Quantum error correction

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Applied Mathematics

Synopsis:

In this course, we will build a theory for the construction of quantum error-correcting codes. In the introductory part, necessary chapters from the classical theory will be summarized, atop of which we then present the quantum analogy. We will show how coherently stored quantum information can be made robust to loss and noise. We conclude the course by arriving at the principle of fault tolerance, based on which quantum computers are able to continuously correct errors arising at runtime and thus achieve correct results even with erroneous bits, gates or measurements.

Requirements:

Syllabus of lectures:

1.Block codes, code density (transmission rate), linear codes, Hamming codes, syndrome decoding.

2. Singleton bound, Hamming (sphere-packing) bound, Gilbert-Varshamov bound, dual codes.

3. LDPC codes, expander codes.

4. Specifics of quantum information storage and processing: state space, unitary operations, measurement, no-cloning theorem. Quantum gates and universal subsets. Pauli group.

5. Pure and mixed states, decoherence channels. Errors in information transfer, storage, computation. Model of mutually independent, rotationally symmetric single-bit errors. Error correction conditions. Linearity.

6. Stabilizer codes, concept of physical, logical qubits and code distance. Repetition code as an example of a stabilizer code. Concatenation and Shor's [[9,1,3]] code. Non-destructive syndrome measurement.

7. CSS construction, Steane's [[7,1,3]] code.

8. Quantum versions of code distance bounds. Degenerate quantum codes. [[5,1,3]] cyclic code, [[4,2,2]] erasure code, other prominent examples.

9. Clifford group. Stabilizer transformations during operations. Encoders and decoders of stabilizer codes.

10. Operations on one or more logical qubits. Principle of fault tolerance of quantum computations.

11. Gottesman-Knill theorem. Universal quantum computation on encoded qubits. Threshold theorem for universal error-tolerant quantum computation.

12. Important quantum code families. Toric code and other topological constructions. Fault-tolerant computation without concatenation.

13. research directions of quantum error-correcting codes.

Syllabus of tutorials:

1.Block codes, code density (transmission rate), linear codes, Hamming codes, syndrome decoding.

2. Singleton bound, Hamming (sphere-packing) bound, Gilbert-Varshamov bound, dual codes.

3. LDPC codes, expander codes.

4. Specifics of quantum information storage and processing: state space, unitary operations, measurement, no-cloning theorem. Quantum gates and universal subsets. Pauli group.

5. Pure and mixed states, decoherence channels. Errors in information transfer, storage, computation. Model of mutually independent, rotationally symmetric single-bit errors. Error correction conditions. Linearity.

6. Stabilizer codes, concept of physical, logical qubits and code distance. Repetition code as an example of a stabilizer code. Concatenation and Shor's [[9,1,3]] code. Non-destructive syndrome measurement.

7. CSS construction, Steane's [[7,1,3]] code.

8. Quantum versions of code distance bounds. Degenerate quantum codes. [[5,1,3]] cyclic code, [[4,2,2]] erasure code, other prominent examples.

9. Clifford group. Stabilizer transformations during operations. Encoders and decoders of stabilizer codes.

10. Operations on one or more logical qubits. Principle of fault tolerance of quantum computations.

11. Gottesman-Knill theorem. Universal quantum computation on encoded qubits. Threshold theorem for universal error-tolerant quantum computation.

12. Important quantum code families. Toric code and other topological constructions. Fault-tolerant computation without concatenation.

13. research directions of quantum error-correcting codes.

Study Objective:

In this course, we will build a theory for the construction of quantum error-correcting codes. In the introductory part, necessary chapters from the classical theory will be summarized, atop of which we then present the quantum analogy. We will show how coherently stored quantum information can be made robust to loss and noise. We conclude the course by arriving at the principle of fault tolerance, based on which quantum computers are able to continuously correct errors arising at runtime and thus achieve correct results even with erroneous bits, gates or measurements.

Study materials:

1. Gottesman, D.: Stabilizer Codes and Quantum Error Correction

Ph.D. thesis, California Institute of Technology 1997

https://doi.org/10.48550/arXiv.quant-ph/9705052

2. Ball, S.: A Course in Algebraic Error-Correcting Codes

Springer 2020

ISBN 9783030411527

3. Gaitan, F.: Quantum Error Correction And Fault Tolerant Quantum Computing

CRC Press

Note:

The course is presented in English.

Further information:

https://courses.fit.cvut.cz/QNI-QEC

No time-table has been prepared for this course

The course is a part of the following study plans: