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Quantum Computing: A Complete Guide

by Dr. Eleanor Rieffel & Wolfgang Polak

Quantum Mechanics Primer

Quantum States

In quantum mechanics, the state of a system is described by a wave function . For a qubit, this can be written as:

where:

  • and are the basis states
  • and are complex amplitudes
  • (normalization condition)

Superposition

Superposition allows a qubit to be in multiple states simultaneously. For example:

This qubit has equal probability of being measured as 0 or 1.

Entanglement

Entanglement is a quantum phenomenon where qubits become correlated in ways that classical bits cannot. For two entangled qubits:

Measuring one qubit instantly determines the state of the other, regardless of distance.

Measurement and Observables

When we measure a quantum system:

  • The wave function collapses to one of the basis states
  • The probability of each outcome is given by the square of its amplitude
  • After measurement, the system remains in the measured state

Bloch Sphere Representation

A single qubit can be visualized on the Bloch sphere:

where:

  • is the polar angle (0 to )
  • is the azimuthal angle (0 to )