This series will summarize some key points during my Quantum Computing learning journey, mainly from IBM Qiskit Textbook and some other useful resources.

Let's start with Qubits - the basic variable of quantum computers.

Bits to Qubits

Classical states for computation are either 0 or 1

• Bits: only two characters, 0 and 1, we can represent any piece of information.

• Qubits: an extension of the bit to quantum mechanics, a state can be in superposition (i.e., simultaneously in 0 and 1)

  ⇒ Superposition allows performing the calculation on many states at the same time.

  ⇒ Some Quantum algorithms with exponential speed-up

However, once we measure the superposition state, it collapses to one of its states (0 or 1) → it is not that easy to design quantum algorithms, but we can use interference effects

🔮 Qubits

• a quantum variant of the bit
• have exactly the same restrictions as normal bits do: they can store only a single binary piece of information and can only ever give us an output of 0 or 1.
• However, they can also be manipulated in ways that quantum mechanics can only describe.

Before finding out how to represent a qubit, let have a look at the Dirac (bra-ket) notation:

Dirac notation

Used to describe quantum states: \( a, b \in \mathbb{C}^2 \)

• ket: \( |a\rangle = \begin{bmatrix} a_1 \ a_2 \end{bmatrix} \)
• bra: \( \langle b| = |b\rangle ^\dagger = \begin{bmatrix}b_1 \\ b_2 \end{bmatrix} ^\dagger = \begin{bmatrix} b_1^* & b_2^* \end{bmatrix} \)
  where \(b_1^*, b_2^*\) are complex conjugated of \(b_1 , b_2 \), i.e. \(b = c+d.i \Rightarrow b^* = c-d.i \)
• bra-ket (inner product): \( \langle b|a\rangle = a_1b_1^* + a_2b_2^* = \langle a|b\rangle^\dagger \in \mathbb{C} \)
• ket-bra: \( |a\rangle \langle b| = \begin{bmatrix} a_1b_1^* & a_1b_2^* \\ a_2b_1^* & a_2b_2^*\end{bmatrix} \)

Qubit notation

Using 2 orthogonal vectors (the inner product = 0)

$$|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \, \, \, \, |1\rangle =\begin{bmatrix} 0 \\ 1 \end{bmatrix}$$

\( |0\rangle, |1\rangle\) are orthogonal, i.e, $$\langle 0|1\rangle = \begin{bmatrix} 1 & 0 \end{bmatrix}.\begin{bmatrix}0 \\1\end{bmatrix} = 1.0 + 0.1 =0$$

This helps us distinguish them from things like the bit values 0 and 1 or the numbers 0 and 1. It is part of the bra-ket notation, introduced by Dirac.

With vectors, we can describe more complex states than just \(|0\rangle\) and \(|1\rangle\). For example

$$ \begin{aligned} |q_0\rangle & = \tfrac{1}{\sqrt{2}}|0\rangle + \tfrac{i}{\sqrt{2}}|1\rangle \\ & = \tfrac{1}{\sqrt{2}}\begin{bmatrix}1\\0\end{bmatrix} + \tfrac{i}{\sqrt{2}}\begin{bmatrix}0\\1\end{bmatrix}\\ & = \begin{bmatrix}\tfrac{1}{\sqrt{2}}\\0\end{bmatrix} + \begin{bmatrix}0\\\tfrac{i}{\sqrt{2}}\end{bmatrix}\\ & = \begin{bmatrix}\tfrac{1}{\sqrt{2}} \\ \tfrac{i}{\sqrt{2}} \end{bmatrix}\\ \end{aligned} $$

Qubit Statevector

• Since the states \(|0\rangle \) and \(|1\rangle\) form an orthonormal basis, we can represent any 2D vector with a combination of these two states. This allows us to write the state of our qubit in the alternative form:

  \( |q_0\rangle = \tfrac{1}{\sqrt{2}}|0\rangle + \tfrac{i}{\sqrt{2}}|1\rangle\) is a qubit's statevector,

• Qubit Statevector it is not entirely \(|0\rangle\) and is not entirely \(|1\rangle\) ⇒ it is described by a linear combination of the two: 'superposition'

Qubit Measurement

We choose orthogonal bases to describe and measure quantum states. During a measurement onto the bases \(|0\rangle, |1\rangle\), the states will collapse into either state \(|0\rangle\) or \(|1\rangle \) → As those are the eigenstates of \(\sigma_Z\) we call this a Z-measurement

There are infinitely many different bases, but other common ones are:

\(|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\) and \(|-\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle)\) corresponding to the eigenstates of \(\sigma_X\)

\(|+i\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)\) and \(|-i\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)\) corresponding to the eigenstates of \(\sigma_Y\)

Born rule

The probability that a state \(|\psi\rangle\) collapse during a projective measure onto the basis \({|x\rangle, |x\rangle^\dagger}\) to the state \(|x\rangle\) is given by:

• Find the probability of measuring a state \(|\psi \rangle\) in a state \(|x \rangle\):

  \(p(|x\rangle) = | \langle x| \psi \rangle|^2\) and \(\sum\limits_{i} p(x_i) = 1\)

This rule governs how we get information out of quantum states. It is therefore very important for everything we do in quantum computation.

General notation of single-qubit states

Single qubit states can be written down generally as

$$\sqrt{1-p}\vert0\rangle + e^{i\phi}\sqrt{p}\vert1\rangle$$

• \(p\) is the probability that a measurement of the state in the computational basis \({|0\rangle, |1\rangle}\) will have the outcome \(1\)
• \(\phi\) is the phase between the two computational basis states

Single-qubit gates can then be used to manipulate this quantum state by changing either \(p , \phi\) or both.

Reference

• Qiskit Textbook ( https://qiskit.org/textbook/)