Quantum Computing - Week 3, Day 1 Lecture

Introduction

• Lecturer: Mr. Joti Shwarenji
• Welcome to the third week of the one-month short-term quantum course.
• Upcoming guest: Professor Bishin Singh Puna from IIT Rudiki will join upcoming sessions to discuss physical consequences of quantum effects and physical qubits.

Announcements

• Tomorrow's quiz will cover only the previous week's topics.
• Slides are available on the GitHub repository.

Lecture Topics

• Review and consolidation of the past two weeks.
  • Realignment of ideas in preparation for learning a new quantum algorithm next week.

Quantum Algorithms Discussed

• Deutsch-Jozsa Algorithm
  • Covered previously.
• HHL Algorithm
  • To be discussed next week.

Quantum State and Bloch Sphere Representation

• Quantum state: represented as a column vector ( \begin{bmatrix} a \ b \end{bmatrix} ), where ( a ) and ( b ) are complex.
• Condition: (|a|^2 + |b|^2 = 1).
• Bloch Sphere:
  • Represents a single qubit state as a point on a unit sphere.
  • Qubit state can be expressed as ( |\psi\rangle = \cos(\theta/2) |0\rangle + e^{i\phi} \sin(\theta/2) |1\rangle ).
  • The Bloch Sphere was introduced by Felix Bloch for optical states and now used for qubit states.
  • Bloch Sphere represents pure states of qubits.
  • Antipodal points on the Bloch Sphere represent orthonormal states.

Questions and Clarifications

• Understanding of Bloch Sphere, global and relative phases.
• Basis change: every quantum gate operation is a change of basis.
• The significance of Bloch Sphere in providing geometric representation.

Entanglement and Two-Qubit Systems

• Definition of separable and entangled states.
• Controlled gates and their role in creating entanglement.
  • CNOT gate cannot be factored into two single qubit gates, thus used for entanglement.
• Conditions for entanglement:
  • Control qubit must be in superposition.
  • Target qubit must not be in the same basis as control qubit.

Basis Change in Quantum Computing

• Any unitary transformation performs a basis change.
• Basis change rules for single and multi-qubit systems.
• Mathematical treatment of basis changes and its importance in quantum algorithms.

Conclusion

• Basis change is a fundamental concept in quantum computing.
• The session covered both theoretical and practical insights into quantum mechanics and computing.

Upcoming Topics

• Further exploration of quantum algorithms in upcoming lectures.
• Students encouraged to review content and explore additional resources on GitHub.