Fidelities of Quantum ARQ Protocol

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Fidelities of Quantum ARQ Protocol

• Alexei Ashikhmin
• Bell Labs

• 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

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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

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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

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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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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

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Quantum Codes
unitary rotation
k1
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
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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
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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
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Quantum Errors

• 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

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Quantum Errors
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
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Quantum Errors
Similar to the classical case we can define the
weight of error
Obviously
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Quantum Enumerators
is a code with the
orthogonal projector
P. Shor and R. Laflamme
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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 conditional fidelity is the
average value of under the condition that
is projected on
Theorem
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Lemma (representation theory) Let be a
compact group, is a unitary representation of
, and is the Haar measure. Then
Lemma
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Fidelity of the 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 the Quantum ARQ Protocol

• Sketch
• using the MacWilliams identities
  
• we obtain
• using inequalities
  we can
• formulate LP problems for enumerator and
  denominator

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

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Fidelity of the Quantum ARQ Protocol
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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
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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
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Fidelity of the Quantum ARQ Protocol

• Sketch
• Primal LP problem
  
• subject to constrains
  

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• From the dual LP problem we obtain
  

Fidelity of the Quantum ARQ Protocol
Theorem Let and
be s.t.
then
Good solution