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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?
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  • $\begingroup$ Does this answer your question? How to perform Quantum Process Tomography for three qubit gates? $\endgroup$ – glS May 3 at 14:20
  • $\begingroup$ 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. $\endgroup$ – JSdJ May 4 at 7:11
  • $\begingroup$ So do I. Anyway it answers partially. Thus, thanks a lot! $\endgroup$ – Daniele Cuomo May 6 at 13:51
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( 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.

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  • $\begingroup$ 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". $\endgroup$ – Daniele Cuomo May 8 at 12:06
  • $\begingroup$ 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. $\endgroup$ – JSdJ May 8 at 12:19
  • $\begingroup$ 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! $\endgroup$ – Daniele Cuomo May 8 at 15:26

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