# How to prepare a random 1-qubit superposition for data encoding

Let's assume we have a normalized data vector $$\vec{x}= [x_1,x_2]$$. How can I prepare a state $$|\psi\rangle = x_1|0\rangle+x_2|1\rangle$$ for any $$\vec{x}$$. I know that this state is in general not reachable through a single rotation, but is there a systematic way of finding a sequence of unitaries (let's call it $$U_s$$ here), that performs a transformation, such that $$U_s|0\rangle=|\psi\rangle$$?

Thanks!

## 1 Answer

Theorem: Suppose $$U$$ is a unitary operation on a single qubit. Then there exist real numbers $$\alpha, \beta, \gamma, \delta$$ such that $$U = e^{i\alpha} R_z(\beta) R_y(\gamma)R_z(\delta)$$

This is on page 175 of this textbook.

Ignore the global phase, each of these gates can be implement directly on a quantum computer. So given a vector $$\vec{x}$$, you just work-out what $$U$$ needs to be, then find the appropriate angles $$\alpha, \beta, \gamma, \delta$$ . To find $$U$$ that takes $$|0\rangle$$ to $$\vec{x}$$, note that $$U$$ must take the form $$\begin{pmatrix} x_1 & u_{12} \\ x_2 & u_{22} \end{pmatrix}$$. To get the second column, pick some arbitrary vector that is independent from $$\vec{x}$$, then perform the the Gram-Schmidt process.

If you use Qiskit, you can initialize your state directly. For example:

from qiskit.quantum_info import random_state
from qiskit import QuantumCircuit, execute, Aer, IBMQ
provider = IBMQ.load_account()

num_qubits = 1
random_initial_state = random_state(2**num_qubits) #you can place specific vector here
circuits = []
circuit = QuantumCircuit(num_qubits,num_qubits)
circuit.initialize(random_initial_state, 0)

• +1 Note that you can avoid Gram-Schmidt and just set $u_{12} = \overline{x_2}$ and $u_{22} = -\overline{x_1}$. Substitute into dot product to see how this works. – Adam Zalcman Mar 14 at 8:37
• @AdamZalcman Thanks for the comment. You made a good point. That will indeed create a unitary matrix. I was just trying to give a general procedure at how one can systematically do such thing. – KAJ226 Mar 14 at 16:07