Introduction to Quantum Error Correction

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Introduction to Quantum Error Correction
Fault-Tolerant Quantum Logic

• Cherrie Huang

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Why Quantum Error Correction?6

• Cause circuit interacts with the surroundings
• decoherence
• decay of the quantum information stored in the
  device
• Solution Quantum Error Correcting Codes
• protect quantum information against errors.
• perform operations fault-tolerantly on encoded
  states.

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Major Difficulties 9

• 1. No cloning theorem impossible to duplicate an
  arbitrary unknown qubit
• Solution Fight Entanglement with
  entanglement(encode the information that we want
  to protect in entanglement).

impossible
possible
Explain why no cloning is important in this
context
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Major Difficulties 9

• 2. Errors are continuous continuous
  errors?requires infinite resources and infinite
  precision
• Solution Digitalize the errors that circuit
  makes.
• 3. Measurement destroys quantum information
  recovery is impossible if quantum information
  state is destroyed.
• Solution Measure the errors without measuring
  the data.

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Central Idea of QEC

• A small subspace of the Hilbert space of the
  device is designated as the code subspace.
• This space is carefully chosen so that all of the
  errors that we want to correct move the code
  space to mutually orthogonal error subspaces.
• We can make a measurement after our system has
  interacted with the environment that tells us in
  which of these mutually orthogonal spaces the
  system resides, and hence infer exactly what type
  of error occurred.
• The error can then be repaired by applying an
  appropriate unitary transformation.

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Key Ideas of QEC

• Encode the message with redundant information
• Redundancy in the encoded message allows to
  recover the information in the original message.
  
• Measure the errors, not the data.

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General Model of QEC 1

• Deal with errors
• Error detection
• Error correction

1. Errors also in encoding and recovery (they are
   themselves complex quantum computations) ?But,
   fault-tolerant recovery possible if error rate is
   not high (Peter Shor, 1996).
2. Problems to store an unknown quantum state with
   high fidelity for an indefinitely long time and
   problems to do quantum computation?But, possible
   if error rate is below threshold (Manny Knill and
   Raymond Laflamme, 1996).

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Application of what?

• Storage CDs, DVDs, hard drives
• Wireless Cell phones, wireless links
• Satelite and Space TV, Mars rover
• Digital Television DVD, MPEGS layover
• High Speed Modems ADSL, DSL

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Classical Error Correction

• Hierarchy

linear
cyclic
BCH
Bose-Chaudhuri-Hochquenghem
Hamming
Reed-Solomon
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Classical Repetition Code

• Transmission Sending one bit of information
  across the channel.
• Noise flips the bit with the probability p
• Encoding triple each bit 0?000, 1?111,
  C000,111
• Decoding majority voting
• Example 10?111000 101001? 10
• Limitation not possible to recover the
  information correctly if more than one bit is
  flipped.

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Classical Repetition Code

• Analysis

Transmission method 1. Non-encoded transmission 3-bit repetition code
Probability of error p Probability that two or more of the bits are flipped 3p2(1-p)p3
When p0.25 0.25 0.15625 (when plt0.5, this method is better.)
What is this? Explain better this and the whole
table
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QEC The Three Qubit Bit Flip Code

• Example sending one qubit through a channel.
• Noise flips the qubit with the probability p.
• In other words, the state ?gt is taken to state
  X\?gt with the probability p, where X
  (bit flip matrix)
• Encoding
• Decoding Majority Logic
• Limitation may be unable to recover the
  information correctly if more than one bit is
  flipped in some cases.

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QEC The Three Qubit Bit Flip Code

• Encoding
• ?0gt?1gt ? ?000gt?111gt
• Encoding Circuit
• Measuring the ancilla bits reveals the error but
  not the information qubit.2

Explain Why?
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QEC The Three Qubit Bit Flip Code
Pre-assumption of the errors One or none error
occurs

1. Transfer the stored information to the output
   qubit.
2. Limited if more than one error.
3. We dont have enough info of the location of
   errors.

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Analysis OF WHAT?
Add more detailed captions to the table
The error probability can depend significantly on
the initial state.
0gt?000gt 0gt?000gt 0gt?000gt ? ? ?
Encode Decode Prob. (P0.25) Encode Decode Prob OF WHAT.
000gt 0gt (1-0.25)3 0.4219 0.4219
100gt 0gt 0.1406 0.1406
010gt 0gt 0.1406 0.1406
001gt 0gt 0.1406 0.1406
110gt 1gt 0.0469 0.0469
101gt 1gt 0.0469 0.0469
011gt 1gt 0.0469 0.0469
111gt 1gt 0.0156 0.0156
Error Sum Error Sum 0.01563 Error Sum Error Sum 0
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Fault-Tolerant Computation

• General Stages

Preparation/ Encode
Verify
Computation of Error Syndrome
Recovery
Explain better what each block does, especially
verify
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Fault-Tolerant Computation6

• Rules
• Implement gates that can process encoded
  information.
• Control propagation of errors.
• Ensure that recovery from errors is performed
  reliably.

