Qubits

Summary: The fundamental unit of quantum information, capable of existing in superpositions of 0 and 1, represented mathematically as vectors in a two-dimensional Hilbert space.

Sources: quantum-mechanics-basics.ipynb

1. Classical Bits vs. Quantum Qubits

A classical bit represents a switch that is either completely open (0) or closed (1). In contrast, a qubit (quantum bit) can represent 0, 1, or any superposition of both simultaneously. This enables quantum computers to explore massive computational state spaces in parallel.

2. Mathematical State Representation

We represent the state of a qubit using Dirac notation (also called bra-ket notation).

The Basis States

The two computational basis states are denoted as kets:

  • $\vert 0 \rangle$ (representing the ground state)
  • $\vert 1 \rangle$ (representing the excited state)

In vector form, these form an orthonormal basis: \(\vert 0 \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \vert 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\)

Superposition State

Any pure state of a single qubit, $\vert \psi \rangle$, is a linear combination of the basis states: \(\vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}\)

Here, $\alpha$ and $\beta$ are complex numbers representing probability amplitudes.

Normalization Condition

Since the total probability of measurement outcomes must equal 1, the state vector must be normalized: \(\vert \alpha \vert^2 + \vert \beta \vert^2 = 1\)

  • $\vert \alpha \vert^2$: Probability of measuring state $\vert 0 \rangle$.
  • $\vert \beta \vert^2$: Probability of measuring state $\vert 1 \rangle$.

3. The Bloch Sphere

The state of a single qubit can be visualized as a point on the surface of a three-dimensional unit sphere, known as the Bloch Sphere.

Any normalized qubit state can be written using spherical coordinates: \(\vert \psi \rangle = \cos\left(\frac{\theta}{2}\right) \vert 0 \rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right) \vert 1 \rangle\)

  • $\theta$ ($0 \le \theta \le \pi$): Determines the polar angle (latitude), controlling the balance of superposition.
    • $\theta = 0$ corresponds to the north pole: $\vert 0 \rangle$.
    • $\theta = \pi$ corresponds to the south pole: $\vert 1 \rangle$.
  • $\phi$ ($0 \le \phi \le 2\pi$): Determines the azimuthal angle (longitude), representing the relative phase of the qubit.

4. Measurement & State Collapse

Measurement is a destructive process in quantum mechanics:

  1. Superposition: Before measurement, the qubit is in a superposition $\alpha \vert 0 \rangle + \beta \vert 1 \rangle$.
  2. Collapse: Upon measurement, the qubit’s state collapses to either the classical state $\vert 0 \rangle$ (with probability $\vert\alpha\vert^2$) or $\vert 1 \rangle$ (with probability $\vert\beta\vert^2$).
  3. Repeated Sampling: Running a quantum circuit once only returns a single classical bit. To estimate the probability amplitudes $\alpha$ and $\beta$, we must execute the circuit multiple times (referred to as shots) and count the frequency of results.
  • [[quantum-gates]]
  • [[entanglement]]
  • [[ibm-quantum-learning]]