Quantum Error Corection

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Quantum Computation
Dr. Richard B. Gomez rgomez_at_gmu.edu

Introduction to Quantum Computing
Lecture 10
George Mason University School of Computational
Sciences
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Quantum Error CorrectionLecture 10

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Outline

• Sources and types of errors
• Differences between classical and quantum error
  correction
• Basic quantum gates
• Quantum error correcting codes

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Introduction

• Quantum states of superposition (which stores
  quantum information) extremely fragile.
• Quantum error correction more tricky than
  classical error correction.
• In the field of quantum computation, what is
  possible in theory is very far off from what can
  be implemented.
• Complex quantum computation impossible without
  the ability to recover from errors

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What can go wrong

• Internal
• Initial states on input qubits not prepared
  properly.
• Quantum gates used may not be accurate
• Quantum gates may introduce small errors which
  will accumulate.
• External
• Dissipation
• A qubit loses energy to the environment.
• Decoherence

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Decoherence

• Decoherence is the loss of quantum information of
  a quantum system due to its interaction with the
  environment.
• Almost impossible to isolate a quantum system
  from the environment.
• Over time, our quantum system will be entangled
  with the environment.

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• Information encoded in our quantum system will be
  encoded instead in the correlations between the
  quantum system and the environment.
• The environment can be seen as measuring the
  quantum system, collapsing its superposition
  state.
• Hence quantum information (encoded in the
  superposition) is irreversibly lost from the
  qubit.

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Dealing With Decoherence

• Design quantum algorithms to finish before
  decoherence ruins the quantum information.
• Can be difficult as
• Decoherence occurs very quickly.
• Quantum algorithms may be very complex and long.

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Dealing With Decoherence

• Try to lower the rate at which decoherence
  occurs.
• Accomplished by using a right combination of
• Quantum particle type
• Quantum computer size
• Environment

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• Decoherence time refers to the time available
  before decoherence ruins quantum information.
• Decoherence time is affected by the size of the
  system, as well as the environment.
• Decoherence time affected by environmental
  factors like temperature and amount of
  surrounding particles in the environment

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• Time needed for a quantum gate operation as
  important as decoherence time.
• Different types of quantum systems have different
  decoherence time and per gate operation time.

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• The better the decoherence time, the slower the
  quantum gate operations.

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Dealing With Decoherence and other sources of
errors

• Use Quantum Error correcting codes
• Encode qubits (together with extra ancillary
  qubits) in a state where subsequent errors can be
  corrected.
• Allows long algorithms requiring many operations
  to run, as errors can be corrected after they
  occur.

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Single Qubit errors

• Bit flip error
• Do a bit flip using a operator.

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• Phase flip error
• Do a phase flip using a operator.

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• Bit and phase flip error
• Do a bit and phase flip using a operator.

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A simple classical error correction encoding

• 3 bit repetition encoding
• 0 encoded as 000
• 1 encoded as 111
• Assuming only 1 bit error
• Decoding Take majority vote of the 3 bits
• E.g.

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Difficulty of using classical error correction
for correcting Qubits

• No cloning theorem
• Unable to encode as
• Measurement of qubits cause disturbance
• Need to do error correction without measuring the
  value of each qubit.

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• Unable to correct phase errors
• Unable to correct small errors
• For , an error might change a
  and ß by a small order.
• These small errors can accumulate.
• Classical methods only designed to correct large
  discrete errors (i.e. bit flips)

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Some quantum gates

• Control Not (CNOT)
• If q1 (control bit) is 1, then q2 will be
  flipped.
• Similar to classical XOR gate, except its
  reversible

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• Toffoli (C2NOT)
• If q1 and q2 is 1, then q3 will be flipped.
• Similar to classical NAND gate, except its
  reversible
• Note CNOT and Toffoli perform their function
  without measuring the qubits.
• E.g. if each of the 2 control qubits of the
  toffoli gate are in a superposition, those
  superposition will remain intact after passing
  through the gate.

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• Hadamard (H)

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Quantum Error correcting codes

• Correcting single bit flip error using 3 qubits
• Correcting single phase error using 3 qubits
• 9 qubits error correcting code
• 5 qubits error correcting code
• Concatenated code

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Correcting Single Bit Flip

• Use 3 qubits to encode 1 qubit (3,1)

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• For encoding, use 2 extra qubits initially set to
  
• Encoding circuit

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• Assuming at most 1 bit will be flipped and the
  bit flip is just as likely to affect any qubit.
• Decoding circuit

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• The last 2 qubits are called the syndrome and
  their values indicate the error that occurred.
• All possible states at the end of decoding
  circuit

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• Correction circuit

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• (3,1) repetition code circuit

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Correcting single phase flip

• Use Hadamard to convert a phase flip to bit
  flip
• Similarly

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• Proof

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Correcting single phase flip

• Circuit for correcting single bit flip
• Modified circuit to correct single phase flip.

