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In qiskit, how can I initialise a qubit in a complex state, specifically in the state: $$\left|q\right> = \frac{1}{\sqrt2} \left|0\right> + i \frac{1}{\sqrt2} \left|1\right>$$

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You can generate a general quantum state $|\psi\rangle$ in Qiskit by using their Custom package.

from qiskit.aqua.components.initial_states import Custom 
import math 
state_vector = [1, 1j]
Psi = Custom(1,state_vector = state_vector).construct_circuit()
Psi.draw()

Running the above code in Qiskit would give you something like below:

enter image description here


But also notice that your state $|q\rangle = \dfrac{1}{\sqrt{2}}\big( |0\rangle + i | 1\rangle \big) $ is actually resulting from

$$ \dfrac{Z + Y}{\sqrt{2}} |0\rangle = |q\rangle $$

where $Z$ and $Y$ are the Pauli matrices and $|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $.

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You can use initialize function. Here is a code constructing your one qubit state:

from qiskit import ClassicalRegister, QuantumRegister, QuantumCircuit, Aer, execute
import math as m

quantumState = [
    1 / m.sqrt(2) * complex(1, 0),
    1 / m.sqrt(2) * complex(0, 1)]

q = QuantumRegister(1, name = 'q')
c = ClassicalRegister(1, name = 'c')

circuit = QuantumCircuit(q,c)

circuit.initialize(quantumState, [q[0]])

You can prepare multiqubit states as well, for example for $|\psi\rangle = \frac{1}{2}(|00\rangle + |01\rangle+|10\rangle+|11\rangle)$:

from qiskit import ClassicalRegister, QuantumRegister, QuantumCircuit, Aer, execute
import math as m

quantumState = [0.5,0.5,0.5,0.5]

q = QuantumRegister(2, name = 'q')
c = ClassicalRegister(2, name = 'c')

circuit = QuantumCircuit(q,c)

circuit.initialize(quantumState, [q[0],q[1]])
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