Quantum Information and Error Correction
Special Topics Course - Math 577-2 - Spring 2012
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- Quantum computing
- Classic and quantum information theory
- Classic and quantum error correcting codes
Course Contents
In recent years quantum information processing has gained prominence as the prospect of the quantum computer is becoming a reality. Quantum information is stored in a coherent state of a quantum system. Quantum state is, by its nature, fragile. Protecting quantum information from being lost is subject to the laws of both quantum physics and mathematics. In classical theory of error correcting codes, we protect several bits of information by adding a few redundant bits, called parity check bits added to the transmitted message. Elimination of errors in the received message is performed by using syndromes, certain combinations of bits of the received message which allow error detection and correction. The task of protecting quantum information is more formidable and relies upon fascinating phenomena such as entanglement, which also enables communications with speeds greater than light, and suitable analogues of the concepts of parity check bits and syndromes. About 15 years ago Peter Shor invented a quantum circuit (see figure above) consisting of 9 qubits (subsequently improved to 5 qubits) which allows for perfect reconstruction of 1 qubit of quantum information. Today, quantum information is a vibrant field, rich in interaction of ideas from mathematics and quantum physics. We will start with these fundamental ideas, and proceed to study the subject up to its current state.Target audience
Students interested quantum computing and its emerging applications, including future mathematicians, electrical engineers and computer scientists, with both academic and industrial job goals.Textbooks and other course materials
The general textbook, ``Quantum Computation and Quantum Information'' by Michael A.Nielsen and Isaac L.Chuang. ISBN: 0521635039, will be used. Additionally, a number of research papers of the vast number of papers written in the areas of classical and quantum error correcting codes will be used in lectures and student assignments.Prerequisites and Requirements
Familiarity with Hilbert spaces or foundations of Quantum Mechanics. Some experience with algorithms and computing. Student is expected to attend lectures, read and deliver a research paper agreed upon with the instructor and complete 4-5 homework assignments. -
Resources
A diagram of Feynman switch gate
- The original source: Strange Paths
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The diagram:
Feynman switch gate: B detects A without affecting its path.
Teleportation across Danube river
The source with a link to the original article in Nature.
Inverter
Below is an schematic of a NOT gate (inverter) built using 1 transistor
A Lov Grover algorithm discussion
The source is this blog post.
Linearly polarized light
Click on the images below to get to the Wikipedia article.
An explanation of various polarization types with formulas.
Deutch-Jozsa circuit
Software
QInf
What is QInf?
QInf is Maxima-based collection of quantum computaction utilities.
Maxima is a free, comprehensive, open-source Computer Algebra System, capable of many relevent operations already, such as linear algebra and tensor calculations. Altoghether, QInf/Maxima form a complete system for quantum computing.
References
jQuantum
What is jQuantum?
jQuantum is a Java application which allows one to assemble and test quantum circuits. It contains a number of pre-built gates, including Quantum Fourier Transform and Grover gate. It comes with a number of ready-to-use examples which can be loaded and immediately simulated.
References
Lecture notes and textbooks
- Preskill's notes on quantum computation.
- From Classical to Quantum Shannon Theory. A recent textbook/lecture notes by Wilde covering quantum information theory (and the prerequisites).
Early quantum information theory and codes
- Good Quantum Error-Correcting Codes Exist.
- Mixed State Entanglement and Quantum Error Correction.
- Quantum Error Correction Via Codes Over GF(4).
- Quantum Channel Capacity of Very Noisy Channels.
- Stabilizer Codes and Quantum Error Correction.
Other approaches to quantum error correction
- Fault-tolerant quantum computation by anyons. This is topological quantum computation (see also chapter 9 of Preskill's notes).
- Asymptotically Good Quantum Codes. Uses algebraic geometry codes.
- Operator quantum error correction. Unifies topological quantum computation with quantum error correction.
Low Density Parity Check (LDPC) codes and belief propagation decoding
- Sparse Graph Codes for Quantum Error-Correction. One of the first papers to suggest using LDPC codes, which have been very successful in non-quantum coding, for quantum error correction.
- Quantum Error Correction beyond the Bounded Distance Decoding Limit. The best practical quantum LDPC codes?
- Quantum LDPC codes with positive rate and minimum distance proportional to n^1/2. The best theoretical quantum LDPC codes?
- On the iterative decoding of sparse quantum codes. Problems with decoding LDPC codes and some heuristic solutions.
More modern quantum information theory
- Quantum Channel Capacities. A short review of whats known about the different kinds of quantum capacities.