Fidelity of a Quantum ARQ Protocol

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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
  )

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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

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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

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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

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

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Quantum Errors
Depolarizing Channel
Depolarizing Channel
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Quantum ARQ Protocol
If is close to 1 we can
use
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Quantum Enumerators
is a code with the
orthogonal projector
P. Shor and R. Laflamme (1996)
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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

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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

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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
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It follows from the representation theory that
Lemma
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Fidelity of Quantum ARQ Protocol
Quantum Codes of Finite Lengths
We can numerically compute upper and lower bounds
on , (recall that
)
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Fidelity of Quantum ARQ Protocol
For the Steane code that encodes 1 qubit into 7
qubits we have
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Fidelity of Quantum ARQ Protocol
Lemma The probability that will be
projected onto equals
Hence we can consider as a function
of
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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

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Fidelity of Quantum ARQ Protocol
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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
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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