quantum operation is the completely positive trace-preserving (CPTP) linear map which is the main subject of chapter $8$ of Nielsen & ...
Adam Zalcman's user avatar
3 votes
Accepted

Alternatives to process tomography and gate set tomography

I'm one of the devs on the pyGSTi team (one of the main software packages for designing and analyzing gate set tomography experiments). 2-qubit GST can certainly be computationally expensive to ...
Corey Ostrove's user avatar
3 votes
Accepted

In quantum process tomography, how does $\chi$ characterize a quantum process?

The linear map $\mathcal{E}$ is what characterizes a quantum process, $$\rho \rightarrow \mathcal{E}(\rho),$$ but $\mathcal{E}$ can be determined by $\chi$. Using the operator-sum representation, $$\...
ryanhill1's user avatar
  • 2,503
3 votes
Accepted

Can Gate Set Tomography work on Quantum Channels?

"Can gate set tomography be applied to a quantum channel" Yes, because gates are just unitary channels, if you wanted you could just let a qubit idle and undergo decay/dephasing processes ...
chrysaor4's user avatar
  • 1,386
3 votes

How to compute the unitary from the $\chi$ matrix obtained from QPT

As @AdamZalcman has pointed out, the $\chi$ matrix represents a (more general than unitary) Quantum channel. If you were trying to implement a unitary operation, your channel might be close to a ...
JSdJ's user avatar
  • 5,539
3 votes
Accepted

Is a process matrix of rank $1$ unique?

TL;DR: The elements of the process matrix with respect to an operator basis $E_i$ are just the coefficients in the expansion of the channel, viewed as an operator on $\mathcal{H}\otimes\mathcal{H}$, ...
Adam Zalcman's user avatar
2 votes
Accepted

How to measure a general two-qubits gate? Does it help to Bob and Alice?

Long story short: if you want to characterize a circuit or operation, you have to perform Quantum process tomography (QPT), which is a generalization of quantum state tomography, which is used to ...
JSdJ's user avatar
  • 5,539
2 votes

How to describe the evolution of a density matrix using the Choi matrix?

You want to calculate $$ \rho_{out}=2\text{Tr}_0(\rho_{in}^T\otimes I\cdot\rho^{sys}_{choi}). $$
DaftWullie's user avatar
  • 59.2k
2 votes
Accepted

Quantum process tomography, non-trace preserving

Yes, I think you have the correct idea. I'll just add some additional details. As you suggest, "non-trace-preserving" means that the map of interest is applied with some probability which (...
Jacob's user avatar
  • 635
2 votes
Accepted

Can you perform quantum process tomography using an orthonormal basis the contains non Hermitian matrices?

Considering just the single qubit case, the four possible operators you list are \begin{align} |0\rangle\langle 0|,|0\rangle\langle 1|,|1\rangle\langle 0|,|1\rangle\langle 1| \end{align} and like you ...
chrysaor4's user avatar
  • 1,386
2 votes

Does any quantum channel satisfy ${\rm Tr}(\mathcal E^\dagger \mathcal E) \in[0, d^2]$?

Note that the bound $\mathrm{Tr}(\mathcal E^\dagger \mathcal E) \geq 0$ is trivial since $\mathrm{Tr}(\mathcal E^\dagger \mathcal E) = \| \mathcal E \|_2^2$ is the square of the Schatten 2-norm of $\...
Markus Heinrich's user avatar
2 votes

Why does the $\chi$ matrix have $d^4-d^2$ independent parameters?

First, note that for a $k\times k$ Hermitian matrix, there are $k^2$ parameters. This is because there are $k$ diagonal elements, which must be real, and $\frac12 k(k-1)$ complex elements (i.e. 2 real ...
DaftWullie's user avatar
  • 59.2k
2 votes
Accepted

Is tomography of the Choi state sufficient for channel tomography?

Sure, process tomography is intimately related to state tomography, and one way to see it is via the Choi isomorphism, as you point out. See also eg the discussion in section IV of (Mohseni et al. ...
glS's user avatar
  • 25.5k
2 votes

Is there a tight operator frame that is also a POVM?

