Need explanation on how the pauli preparation basis works

[KangHaiYue]

I was looking at PauliPreparationBasis for ProcessTomography. However, it is unclear to me how those bases will eventually be used to construct the channel matrix elements (i.e. for arbitrary basis |n><m|). Would anyone be able to add the citation to the paper where this comes from or explain that in the document?

[wshanks]

The preparation basis is somewhat described in the Standard Quantum Process Tomography section of this review by Mohsenl et al 2007. It is just the simplest set of preparations to perform, at least for superconducting qubits -- ground, excited, and then two orthogonal states on the equator of the Bloch sphere.

I am not sure at which level you are asking about how the preparation states are used. The tomography code generally follows the maximum likelihood estimation approach described in the Wikipedia article on quantum tomography. Using the linearly independent state preparations and measurements, the code sets up a large optimization problem to solve for the matrix that best matches the measurement data.

[KangHaiYue]

The preparation basis is somewhat described in the Standard Quantum Process Tomography section of this review by Mohsenl et al 2007.

Thank you Will for providing this information. This is exactly what I was confused about: According to this paper, |m><n| is constructed from |m><m|, |n><n|, |+><+|, and |-><-|. The former two are pretty clear as they can be prepared using 'Zp' and 'Zm' from the screenshot of the table above. However, |+>=(|m>+|n>)/sqrt(2) is an entangled state for many qubits (and so does for |->), which is not equivalent to n copies of |+> for single-qubit. But from that table, it seems the preparation is only local (Xp and Yp). Moreover, I think they probably missed an 'i' before |-><-| in the paper: |m><n| = |+><+|+i|-><-|-(1+i)/2*(|m><m|+|n><n|).

[wshanks]

Interesting. I see your point about many qubits. I had only been thinking about the single qubit case. The PauliPreparationBasis table only shows the case for a single qubit. When there are multiple qubits, those preparations are performed on each qubit independently, so for two qubits there will be 16 preparations corresponding to indices 0-3 on each qubit. I think this can be seen in TomographyExperiment._basis_indices here:

qiskit-experiments/qiskit_experiments/library/tomography/tomography_experiment.py

Lines 249 to 251 in 519bb53

	if self._prep_circ_basis:
	prep_shape = self._prep_circ_basis.index_shape(self._prep_physical_qubits)
	prep_elements = product(*[range(i) for i in prep_shape])

which is called by the circuits() method. I suppose one could define a basis that returned a shape that was not tuple(2 for _ in range(len(num_qubits))) but the only current definition of index_shape is in the LocalPreparationBasis class:

qiskit-experiments/qiskit_experiments/library/tomography/basis/local_basis.py

Lines 105 to 106 in 519bb53

	def index_shape(self, qubits: Sequence[int]) -> Tuple[int, ...]:
	return len(qubits) * (self._size,)