Quantum%20Computing%20Lecture%2022

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Quantum ComputingLecture 22

• Michele Mosca

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Correcting Phase Errors

• Suppose the environment effects error
• on our quantum computer, where

This is a description of errors in phase because
we use powers of operator Z
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Quantum Error Correction

• We can encode
• Consider error term acting on the logical 0
  gives

Z error in upper bit
Such error arriving in decoder is shown next slide
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Quantum Error Correction
error
Please observe repetitions of these patterns
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Equivalently , cancelling pairs of H inside the
diagram we get
Final circuit for correcting phase errors
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Quantum Error Correction

• If the error effected on the system in state
  is of the form

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Quantum Error Correction

• and if the state only consists of mixtures of
  superpositions of codewords and
• then the correction procedure (call it )
  will map

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Correcting both phase errors and bit flip errors

• Consider the codewords of Shors code
• We can easily correct any single X- error in one
  of the 3 three-bit parts
• We can then also correct a single Z- error on one
  of the 9 qubits.
• This means we can also correct Y-errors on one of
  the 9 qubits

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Quantum Error Correction

• Theorem 10.2 Suppose C is a quantum code and
  is the error-correction operation constructed
  in the proof of Theorem 10.1 to recover from a
  noise process with operation elements
  . Suppose is a quantum operation with
  elements which are linear combinations of
  the . Then the error correction operation
  also corrects the effects of the noise
  process on the code C.

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Correcting any error

• Since any error operator Ek can be written as a
  linear combination of I,X,Z and Y, then the same
  procedure will correct ANY error acting on just 1
  of the 9 qubits.
• If
• where is a quantum operator whose
  operator terms are correctable with correction
  operator , then

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Correcting any error

• Theorem 10.1 (Quantum Error Correction
  Conditions) Let C be a quantum code, and let P be
  the projector onto C. Suppose is a quantum
  operation with operation elements A necessary
  and sufficient condition for the existence of an
  error-correction operation correcting on
  C is that
• for some Hermitian matrix of complex
  numbers.
• (no mention of efficiency)

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Degenerate Codes

• Consider the 9-qubit code.
• A single Z-error on the first qubit of a codeword
  produces the same outcome as a single Z-error on
  either the 2nd or 3rd qubit.
• The correction procedure will correct these
  errors regardless
• A degenerate code is one where two correctable
  errors produce the same effect on the codewords
  (this is impossible with classical codes).

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Quantum Hamming Bound

• Any non-degenerate quantum error correcting code
  that encodes k logical qubits into n qubits and
  can correct errors on up to t qubits must have
• If tk1, we get (there exists a 5-qubit
  code that accomplishes this)