Quantum%20Error%20Correction

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Quantum Error Correction

• Joshua Kretchmer
• Gautam Wilkins
• Eric Zhou

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Error Correction

• Physical devices are imperfect
• Interactions with the environment
• Error must be controlled or compensated for
• One step has probability to succeed p
• t steps has probability to succeed pt

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Classical Error Correction

• Error Model
• Channels provide description of the type of error
• Encoding
• Extra bits added to protect logical bit
• String of bits ? codeword
• Redundancy
• Error Recovery
• Recovery operation
• Measure bits and re-set all values to majority
  vote

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Classical 3-Bit Code Bit Flip Error

• Bit flip channel bit is flipped with prob. p lt
  1/2
• Encoding
• Error Recovery
• 000 ? (000, (1-p)3),
• (001, p(1-p)2), (010, p(1-p)2), (100, p(1-p)2)
• (011, p2(1-p)), (110, p2(1-p)), (101, p2(1-p))
• (111, p3)
• Prob(unrecoverable error) 3p2(1-p)p3 3p2-2p3

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Problems with QEC

• No cloning theorem
• Cant copy an arbitrary quantum state
• Entanglement
• Measurement
• Cannot directly measure a qubit
• Error syndrome
• Quantum evolution is continuous

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Quantum 3-Bit Code Bit Flip Error
?gt
?gta0gtb1gt 0gt 0gt
M
0gt 0gt
Encoding
Decode
X
Error Channel
M
Diagnose and Correct

• Encoding ? a000gtb111gt
• Error channel
• Noise acts on each qubit independently
• Probability noise does nothing 1 - p
• Probability noise applies ?x p lt 1/2

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Quantum 3-Bit Code Bit Flip Error

• After channel ? 8 possible results
• State Probability
• a000gtb111gt (1-p)3
• a100gtb011gt p(1-p)2
• a010gtb101gt p(1-p)2
• a001gtb110gt p(1-p)2
• a110gtb001gt p2(1-p)
• a101gtb010gt p2(1-p)
• a011gtb100gt p2(1-p)
• a111gtb000gt p3

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Quantum 3-Bit Code Bit Flip Error

• After CNOTs ? 4 possible results
• State Probability
• a000gtb111gt00gt (1-p)3
• a100gtb011gt10gt p(1-p)2
• a010gtb101gt01gt p(1-p)2
• a001gtb110gt11gt p(1-p)2
• a110gtb001gt01gt p2(1-p)
• a101gtb010gt10gt p2(1-p)
• a011gtb100gt11gt p2(1-p)
• a111gtb000gt00gt p3

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Quantum 3-Bit Code Bit Flip Error

• Measure 2 ancilla qubits ? error syndrome
• Measured syndrome action
• 00 do nothing
• 01 apply ?x to 3rd qubit
• 10 apply ?x to 2nd qubit
• 11 apply ?x to 1st qubit
• Designed to correct if theres an error in 1 or
  no qubits
• Error in 2 or 3 qubits is an uncontrollable error

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Quantum 3-Bit Code Bit Flip Error

• Failing probability ? pu 3p2(1-p)p3
• 3p2-2p3 O(p2)
• Fidelity ? success probability 1- pu 1- 3p2
• Without error correction pu O(p)

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Quantum 3-Bit Code Phase Error

• Random rotation of qubits about z-axis
• Continuous error
• P(??) ei?? 0 cos(??)I isin(??)?z
• 0 e-i??
• ? - fixed quantity stating typical size of
  rotation
• ? - random angle

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Quantum 3-Bit Code Phase Error

• Apply H to each qubit at either end of the
  channel
• HIH HH I H?zH ?x
• ? HPH cos(??)I isin(??)?x
• Same result from bit flip code
• Fidelity 1 - 3p2
• p ltsin2(??)gt ? (2??)2/3 for ?ltlt1

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General Quantum Error

• Errors occur due to interaction with environment
• 0gtEgt ? ?10gtE1gt ?21gtE2gt
• 1gtEgt ? ?31gtE3gt ?40gtE4gt
• (?00gt ?11gt)Egt ?
• ?0?10gtE1gt ?0?21gtE2gt
• ?1?31gtE3gt ?1?40gtE4gt

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General Quantum Error

• (?00gt ?11gt)Egt ?
• 1/2(?00gt ?11gt)(?1E1gt ?3E3gt)
• 1/2(?00gt - ?11gt)(?1E1gt - ?3E3gt)
• 1/2(?01gt ?10gt)(?2E2gt ?4E4gt)
• 1/2(?01gt - ?10gt)(?2E1gt - ?4E4gt)
• ?00gt ?11gt ?gt
• ?00gt - ?11gt Z?gt
• ?01gt ?10gt X?gt
• ?01gt - ?10gt XZ?gt

