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

I'm struggling with the framework for quantum process tomography on Qiskit.

The final step of such a framework is running fit method of ProcessTomographyFitter class. Documentation states that such function gives a Choi matrix as output. Nevertheless, I'd want the Chi matrix to define the superoperator of a circuit. Specifically, I'm interested in understanding how a 2-qubit circuit affects a single qubit.

Thus, my questions are:

• What is the relationship between Choi and Chi matrix?
• When do they coincide?
• How to obtain Chi from Choi matrix?
• Does this answer your question? How to perform Quantum Process Tomography for three qubit gates? – glS May 3 at 14:20
• I feel that that question is not a complete answer to this question, as it does not describe how to obtain the chi matrix from the Choi matrix or when they are the same. – JSdJ May 4 at 7:11
• So do I. Anyway it answers partially. Thus, thanks a lot! – Daniele Cuomo May 6 at 13:51

( 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 found in this previous answer. Consider the maximally entangled state $$|\Omega \rangle = \sum_{\mathrm{i}}|\mathrm{i}\rangle \otimes |\mathrm{i}\rangle$$, where $$\{|\mathrm{i}\rangle\}$$ forms a basis for the space on which $$\rho$$ acts. (Note that we thus have a maximally entangled state of twice as many qubits). The Choi matrix is the state that we get when on one of these subsystems $$\Lambda$$ is applied (leaving the other subsystem intact): $$\begin{equation} \rho_{\mathrm{Choi}} = \big(\Lambda \otimes I\big) |\Omega\rangle\langle\Omega|. \end{equation}$$ As the Choi matrix is a state, it must be positive semidefinite (corresonding the the CP constraint) and must be unity trace (corresponding to the TP constraint).

The process- or $$\chi$$-matrix comes from the fact that we can write our map as a double sum: $$\begin{equation} \Lambda(\rho) = \sum_{m,n} \chi_{mn}P_{m}\rho P_{n}^{\dagger}, \end{equation}$$ where $$\{P_{m}\}$$ & $$\{P_{n}\}$$ form a basis for the space of density matrices; we use the Pauli basis $$\{I,X,Y,Z\}^{\otimes n}$$ (thereby omitting the need for the $$\dagger$$ at $$P_{n}$$). The matrix $$\chi$$ now encapsulates all information of $$\Lambda$$; the CP constraint reads that $$\chi$$ must be positive semidefinite, and the trace constraint reads that $$\sum_{m,n}\chi_{mn}P_{n}P_{m} \leq I$$ (with equality for TP).

## Computing one from another

From this, we get the following two identities: $$\begin{equation} \begin{split} \rho_{\mathrm{Choi}} &= \sum_{m,n} \chi_{m,n} |P_{m}\rangle\rangle\langle\langle P_{n}|, \\ \chi_{m,n} &= \langle\langle P_{m} | \rho_{\mathrm{Choi}} |P_{n}\rangle\rangle, \end{split} \end{equation}$$ where $$|P_{m}\rangle\rangle$$ is the 'vectorized' version of $$P_{m}$$, which is essentially just the columns of $$P_{m}$$ stacked on top of each other, giving a vector. That answers question 3.

Again I shamelessly 'self-promote': in the first appendix of my thesis I work through proofs of all these relations. The most intuitive way is by using the Kraus decomposition as an intermediary, but it is not needed.

## Relationship between the two

From this, you can see that the Choi matrix and the chi matrix do indeed have some relationship: The Choi matrix is the $$\chi$$ matrix, when the used based is not the Pauli basis but the Bell basis. That, I believe, answers question 1 & 2.

• Thank you! What I believe is missing now is how all this math behind QPT and Quantum Channels is applied in Qiskit. The Qiskit's method works with Pauli basis, and it returns a Choi-matrix, but it seems corresponding to a Chi-matrix. Speaking more in programming language, the method returns a Choi object that is a 4x4 matrix (since I'm interested in the superoperator from one qubit to one qubit for a 2-qubit circuit). From documentation of the Chi object we have that Chi-matrix "is related to the Choi representation by a change of basis of the Choi-matrix into the Pauli basis". – Daniele Cuomo May 8 at 12:06
• You're welcome!I am not 100 percent sure what is the problem here anymore. The fit method indeed returns a Choi matrix (at least that is what it did when I used it ~1.5 years ago). If you want to have the chi matrix, you can just calculate it yourself using the equations I provided. Or, if you check the source code for the chi matrix class (qiskit.org/documentation/_modules/qiskit/quantum_info/operators/…), you can pass the Choi matrix to the init function as the 'data' variable; thereby obtaining the chi matrix for that channel. – JSdJ May 8 at 12:19
• That's what I thought too. But if I do QPT over a basic circuit with, for example, just an X-gate, I obtain a certain Choi-matrix. Treating that matrix as it were a Chi-matrix, we can see that it transforms quantum states coherently with the declared circuit. If, instead, I do what you just suggested, the resulting superoperator leads to unexpected results. That's why I think that the intial result is the Chi-matrix, but I'm still not sure about the reason. P.S. your thesis is helpful, thank you! – Daniele Cuomo May 8 at 15:26