Introduction to Quantum Computing
Course Number: COMS 4281
Zoom link: https://columbiauniversity.zoom.us/j/94433279768 (Passcode: available on CourseWorks, or e-mail me)
Piazza link: http://piazza.com/columbia/spring2021/comsw4281
Date/Time: MW 2:40-3:55pm
First meeting: January 11
This Week’s Office Hours
This class is an introduction to the theory of quantum computing and quantum information. Topics covered include:
- The fundamental postulates of quantum information theory
- Entanglement and nonlocality
- The quantum circuit model
- Basic quantum protocols, such as quantum teleportation and superdense coding
- Basic quantum algorithms, such as Simons’ algorithm, the Quantum Fourier Transform, Phase Estimation, Shor’s Factoring algorithm, Grover search, amplitude amplification
- Quantum error correction and fault-tolerance
- (Time permitting) Quantum cryptography, quantum advantage/quantum supremacy, quantum complexity theory
The goal of the course is to provide a rigorous foundation for future research/studies in quantum computing and quantum information, and along the way provide students with an understanding of the state of the field, and where it’s headed.
No background in quantum physics is required. However, having familiarity and comfort with abstract linear algebra is a must.
All problem sets can be found at this Overleaf link. Problem Set $n$ is in the file “ps$n$.tex”. You can also use the tex file as your LaTeX template for your solutions.
- Problem Set 0 - due January 19, 11:59pm EST
- Problem Set 1 - due January 31, 11:59pm EST
- Problem Set 2 - due
February 14February 16, 11:59pm EST
- Problem Set 3 - due March 14, 11:59pm EST
- Problem Set 4 - due
March 28March 30, 11:59pm EST
- Problem Set 5 - due
April 11April 18, 11:59pm EST
These worksheets are meant to help you get practice with Dirac notation and basic linear algebraic manipulations used in the class. These are not collected nor graded. The TAs are more than happy to help you with them in Office Hours.
- Week 1
- January 11. Overview of the quantum computing and quantum information; class administrivia, and basic postulates of quantum information. [Slides] [Video]
- January 13. Classical versus quantum bits. How to safely test a bomb (with high probability). Composite quantum systems. No-cloning. [Slides] [Video]
- Supplementary reading: Nielsen and Chuang, Sections 2.1 and 2.2.
- Week 2
- Week 3
- January 25. Measurement in other bases. Projective measurements. Heisenberg's Uncertainty Principle. [Whiteboard] [Video]
- January 27. Entanglement and quantum correlations. The EPR Paradox and Bell's Theorem. CHSH game. [Whiteboard] [Video]
- Supplementary reading: Nielsen and Chuang, Sections 2.2.5, and lecture notes.
- Week 4
- February 1. Holevo's theorem. Superdense coding. Introduction to the quantum circuit model. [Whiteboard] [Video]
- February 3. Universal gate sets. Solovay-Kitaev theorem. Implementing classical computation in quantum circuits. Deutsch-Josza algorithm. [Whiteboard] [Video]
- Supplementary reading: Nielsen and Chuang Sections 1.4.4 and 4.5. See also these lecture notes.
- Week 5
- Week 6
- Week 7
- Week 8
- Week 9
- Week 10
- March 22. Hamiltonians. Spectral theorem and functions of Hermitian matrices. Physics perspective. Encoding CSPs as local Hamiltonians. [Whiteboard] [Video]
- March 24. Hamiltonian simulation algorithms. HHL algorithm for solving linear systems. [Whiteboard] [Video].
- Supplemental reading: notes and Section 2.4 of these notes for information about Hamiltonians. notes on the HHL algorithm.
- Week 11
- Week 12
- April 5. Noise and errors in classical computation. Errors in quantum computation. Bitflip code, phaseflip code, Shor's 9 qubit code. [Whiteboard] [Video]
- April 7. Finish up discussion of Shor's code. Why correcting X/Y/Z errors is enough. Other error correcting codes. Fault tolerance and the threshold theorem. [Whiteboard] [Video]
- Supplemental reading: notes
- Week 13