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Fault-Tolerant Computation6

• 1st Law Dont use the same bit twice.

Bad Error propagates, so infection spreads.
Good!
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Fault-Tolerant Computation6

• 2nd Law Copy/measure the errors, not the data.
• Copy the information from the data to the
  ancilla.
• Measure the ancilla to find an error syndrome.
• Based on the error syndrome, we perform the
  required recovery.

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Fault-Tolerant Computation6

• 3rd Law Verify when you encode a known quantum
  state.

A nondestructive measurement is performed (twice
performed above) to verify that the encoding was
successful.
More explanation needed
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Fault-Tolerant Computation

• 4th Law Repeat the operations

More explanation needed
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Fault-Tolerant Computation

• 5th Law Use the right code

More explanation needed
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Error Correction in The Three Qubit Code

• 0gt?000gt, 1gt?111gt
• Error Correction

More explanation needed
Measurement(M1, M2) Action
(0,0) None
(1,0) Flip the second bit
(0,1) Flip the third bit
(1,1) None
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Example The Shor Code

• Also known as the 9-qubit code
• Combination of the three qubit phase flip codes
  and bit flip codes.
• Seen as a two-level concatenated code.3
• One qubit is encoded into 9 qubits
• The data is no longer stored in a single qubit,
  but instead spread out among nine of them.8
• Correction of bit flips majority voting.

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Assumptions of the Shor Code

• For simplicity, we assume that any qubit error
  consists in the application of bit flip error,
  phase flip error, and/or combination of these
  two.
• X (bit flip error)
• Z (phase flip error)
• Y iXZ (combination of bit flip and
  phase flip error)

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Preparation in The Shor Code
Block 1
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Majority Logic in The Shor Code3
Explain decoding and recovery, how majority
works, may be you need more slides for this
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Bit Flip Correction

• Bit flip switch 0gt and 1gt
• Describe the error as bit flip matrix X
• Correction
• For a block, compare the first two qubits, and
  compare the first with the third.
• If the first was flipped, it will disagree with
  the third.
• If the second was flipped, the first and third
  will agree.

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Phase Flip Correction

• Example
• Describe the error as phase flip matrix Z
• Correction
• By comparing the sign of the first block of three
  with the second block of three, we can see that a
  sign error has occurred in one of those blocks.
• Then, by comparing the signs of the first and
  third blocks of three, we can narrow down the
  location of phase error and flip it back.

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Simultaneous Bit and Phase Flip Error

• Describe the error as YiXZ
• Correction We can fix the bit flip first, and
  then fix the phase flip for the simultaneous bit
  and phase flip error, even if they are on
  different qubits.

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Stabilizer Coding in The Shor Code8

• Bit Flip Error
• Equivalent to measure the eigenvalues of Z1Z2 and
  Z1Z3.
• For example, if the first two qubits are the
  same, the eigenvalue of Z1Z2 is 1 otherwise,
  the value is 1.
• Phase Flip Error
• Equivalent to measure the eigenvalues of
  X1X2X3X4X5X6 and X1X2X3X7X8X9.
• If the signs agree, the eigenvalues will be 1
  otherwise, the values is 1.

Remind on an example what are eigenvalues ,
define them
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Stabilizer Coding in The Shor Code

• In order to totally correct the code, we must
  measure the eigenvalues of a total of eight
  operators.

M1 Z Z I I I I I I I
M2 Z I Z I I I I I I
M3 I I I Z Z I I I I
M4 I I I Z I Z I I I
M5 I I I I I I Z Z I
M6 I I I I I I Z I Z
M7 X X X X X X I I I
M8 X X X I I I X X X
Explain what we see here
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Another Phase Error Correction in The Shor Code

1. Hadamard Transformation on each qubit.
2. The qubits taken 1, 4, and 7 (or 2,5,8 or
   3,6,9).

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Stabilizer Code

• Many quantum states can be more easily described
  by working with the operators that stabilize them
  than by working explicitly with the state itself.
  
• ?gt
• ?X1X2?gt ?gt and Z1Z2?gt ?gt
• ??gt is stabilized by the operators X1X2 and
  Z1Z2.
• ??gt is the unique quantum state which is
  stabilized by these operators X1X2 and Z1X2.

