Fidelity of a Quantum ARQ Protocol Alexei Ashikhmin

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 )

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

Classical ARQ Protocol Binary Symmetric Channel • 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

Classical ARQ Protocol Binary Symmetric Channel • Syndrome • is the distance distribution of • The conditional probability of undetected error Ø For the best code of rate Ø If as there exists a linear code s. t.

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

Quantum Errors Depolarizing Channel

Quantum ARQ Protocol ARQ protocol: – – – If The fidelity 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 close to 1 we can use is the average value of

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

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

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 can correct any single ( since therefore this code ) error

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

It follows from the representation theory that Lemma

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

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

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

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

Fidelity of Quantum ARQ Protocol

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

Fidelity of Quantum ARQ Protocol Existence bound 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