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Initial Problems Avoided

• No cloning involved in encoding
• Able to diagnose the error without damaging the
  quantum information.
• Able to correct errors without knowing state of
  qubit.
• Able to correct bit flip or phase flip error
  depending on the circuit used.

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• Able to correct small errors
• Example Assume encoded qubit damaged such that
• 0.7 probability of getting no errors
• 0.3 probability of getting 1st bit flipped

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• After the circuit, 1st qubit will always be
• The decoding circuit maps the state into either
  one with no error, or one with an error which we
  know how to correct.

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9 qubits error correcting code

• The 2 codes earlier corrects either bit flips or
  phase flips.
• Shors 9 qubits error correcting code combines
  both codes.
• It can correct any arbitrary single qubit error

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Encoding

• Use 9 qubits to encode 1 qubit (9,1)

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• Encoding circuit

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• Assuming at most 1 qubit error and the error is
  just as likely to affect any qubit.
• The decoding and correction circuit

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• Example Assume encoded qubit damaged such that

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The (9,1) circuit
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5 Qubits Error Correcting Code

• Shors code uses 9 qubits to encode 1 qubit, but
  more efficient codes exist.
• Given our error model where errors can be any of
  the Pauli matrices applied to
  any qubit.
• To recover from 1 qubit errors, we need a minimum
  of 5 qubits to encode 1 qubit.
  

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• Argument
• Supposing we encode 1 qubit using n qubits.
• We can have n-1 syndrome bits, the values of
  which tells us the exact error that occurred.
• Hence 2n-1 errors can be represented by the
  syndrome bits
• We have n qubits, and so 3n possible errors.
  Consider also the case of no errors.
• Hence,
• Least value of n is 5.

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Encoding

• Use 5 qubits to encode 1 qubit

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• Encoding circuit

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• If qubits 2,3 and 4 are 1, flip the phase
• If qubits 2 and 4 is 0 and qubit 3 is 1, flip
  the phase
• If qubit 1 is 1, flip qubits 3 and 5

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• Assuming at most 1 qubit error and the error is
  just as likely to affect any qubit.
• The decoding circuit is the encoding circuit in
  reverse

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• Example Assume encoded qubit damaged such that

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• Re-express equation to prepare for Hadamard
  transform

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• Qubits 1,2,4 and 5 are the syndrome bits which
  indicate the exact error that occurred and the
  current state of qubit 3.

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Syndromes Table
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• According to syndrome table, the 3rd qubit is in
  state .
• So apply a phase flip and a bit flip to obtain
  the protected qubit .

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The 5 qubits error correcting circuit
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Concatenated Code

• 1 qubit can be encoded using 5 qubits.
• Each of the 5 qubits can be further encoded using
  5 qubits.
• Continue doing this until some number of
  hierarchical levels is reached.

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• Illustration
• We will use the 5 qubit encoding.
• Assume probability of single qubit error is e and
  that errors are uncorrelated.

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• For 2 levels, number of qubits required is 52
  25
• This encoding will fail when 2 or more sub blocks
  of 5 qubits cannot recover from errors.
• Hence probability of recovery failure is in order
  of (e2)2 e4
• e4 lt e2. 2 levels encoding has better probability
  of error recovery than 1 level if e is small
  enough

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• For 3 levels, number of qubits required is 53
  125
• This encoding will fail when 2 or more sub blocks
  of 25 qubits cannot recover from errors.
• Hence probability of recovery failure is in order
  of (e4)2 e8
• e8 lt e4. 3 levels encoding has better probability
  of error recovery than 2 levels if e is small
  enough.

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• In general for L levels,
• Number of qubits required is 5L
• Probability of recovery failure is in the order
  of
• Advantages of concatenated code
• If probability of individual qubit error, e, is
  pushed below a certain threshold value, adding
  more levels will reduce probability of recovery
  failure.
• I.e. we can increase the accuracy of our encoding
  indefinitely by adding more levels.
• Error correction is simple using a divide and
  conquer strategy.

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• Disadvantages of concatenated coding
• If probability of individual qubit error, e, is
  above the threshold value, adding more levels
  will make things worse.( I.e. probability of
  recovery failure will be higher)
• Exponential number of qubits needed.
• Note
• Threshold value depends on
• Type of encoding used
• Types of errors that occurs.
• When the errors are likely to occur (during qubit
  storage, or gate processing)