Quantum Computing Course Overview

Jun 9, 2024

Quantum Computing Course Overview

Instructor Introduction

  • Michael from Quantum Sore created the course
  • Aimed to provide a solid foundation without relying on analogies

Course Structure

  1. Basic Mathematics
    • Complex numbers
    • Basic Linear Algebra
  2. Mechanics of Quantum Computers
    • Qubits and their mathematical representation
    • Single and multiple qubit operations
    • Quantum entanglement and phase kickback
  3. Quantum Algorithms
    • Analyzing popular quantum algorithms
    • Demonstrating quantum computational power

Essential Mathematics Required

Imaginary and Complex Numbers

  • Imaginary Numbers: Numbers involving the square root of -1, represented as i
    • √-4 = ±2i
  • Complex Numbers: Combination of real and imaginary numbers, represented as a + bi.
    • Magnitude calculated using Pythagoras' theorem: √(a^2 + b^2)
    • Polar form: r(cosθ + i*sinθ)
    • Exponential form: r * e^(iθ)

Basic Linear Algebra

  • Matrix Operations
    • Addition/Subtraction: Add/Subtract corresponding elements
    • Scalar Multiplication: Each element multiplied by the scalar
    • Matrix Multiplication: Dot product of rows and columns
  • Vector Representation
    • Column Vectors for state representation
    • Identity and Inverse Matrices
    • Complex Conjugate: Flip the sign of imaginary parts
    • Transpose: Swap rows and columns
    • Unitary Matrices: Maintain vector length (UU† = I)
    • Hermitian Matrices: Equal to their own conjugate transpose (H = H†)

Quantum Mechanics in Computing

Qubits

  • Physical Representation: Quantum particles (e.g., photons) in two states
  • Mathematical Representation: Column vectors |0⟩ = [1,0], |1⟩ = [0,1]
  • Superposition: Qubit being in both states simultaneously
  • Measurement: Collapses qubit into |0⟩ or |1⟩ with certain probabilities
    • Probability of state |0⟩ ≈ |α|^2 and |1⟩ ≈ |β|^2 for a state α|0⟩ + β|1⟩
    • Probabilities must sum to 1
  • Dirac Notation: Summing matrices with scalar factors for state simplification
  • Bloch Sphere Representation: Visualizing qubits' state and phase

Quantum Gates and Circuits

  • Single-Qubit Gates
    • X Gate: Flips |0⟩ and |1⟩
    • Y Gate: Rotates 180° around the y-axis
    • Z Gate: Rotates 180° around the z-axis
  • Multi-Qubit Gates
    • CNOT Gate: Applies X gate if control qubit is |1⟩
    • Toffoli Gate: Applies X gate with two control qubits
  • Circuit Diagram Representation: Visual depiction of operations on qubits
  • Special States on Bloch Sphere: Plus, Minus, I, and -I states

Phase and Entanglement

  • Relative Phase: Impact on computational power
  • Entanglement: Measurement of one qubit affects the other
    • Maximally Entangled States: Bell States
    • Partially Entangled States: Affect probabilities of measurement
  • Phase Kickback: Used in quantum algorithms

Quantum Algorithms

Superdense Coding

  • Enables transmission of 2 classical bits using 1 qubit with pre-shared entanglement

Deutsch's Algorithm

  • Determines if a function is constant or balanced with one query
    • Uses Hadamard and Uf gates
    • Measurement indicates function type based on final qubit state

Deutsch-Jozsa Algorithm

  • Generalized version of Deutsch's algorithm for functions accepting n-bit inputs
    • Uses Hadamards to create superposition states

Bernstein-Vazirani Algorithm

  • Finds a secret string S using phase Oracle
    • Queries function once to determine secret string

Shor's Algorithm (Overview)

  • Factorizes large numbers by finding period of modular exponentiation
    • Utilizes Quantum Fourier Transform and classical post-processing

Quantum Fourier Transform

  • Encodes numbers into phase on Bloch sphere
    • Representation with phases instead of binary

Quantum Phase Estimation

  • Estimates phase to find eigenvalue of an eigenvector
    • Central algorithm component in Shor's and other algorithms

Summary

  • This course provides a solid foundational understanding
  • Emphasis on mathematical accuracy and fundamental principles without analogies