Quantum Computing Lecture 22 - PowerPoint PPT Presentation

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Quantum Computing Lecture 22

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Title: Quantum Computing Lecture 22


1
Quantum ComputingLecture 22
  • Michele Mosca

2
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
3
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
4
Quantum Error Correction
error
Please observe repetitions of these patterns
5
Equivalently , cancelling pairs of H inside the
diagram we get
Final circuit for correcting phase errors
6
Quantum Error Correction
  • If the error effected on the system in state
    is of the form

7
Quantum Error Correction
  • and if the state only consists of mixtures of
    superpositions of codewords and
  • then the correction procedure (call it )
    will map

8
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

9
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.

10
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

11
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)

12
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).

13
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)
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