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General Quantum Error

• (?00gt ?11gt)Egt ?
• 1/2(?gt)(?1E1gt ?3E3gt)
• 1/2(Z?gt)(?1E1gt - ?3E3gt)
• 1/2(X?gt)(?2E2gt ?4E4gt)
• 1/2(XZ?gt)(?2E1gt - ?4E4gt)
• Error basis I, X, Z, XZ
• ?gtL?gte ? ?(?i?gtL)?igte
• ?gtL ? general superposition of quantum codewords
• ?i ? error operator tensor product of pauli
  operators

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Correction of General Errors

• ?gtL?gte ? ?(?i?gtL)?igte
• ?gtL - orthonormal set of n qubit states
• To extract syndrome attach an n-k qubit ancilla
  a to system ? perform operations to get
  syndrome ? sigta
• ? 0gta?(?i?gtL)?igte ? ?sigta(?i?gtL)?igte
• Measure si to determine ?i-1 ? correct for error
• ? sigta(?i?gtL)?igte ? sigta(?gtL)?igte

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Shors Algorithm

• Each qubit is encoded as nine qubits

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Shors Algorithm

• Assume decoherence on first bit of first triple,
  becomes

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Shors Algorithm
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Shors Algorithm

• No error
• Z error
• X error
• ZX Y error

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Shors Algorithm

• Success Rate
• Works if only one qubit decoheres
• If probability of a qubit decohering is p
• Probability of 2 or more out of 9 decohering
  is1-(18p)(1-p)8?36p2
• Therefore probability that 9k qubits can be
  decoded is (1-36p2)k

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Shors Algorithm

• More on decoherence
• Decoherence probability increases with time
• Use watchdog effect to periodically reset quantum
  state
• Unfortunately, each reset introduces small amount
  of extra error
• Therefore cannot store indefinitely

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Steanes Algorithm

• Basis 1 is 0 ?, 1 ?
• Also called basis F, or flip basis
• Basis 2 is 0 ? 1 ?, 0 ?- 1 ?
• Also called basis P, or phase basis

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Steanes Algorithm

• The word 0000 ? consisting of all zeroes in
  basis 1 is equal to a superposition of all 2n
  possible words in basis 2, with equal
  coefficients.
• If the jth bit of each word is complemented in
  basis 1, then all words in basis 2 in which the
  jth bit is a 1 change sign.
• Hamming Distance
• The number of places two words of the same length
  differ
• Minimum Distance
• Smallest Hamming distance between any two code
  words in a code

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Steanes Algorithm

• A code of minimum distance d allows (d-1)/2 to
  be corrected
• If less than d/2 errors occur, the correct
  original code word that gave rise of the
  erroneous word can be identified as the only code
  word at a distance of less than d/2 from the
  received word.
• n,k,d is a linear set of 2k code words each of
  length n, with minimum distance d

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Steanes Algorithm

• Parity Check Matrix
• Matrix H of dimensions (n-k) by n, where Hv 0
  iff v is in the code C
• Generator Matrix
• Matrix G of dimensions n by k, basis for a linear
  code
• w cG, where w is a unique codeword of linear
  code C, and c is a unique row vector
• For a linear code C in basis 1, a superposition
  with equal coefficients, then in basis 2 the
  words of the superposition form the dual code of
  C
• The Parity Check Matrix of C is the Generator
  Matrix for its dual code

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Steanes Algorithm

•  Let a ? and b ? be expressed as 7, 3, 4 in
  basis 1

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Steanes Algorithm

• a ? and b ? are non-overlapping, and have
  distance of 3
• Find bit flip with parity check 
• Switch to basis 2
• c ? a ? b ?
• Contains only even parity words of a 7,4,3 code
• d ? a ? -b ?
• Contains only odd parity words
• Distance between c ? and d ? is at least 3
• Phase error can be found with a parity check

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Implications for Physical Realizations of Quantum
Computers
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Why Do We Need It?

• Quantum computers are very delicate.
• External interactions result in decoherence and
  introduction of errors.

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Fault-tolerance

• Especially important when considering physical
  implementations.
• Must consider errors introduced by all parts,
  including gates.
• Incorrect syndromes introduce errors.

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Impact on Physical Systems

• Increased size
• Level of coherence determines increase

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Alternative to Error Correction

• Topological Quantum Computing
• Involves particles called anyons that form
  braids, whose topology determines quantum state.

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Topological Quantum Computing
Slight perturbations to system cause braids to be
deformed, but only large disturbances result in
them being cut or joined.
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Summary

• Error correction is vital for physical
  realizations of trapped particle quantum
  computers.
• Allows reliable quantum computation without
  requiring extremely high levels of coherence.