Title: Fidelity of a Quantum ARQ Protocol
1Fidelity 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
) -
2Classical 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 -
3Classical 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 -
4Classical 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
6Quantum Errors
Depolarizing Channel
Depolarizing Channel
7Quantum ARQ Protocol
If is close to 1 we can
use
8Quantum Enumerators
is a code with the
orthogonal projector
P. Shor and R. Laflamme (1996)
9Quantum 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
-
-
-
-
10Quantum 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
11Fidelity 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
13Fidelity of Quantum ARQ Protocol
Quantum Codes of Finite Lengths
We can numerically compute upper and lower bounds
on , (recall that
)
14Fidelity of Quantum ARQ Protocol
For the Steane code that encodes 1 qubit into 7
qubits we have
15Fidelity of Quantum ARQ Protocol
Lemma The probability that will be
projected onto equals
Hence we can consider as a function
of
16Fidelity 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
-
-
-
-
17Fidelity of Quantum ARQ Protocol
18Fidelity 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