Fidelities of Quantum ARQ Protocol

1. Fidelities of Quantum ARQ Protocol • Classical Automatic Repeat Request (ARQ) Protocol • Qubits, von Neumann Measurement, Quantum Codes • Quantum Automatic Repeat Request (ARQ) Protocol • Quantum Errors • Quantum Enumerators • Fidelity of Quantum ARQ Protocol • Quantum Codes of Finite Lengths • The asymptotical Case (the code length ) Some results from the paper “Quantum Error Detection”, by A. Ashikhmin,A. Barg, E. Knill, and S. Litsyn are used in this talk Alexei Ashikhmin Bell Labs

2. Classical ARQ Protocol Noisy Channel • is a parity check matrix of a code • Compute syndrome • If we detect an error • If , but we have an undetected error

3. Qubits qubits • The state (pure) of qubits is a vector • Manipulating by qubits, we effectively manipulate by • complex coefficients of • As a result we obtain a significant (sometimes exponential) • speed up

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

5. is the orthogonal projection on von Neumann Measurement and orthogonal subspaces, is the orthogonal projection on • is projected on with probability • is projected on with probability • We know to which subspace was projected

6. is the code space Quantum Codes unitary rotation k+1 n 1 2 1 2 k n … … … information qubits in state quantum codeword in the state redundant qubits in the ground states the joint state: is the code rate

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 is fidelity If is close to 1 we can use

8. Conditional Fidelity Quantum ARQ Protocol Recall that the probability that is projected on is equal to The conditional fidelity is the average value of under the condition that is projected on

9. Quantum computer is unavoidably vulnerable to errors Any quantum system is not completely isolated from the environment Uncertainty principle – we can not simultaneously reduce: laser intensity and phase fluctuations magnetic and electric fields fluctuations momentum and position of an ion The probability of spontaneous emission is always greater than 0 Leakage error – electron moves to a third level of energy Quantum Errors

10. Quantum Errors means the absence of error Depolarizing Channel (Standard Error Model) Depolarizing Channel are the flip, phase, and flip-phase errors respectively This is an analog of the classical quaternary symmetric channel

11. Quantum Errors Similar to the classical case we can define the weight of error: Obviously

12. Quantum Enumerators is a code with the orthogonal projector P. Shor and R. Laflamme:

13. 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

14. 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

15. Fidelity of Quantum ARQ Protocol Recall that the probability that is projected on is equal to The conditional fidelity is the average value of under the condition that is projected on Theorem

16. Lemma (representation theory) Let be a compact group, is a unitary representation of , and is the Haar measure. Then Lemma

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

18. Fidelity of the Quantum ARQ Protocol • Sketch: • using the MacWilliams identities • we obtain • using inequalities we can • formulate LP problems for enumerator and denominator

19. Fidelity of the Quantum ARQ Protocol For the famous Steane code (encodes 1 qubit into 7 qubits) we have:

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

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

22. Fidelity of the Quantum ARQ Protocol

23. Fidelity of the Quantum ARQ Protocol The Asymptotic Case Theorem ( threshold behavior ) Asymptotically, as , we have (if Q encodes qubits into qubits its rate is ) Theorem (the error exponent) For we have

24. Existence bound Fidelity of the Quantum ARQ Protocol Theorem There exists a quantum code Q with the binomial weight enumerators: Substitution of these into gives the existence bound on Upper bound is much more difficult

25. Fidelity of the Quantum ARQ Protocol • Sketch: • Primal LP problem: • subject to constrains:

26. From the dual LP problem we obtain: Fidelity of the Quantum ARQ Protocol Theorem Let and be s.t. then Good solution: