Quantum Error Correction a tutorial A.M. Steane Oxford University

1. Quantum Error Correction • a tutorial • A.M. Steane • Oxford University

2. Hiding qubits from the noise y y ground state throw away exact!

3. Classical information theory

4. The simple repetition idea can achieve good quality (i.e. low noise) communication, but it makes inefficient use of the bits. • To do better, we look in detail at how bit flips affect bit strings.

5. Classical Error Correcting Codes 110 111 100 010 011 000 001 A code is also a group under addition

6. Example: • G=1010101 H = 1010101 • 0110011 0110011 • 0001111 0001111 • 1111111 • C = 0000000, 1010101, 0110011, 1100110, • 0001111, 1011010, 0111100, 1101001 • 7 bits, 23 = 8 members, minimum weight 4 Minimum distance d) correct up to (d-1)/2 errors. For a linear code, the minimum distance between members is also the minimum weight.

7. Dual codes Some codes contain their duals.

9. Quasi-classical code

11. }  correct } • “correction” results in

12.  to pdf