1. Fidelity of a Quantum ARQ Protocol Alexei Ashikhmin Bell Labs  Classical Automatic Repeat Request (ARQ) Protocol  Quantum Automatic Repeat Request (ARQ) Protocol  Fidelity of Quantum ARQ Protocol Quantum Codes of Finite Lengths The asymptotical Case (the code length )

2. is a classical linear code If is a parity check matrix of then for any Compute syndrome If we detect an error If, but we have an undetected error Classical ARQ Protocol Binary Symmetric Channel

3. Syndrome is the distance distribution of is the channel bit error probability The probability of undetected error is equal to for good codes of any rate we have as If, but we have an undetected error Classical ARQ Protocol Binary Symmetric Channel

4. Syndrome is the distance distribution of The conditional probability of undetected error  For the best code of rate as  If there exists a linear code s. t. If, but we have an undetected error Classical ARQ Protocol Binary Symmetric Channel

5. In this talk all complex vectors are assumed to be normalized, i.e. All normalization factors are omitted to make notation short

6. Depolarizing Channel Quantum Errors

7. Quantum ARQ Protocol ARQ protocol: –We transmit a code state –Receive –Measure with respect to and –If the result of the measurement belongs to we ask to repeat transmission –Otherwise we use If is close to 1 we can use The fidelity is the average value of

8. Quantum Enumerators P. Shor and R. Laflamme (1996): is a code with the orthogonal projector

9. and are connected by quaternary MacWilliams identities where are quaternary Krawtchouk polynomials: The dimension of is is the smallest integer s. t. then can correct any errors Quantum Enumerators

10. In many cases are known or can be accurately estimated (especially for quantum stabilizer codes) For example, the Steane code (encodes 1 qubit into 7 qubits): Quantum Enumerators and therefore this code can correct any single ( since ) error

11. Fidelity of Quantum ARQ Protocol Theorem The fidelity is the average value of is the projection onto and Recall that the probability that is projected on is equal to

12. It follows from the representation theory that Lemma

13. Quantum Codes of Finite Lengths We can numerically compute upper and lower bounds on, (recall that ) Fidelity of Quantum ARQ Protocol

14. For the Steane code that encodes 1 qubit into 7 qubits we have Fidelity of Quantum ARQ Protocol

15. Lemma The probability that will be projected onto equals Hence we can consider as a function of Fidelity of Quantum ARQ Protocol

16. Let be the known optimal code encoding 1 qubit into 5 qubits Let be a “silly” code that encodes 1 qubit into 5 qubits defined by the generator matrix: is not optimal at all Fidelity of Quantum ARQ Protocol

18. (if Q encodes qubits into qubits its rate is ) Theorem ( threshold behavior ) Asymptotically, as, we have for   If then there exists a stabilizer code s.t. Theorem (the error exponent) For we have The Asymptotic Case Fidelity of Quantum ARQ Protocol

19. Existence bound Fidelity of Quantum ARQ Protocol Theorem (Ashikhmin, Litsyn, 1999) There exists a quantum stabilizer code Q with the binomial quantum enumerators: Substitution of these into gives the existence bound on Upper bound is more tedious