How to create a qubit in arbitrary state in qiskit

Question

A QuantumCircuit() command creates a qubit with zero state always.

I've seen some instructions about using Arbitrary Initialization but it is not accessible and returns 404 error.

Answer 1

Any one-qubit state can be described as (up to a global phase):

$$|\psi \rangle = \cos \left(\frac{\theta}{2} \right)|0\rangle + \sin \left(\frac{\theta}{2} \right) e^{i\varphi}|1\rangle$$

If we start from $|0\rangle$ state arbitrary quantum state can be generated (up to a global phase) with $R_y(\theta)$ and $R_z(\varphi)$ unitary operators (the order is important). Firstly we apply $R_y(\theta)$

$$R_y(\theta) |0\rangle = \cos \left( \frac{\theta}{2} \right)|0\rangle + \sin \left( \frac{\theta}{2}\right)|1\rangle$$

Then $R_z(\varphi)$

$$R_z(\varphi) \left(\cos\left(\frac{\theta}{2}\right)|0\rangle + \sin\left(\frac{\theta}{2} \right)|1\rangle \right) = \cos\left(\frac{\theta}{2}\right)e^{-i\frac{\varphi}{2}}|0\rangle + \sin\left(\frac{\theta}{2}\right)e^{i\frac{\varphi}{2}}|1\rangle$$

Disregarding the global phase we will obtain the following state:

$$|\psi \rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + \sin\left(\frac{\theta}{2}\right)e^{i\varphi}|1\rangle$$

That is an arbitrary state that we wanted to create. There are also other ways (gates) to this transformation.

The code will look like this:

circuit.ry(theta, qubit[0])
circuit.rz(phi, qubit[0])

It can also be done only with one $u3(\theta, \varphi, 0)$ gate:

$$u3(\theta, \varphi, 0) |0\rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + \sin\left(\frac{\theta}{2}\right)e^{i\varphi}|1\rangle$$

circuit.u3(theta, phi, 0, qubit[0])