Tight frames in general A general set of vectors $v_k\in V$ is a frame if there are constants $A,B\in\mathbb{R}$ such that $$A\|v\|^2 \le \sum_k |\langle v_k,v\rangle|^2 \le B \|v\|^2$$ for all $v\in ...
glS's user avatar
  • 25.5k
2 votes
Accepted

I have two Choi matrix I suspect be equivalent. Can I manipulate them?

There is a one-to-one relation between Choi states$^1$ and quantum channels (Choi-Jamiołkowski isomorphism). So, you cannot transform the Choi matrix without changing the channel it represents. If the ...
FDGod's user avatar
  • 2,401
2 votes

When should I use the Choi matrix and when should I use the $\chi$ matrix?

The two descriptions are entirely equivalent. It doesn't matter which you use when, it's just a case of using whichever description you personally find to be mathematically the most convenient.
DaftWullie's user avatar
  • 59.2k
2 votes
Accepted

Process matrix of CNOT gate

Process matrix definition Recall that the process matrix of a channel $\mathcal{E}(\rho)=\sum_kK_k\rho K_k^\dagger$ with respect to an operator basis $A_i$ is obtained by expressing the Kraus ...
Adam Zalcman's user avatar
2 votes

Why is the Choi matrix different from the analytic form for a depolarizing channel?

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 $\...
Frederik vom Ende's user avatar
1 vote

When should I use the Choi matrix and when should I use the $\chi$ matrix?

This is just a comment, but it's too long for a comment, so writing as an answer. As I haven't read the paper you are asking about, I cannot answer as to particularly why that paper is using the ...
FDGod's user avatar
  • 2,401
1 vote

How can I compute the probability distribution by using channel through Choi matrix in quantum process tomography?

The Choi is related to the channel via $$\Lambda = \sum_{ij} (E_{ij}\otimes \mathcal E(E_{ij})) = (I\otimes \mathcal E)\mathbb{P}_+,$$ using the shorthand notation $E_{ij}\equiv |i\rangle\!\langle j|$ ...
glS's user avatar
  • 25.5k
1 vote
Accepted

How to calculate the Pauli Noise model of a physical gate operation?

In general, you cannot represent an arbitrary unitary operation in terms of a Pauli noise model (if you could, then you would be able to simulate it efficiently on a classical computer!). However, if ...
jchadwick's user avatar
  • 451
1 vote
Accepted

Why does the $\chi$ matrix have $d^4-d^2$ independent parameters?

Consider the following observations: Let $V,W$ be finite-dimensional vector spaces. The set of linear functions $A:V\to W$, which I'll denote with $\operatorname{Lin}(V,W)$, is in itself a vector ...
glS's user avatar
  • 25.5k
1 vote

How is the $\beta$-matrix interpreted in single qubit QPT?

I am suggesting a way of indexing the 16 x 16 $\beta$ matrix, but I am not sure if it corresponds to your Eq(*) \begin{equation} \beta = \frac{1}{4} \left[ \begin{matrix} \beta_{11}^{00} & \beta_{...
Sachindra Kumar's user avatar
1 vote
Accepted

Why is $\chi$ not uniquely determined by $\sum_{mn}\beta_{jk}^{mn}\chi_{mn}=\lambda_{jk}$?

Because it's a linear system, and a linear system can in general have (infinitely) many solutions. If you want to find $x$ such that $Ax=b$ for some matrix $A$ and vector $b$, then the set of ...
glS's user avatar
  • 25.5k
1 vote
Accepted

How to derive the number of independent parameters in the $\chi$ matrix from the Choi matrix?

Counting the number of free parameters of Choi operators The number of independent parameters in the $\chi$ matrix is identical to the number of independent parameters in the Choi matrix, or in the ...
glS's user avatar
  • 25.5k
1 vote

How to describe the evolution of a density matrix using the Choi matrix?

Let $\Phi$ be a channel acting on a state $\rho$ (or more generally, a map acting on a linear operator; we don't actually need restrict to CPTP maps and states for these calculations). Let $J(\Phi)$ ...
glS's user avatar
  • 25.5k

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