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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• is a parity check matrix of a code
• Compute syndrome
• If we detect an error
• If , but we have an undetected error
Classical ARQ Protocol
Noisy Channel
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• 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
qubits
Qubits
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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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• is projected on with probability
• is projected on with probability
• We know to which subspace was projected
von Neumann Measurement
and orthogonal subspaces,
is the orthogonal projection on is the orthogonal projection on
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1 2 k… k+1 n…
information qubitsin state
n1 2 …
quantum codewordin the state
unitary rotation
Quantum Codes
redundant qubitsin the ground states
is the code space
is the code rate
the joint state:
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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
Quantum ARQ Protocol
is fidelity
If is close to 1 we can use
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Conditional Fidelity
The conditional fidelity is the average value ofunder the condition that is projected on
Recall that the probability that is projected on is equal to
Quantum ARQ Protocol
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• 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
Quantum Errors
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Depolarizing Channel (Standard Error Model)
Depolarizing Channel
means the absence of error
are the flip, phase, and flip-phase errors respectively
This is an analog of the classical quaternary symmetric channel
Quantum Errors
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Similar to the classical case we can define the weight of error:
Obviously
Quantum Errors
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Quantum Enumerators
P. Shor and R. Laflamme:
is a code with the orthogonal projector
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• 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
Quantum Enumerators
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• 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):
Quantum Enumerators
• and therefore this code can correct any single ( since ) error
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Fidelity of Quantum ARQ Protocol
Theorem
The conditional fidelity is the average value ofunder the condition that is projected on
Recall that the probability that is projected on is equal to
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Lemma (representation theory) Let be a compact group, is aunitary representation of , and is the Haar measure. Then
Lemma
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Quantum Codes of Finite Lengths
We can numerically compute upper and lower bounds on , (recall that )
Fidelity of the Quantum ARQ Protocol
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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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For the famous Steane code (encodes 1 qubit into 7 qubits) we have:
Fidelity of the Quantum ARQ Protocol
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Lemma The probability that will be projected onto equals
Hence we can consider as a function of
Fidelity of the Quantum ARQ Protocol
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• 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
Fidelity of the Quantum ARQ Protocol
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Fidelity of the Quantum ARQ Protocol
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Theorem ( threshold behavior ) Asymptotically, as , we have
Theorem (the error exponent) For we have
The Asymptotic Case Fidelity of the Quantum ARQ Protocol
(if Q encodes qubits into qubits its rate is )
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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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Fidelity of the Quantum ARQ Protocol • From the dual LP problem we obtain:
Theorem Let and be s.t.
then
Good solution: