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
• 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
 For the best code of rate
 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
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 MacWilliams 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

 If
(if Q encodes
then there exists a stabilizer code s.t.
qubits into
qubits its 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