3
$\begingroup$

Could anyone please explain how do I do state tomography when using sampling (on real device or QASM) in Qiskit? I know there's a special method for this, but I could not find a working example.

More precisely, I would be interested in two cases: when the full state tomography is performed (all the $2^n$ amplitudes are measured) and when it's known apriori that the wave function has support in a certain subspace (say, only $|10\rangle$ and $|01\rangle$ for a two-qubit system).

BTW, is $3^n$ a lower bound on the number of operators which have to be measured in order to perform the state tomography? ($X$, $Y$, $Z$ for each qubit.)

Improve this question
$\endgroup$
2
  • $\begingroup$ I don't have time to write a proper answer right now on the entire question, but at least there used to be a wonderful qiskit tutorial notebook on QST (although I can't find it anymore). I'll try to answer later if there is no answer yet. $\endgroup$
    – JSdJ
    Aug 3, 2020 at 11:17
  • $\begingroup$ The $3^{n}$ bound is not a strict bound; in principle you could do with $4^{n}$ state populations (the $3^{n}$ operators of course give $6^{n}$ states) when you use a SIC-POVM - but in practice, it's just easier to use the $3^{n}$ Pauli operators. $\endgroup$
    – JSdJ
    Aug 3, 2020 at 11:19

1 Answer 1

Reset to default
4
$\begingroup$

I would suggest you use the code from the tutorial about quantum state tomography, adapting it to a real device of your choice. You can find the updated tutorial here

Caveat: as state tomography requires 3^n circuits, you will need probably to find a method of batch processing of these circuits if they exceed the job circuit limit of your real device. See the code here

“This performs measurement in the Pauli-basis resulting in :math:`3^n circuits for an n-qubit state tomography experiment.”

For an example of results of « full state tomography » on real devices (Melbourne and ibmqx4) for up to 5 qubits, I suggest you have a look at the end of my own qiskit tutorial here

For the exploration of a certain subspace on a real device, I have some doubt about the approach as noise will inevitability produce a result somewhere in the entire Hilbert space and not confined to the chosen subspace.

However, you may be interested by this recent paper

and by its presentation in Phys.org

I quote from this presentation written by Ingrid Fadelli

“By combining statistical learning and unitary t-design theory, the researchers were able to devise a rigorous and efficient procedure that allows classical machines to produce approximate classical descriptions of quantum many-body systems. These descriptions can be used to predict several properties of the quantum systems that are being studied by performing a minimal number of quantum measurements.”

So, you are surely right in proposing that full tomography can be replaced by alternate methods using a lower number of measurements.

For your last question about the 3^n bound, I see that JSdJ already answered to you.

Improve this answer
$\endgroup$
4
  • $\begingroup$ just fyi, you can add links typing e.g. [click here](http://www.google.com), which gives you click here $\endgroup$
    – glS ♦
    Aug 10, 2020 at 11:12
  • $\begingroup$ As a small sidenote, there is another reason why only considering a subspace during reconstruction is tricky, as not (ac)counting for projections on other subspaces as valid measurement outcomes results in essentially 'lost' measurements (i.e. you performed a measurement but did not record the outcome). Reconstructing for $\rho$ will then not only disregard the information regarding these subspaces, but will also not be trace-class, meaning that the trace of the state will not be $1$. $\endgroup$
    – JSdJ
    Aug 10, 2020 at 13:05
  • $\begingroup$ @gIS thanks for your remark. Accordingly I edited the text using mini-Markdown formatting. $\endgroup$
    – Pierre
    Aug 10, 2020 at 20:52
  • $\begingroup$ @JSdJ Thank for your sidenote which expresses more precisely the concerns about only considering a subspace. $\endgroup$
    – Pierre
    Aug 10, 2020 at 20:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.