ProcessTomography sometimes computes the target channel incorrectly

[chriseclectic]

Informations

• Qiskit Experiments version: 0.2
• Python version: All
• Operating system: All

What is the current behavior?

The process tomography measurement computes the target state for analysis incorrectly when not all qubits are prepared and measured, and the prepared and measured qubits differ. This results in the wrong value being returned for the "process_fidelity" analysis result since it is comparing the fit to the incorrect channel resulting in a low fidelity even for a a correct fit.

Steps to reproduce the problem

Run an experiment like

qc = QuantumCircuit(3)
qc.h(0)
qc.x(1)
qc.s(2)

qptexp = tomo.ProcessTomography(
    qc, preparation_qubits=[0], measurement_qubits=[1])
qptdata = qptexp.run(AerSimulator(), shots=10000).block_for_results()

The fitted channel is close to the true target

      [[0.,  0., 0.,  0.],
       [0.,  1., 0.,  0.],
       [0., 0.,  0., 0.],
       [0.,  0., 0.,  1.]])

However the automatically calculated target is the completely depolarizing channel

     [[ 0.5,  0. , -0. ,  0. ],
       [ 0. ,  0.5,  0. , -0. ],
       [-0. ,  0. ,  0.5,  0. ],
       [ 0. , -0. ,  0. ,  0.5]]

Resulting in an incorrectly reported process fidelity of 0.5

What is the expected behavior?

The target channel should be computed correctly for circuits with no measurements in them

Suggested solutions

This error is due to how the reduced target channel is calculated via partial trace of the full Choi matrix. This partial tracing is only correct for measurement indices, preparation indices should be projected onto the |0><0| state (the initial state of the circuit for those qubits not being prepared via tomography) rather than being traced over which is equivalent to evolving the maximumally mixed initial state I/2.