Different statevectors with different quantity of complex numbers for same single-qubit states in Qiskit

Question

I have several questions, but before ask, I want to write some theoretical. As we know, we can represent any single-qubit quantum state by the next representation:

$$ |\psi\rangle=c_0|0\rangle+c_1|1\rangle, $$ using two numbers, $c_0$ and $c_1$, called probability amlitudes.

We can put these probability amplitudes to vector-column:

$$ \begin{bmatrix} c_0 \\ c_1 \\ \end{bmatrix} $$

In other side, we can represent any single-qubit quantum state next:

$\theta$ and $\phi$ represents next angles in Bloch sphere:

As we can see, enough only one complex number to represent single-qubit quantum state, using $|0\rangle$ and $|1\rangle$ base. In this case $c_0 = \cos{\frac{\theta}{2}}$ and $c_1 = e^{i\phi}\sin{\frac{\theta}{2}}$. By watching Bloch Sphere and using probability amplitudes we can write 6 the most usual quantum states:

$|0\rangle = \cos{\frac{0}{2}}|0\rangle + e^{i0}\sin{\frac{0}{2}}|1\rangle = 1|0\rangle+0|1\rangle=\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$

$|1\rangle = \cos{\frac{\pi}{2}}|0\rangle + e^{i0}\sin{\frac{\pi}{2}}|1\rangle = 0|0\rangle+1|1\rangle=\begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}$

$|+\rangle = \cos{\frac{\pi}{4}}|0\rangle + e^{i0}\sin{\frac{\pi}{4}}|1\rangle = \frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle=\begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \\ \end{bmatrix}$

$|-\rangle = \cos{\frac{\pi}{4}}|0\rangle + e^{i\pi}\sin{\frac{\pi}{4}}|1\rangle = \frac{1}{\sqrt{2}}|0\rangle+(-1)*\frac{1}{\sqrt{2}}|1\rangle=\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle=\begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ \end{bmatrix}$

$|+i\rangle = \cos{\frac{\pi}{4}}|0\rangle + e^{i\frac{\pi}{2}}\sin{\frac{\pi}{4}}|1\rangle = \frac{1}{\sqrt{2}}|0\rangle+i*\frac{1}{\sqrt{2}}|1\rangle=\begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{i}{\sqrt{2}} \\ \end{bmatrix}$

$|-i\rangle = \cos{\frac{\pi}{4}}|0\rangle + e^{i\frac{3\pi}{2}}\sin{\frac{\pi}{4}}|1\rangle = \frac{1}{\sqrt{2}}|0\rangle+(-i)*\frac{1}{\sqrt{2}}|1\rangle=\frac{1}{\sqrt{2}}|0\rangle-i*\frac{1}{\sqrt{2}}|1\rangle=\begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{i}{\sqrt{2}} \\ \end{bmatrix}$

Let me now use Qiskit. We can use U3 universal matrix to create any single-qubit quantum state. For example, if I use

qc = QuantumCircuit(1)
qc.u3(pi/2,pi/2,0,0)

we will have $|+i\rangle$ state with next statevector:

(1)

However, if we do next code:

qc = QuantumCircuit(1)
qc.h(0)
qc.rz(pi/2,0)

we will have next statevector:

(2)

As I understand, here is used global phase. But how can I connect these two statevectors with each other? And why it's difference between each other? Honestly, I had experience only with single-qubit representation, when we use only one complex number, and $i$ can be only in second coefficient of Jones vector. But when I see two complex numbers - I am little bit confused.

Additionally, if we start from $|+\rangle$ state and rotate state around Z for $\frac{\pi}{4}$ angle, we can do it by using "rc" gate:

qc = QuantumCircuit(1)
qc.h(0)
qc.rz(pi/4,0)

and we will have next result:

Or we will use next gate:

qc = QuantumCircuit(1)
qc.h(0)
qc.u1(pi/4,0)

and we will have next result:

Again, in first case we will have two complecs numbers, in second case only one, but states are same.

Could you, please, explain these differences?

Thank you!

Answer 1

As you say, the difference is in the global phase. Let me explain using the first of your examples, $$ \left[\begin{array}{cc} \frac{1-i}{2} & \frac{1+i}{2} \end{array}\right]. $$ Mathematically, this is the same as $$ \left[\begin{array}{cc} \frac{e^{-i\pi/4}}{\sqrt{2}} & \frac{e^{i\pi/4}}{\sqrt{2}} \end{array}\right]=\left[\begin{array}{cc} e^{-i\pi/4}\times\frac{1}{\sqrt{2}} & e^{-i\pi/4}\frac{e^{i\pi/2}}{\sqrt{2}} \end{array}\right]=e^{-i\pi/4}\times\left[\begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{e^{i\pi/2}}{\sqrt{2}} \end{array}\right]=e^{-i\pi/4}\times\left[\begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} \end{array}\right]. $$ So, if you neglect the global phase factor (any factor multiplying the whole state, with modulus 1), you see that $$ \left[\begin{array}{cc} \frac{1-i}{2} & \frac{1+i}{2} \end{array}\right]. $$ and $$ \left[\begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} \end{array}\right] $$ are the same thing.