Projective measurement operation in Qiskit

Question

I would like to implement the operation $\pi = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ on qiskit but I don't know how to do that.

If I use the reset gate, but I use it for example on the first qubit of a Bell state $ |\psi\rangle = \frac{1}{\sqrt 2}(|00\rangle + |11\rangle)$, then the result will be $|00\rangle$ half of the times and $|01\rangle$ the other half (randomly). Instead I should obtain always $|00\rangle$.

If I use a Swap gate with an auxiliary qubit, my final state will be $ |\psi\rangle = \frac{1}{\sqrt 2}(|00\rangle + |01\rangle)$ instead of $|00\rangle$.

The Bell state is just an example, in general I need to work on an arbitrary state.

Thank you!

Answer 1

As narip commented, the matrix is not unitary and, therefore, not a valid "gate". Same applies to reset, so "reset gate" sounds a bit like an oxymoron to me.

If you just want to create an operator, you can:

import qiskit.quantum_info as qi
op = qi.Operator([[1, 0],
                  [0, 0]])

Notice that,

> op.is_unitary()
False

Answer 2

I have thought of a way to solve this problem.

If I have a QuantumCircuit "qc", I can obtain the quantum state of that circuit by using

statevector = np.array(Statevector(qc))

then I can use numpy.matmul to perform a matrix multiplication between this state and the projector that I want to apply. After that, I just normalized the state dividing it by its norm.

If I have a multi-qubit state and I want to perform the projector only on one qubit, I can use the numpy.kron function to perform the tensor product between the projector and the identity matrix that I want to apply to the other qubits. Be careful in this process, the statevector of Qiskit gives the result using the reverse order of the qubits.

After this, I just need to initialize a new circuit with my projected quantum state

qc = QuantumCircuit(Number_of_qubits)
qc.initialize(list(statevector), qc.qubits)

This kind of operation can be useful to implement quasiprobability decomposition for example.