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Stabilizer Code

• In making continuous weak measurements on our
  system, we would like to choose the measurements
  in such a manner that we gather as much
  information about the errors as possible while
  disturbing the logical qubits as little as
  possible?quantum error correcting code.
• Stabilizer formalism provides a way to easily
  characterize many of the error correcting codes.
• Pauli group Pn ?1, ?i?I,X,Y,Z?n

Give examples of Pauli group operators
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Stabilizer Code4

• There exist a set of operators in Pn, called the
  stabilizer generators and denoted by g1, g2, ...,
  gr.
• They are such that every state in C is an
  eigenstate with eigenvalue 1 of all the
  stabilizer generators.
• That is, gi?gt ?gt for all i and for all states
  ?gt in C.
• Moreover, these stabilizer generators are all
  mutually commuting.

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Stabilizer Code4a

• The stabilizer code error correction procedure
  involves
• 1) simultaneously measuring all the stabilizer
  generators and then
• 2) inferring what correction to apply from the
  measurement results.
• The formalism states that the stabilizer
  measurement results indicate a unique correction
  operation.

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ExampleThe Three Qubit Bit Flip Code
ZZI IZZ Error Correction Unitary Action
1 1 None None None
-1 1 Bit 1 flipped XII Flip bit 1
-1 -1 Bit 2 flipped IXI Flip bit 2
1 -1 Bit 3 flipped IIX Flip bit 3
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The Classical 7,4,3 Hamming Code

• Transmit the block 0011

0
0
1
1
Parity bits comes from the rule that the total
number of 1s contained in each circle should be
even.
0
1
1
0
1
1
0
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Steanes Code

• One qubit is encoded into seven qubits.
• Logic 0 those with even number of 1s
• Logic 1 those with odd number of 1s

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Encoder for the Steanes Code
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Computation of Bit Flip Syndrome
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Computation of Phase Errors
R Hadamard Rotation
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Fault-Tolerant Logic Gate

• The fault-tolerant quantum gates and measurements
  must prevent a single error from propagating to
  more than one error in any code block.
• Therefore the small correctable errors will not
  grow to exceed the correction capability of the
  code. 7

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One-Bit Teleportation

• One-teleportation is based on Swap gate

I do not understand . This is not swap Explain
why we need one-bit teleportation
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Several Facts to derive one-bit teleportation

• Fact 1 X HZH where

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Several Facts to derive one-bit teleportation

• Fact 2 When the control qubit is measured, a
  quantum-controlled gate can be replaced by a
  classical controlled operation.

U is performed if the measurement result is 1.
Why it is so?
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Z-teleportation

• The two bits are disentangled before the second
  Hadamard gate.
• Therefore the second qubit can be measured
  before the second Hadamard gate without affecting
  the unknown state in the first qubit.

H?gt
Why it is so?
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X-teleportation
Why it is so?
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Fault-Tolerant Toffoli Using One-bit Teleportation
Non-FT Gate
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Fault-Tolerant Toffoli Gate
0gt
x gt
H
X
0gt
y gt
X
H
Z
0gt
Z
x gt
y gt
z gt
H
Mistake
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Alternative FT Toffoli Gate
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Anticommute Commute

• Anticommute A, BABBA0
• Commute A,BAB-BA0
• Two commuting matrices can be simultaneously
  diagonalized.
• This means that we can measure the eigenvalue of
  one of them without disturbing the eigenvalues of
  the other.
• Conversely, if two operators do not commute,
  measuring one will disturb the eigenvectors of
  the other, so we cannot simultaneously measure
  non-commuting operators.

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References

• 1 Web resouce from http//www-2.cs.cmu.edu/afs/c
  s/project/pscico-guyb/realworld/www/
• 2 Quantum Codes, Class slides from
  http//math.uwyo.edu/moorhous/quantum
• 3 Quantum Physics, abstractquant-ph/0211071
• From Jumpei NIWA view email Date Wed, 13 Nov
  2002 084346 GMT (584kb) Simulating the Effects
  of Quantum Error-correction Schemes
• Authors Jumpei Niwa, Keiji Matsumoto, Hiroshi
  ImaiComments 13 pages, 25 figures
• 4 A Practical Scheme for Error Control using
  Feedback, Mohan Sarovar,1, . Charlene Ahn,2,
  Kurt Jacobs,3, and Gerard J. Milburn1,
• 1Centre for Quantum Computer Technology, and
  School of Physical Sciences,
• The University of Queensland, St Lucia, QLD 4072,
  Australia
• 2Institute for Quantum Information, California
  Institute of Technology, Pasadena, CA 91125, USA
• 3Centre for Quantum Computer Technology, Centre
  for Quantum Dynamics,
• School of Science, Grith University, Nathan, QLD
  4111, Australia
• 6 Reliable Quantum Computers, J. Preskill,
  California Institute of Technology
• 7 Towards Robust Quantum Computation,
  dissertation, Debbie W. Leung, 2000
• 8 Stabilizer Codes and Quantum Error
  Correction, Thesis, Daniel Gottesman,
  California,Institute of Technology, 2001

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References

• 9 Quantum Computation and Quantum Information,
  M.A. Nielsen I. L. Chuang, 2000