Fidelity of a Quantum ARQ Protocol

1. Fidelity of a Quantum ARQ Protocol • 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 ) Alexei Ashikhmin Bell Labs

2. Binary Symmetric Channel Classical ARQ Protocol • 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

3. Binary Symmetric Channel Classical ARQ Protocol • 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

4. Binary Symmetric Channel Classical ARQ Protocol • 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

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

6. Quantum Errors Depolarizing Channel Depolarizing Channel

7. 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 The fidelity is the average value of Quantum ARQ Protocol If is close to 1 we can use

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

9. Quantum Enumerators • 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

10. Quantum Enumerators • 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): • and therefore this code can correct any single ( since ) error

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

12. It follows from the representation theory that Lemma

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

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

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

16. Fidelity of Quantum ARQ Protocol • 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

17. Fidelity of Quantum ARQ Protocol

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

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