Classical and Quantum Error Correction Chien Hsing James Wu David Gottesman Andrew Landahl
Classical and Quantum ErrorCorrection
Chien Hsing James WuDavid GottesmanAndrew Landahl
Outline
• Classical and quantum channels• Overview of error correction• Classical linear codes• Quantum codes• Conclusions
Two types of channelsTwo types of channelsare discussed:are discussed:
1
2
XOR
Standardaddition
QuantumQuantum Channel Models
Pauli rotations in each qubit
Computing Power versus ErrorComputing Power versus ErrorControlControl
Basic Concepts in Error Control
Error ControlError Control Everywhere
History of Classical ErrorCorrection Codes (ECC)
Encoding is amapping Please remember our
hypercube illustration ofcodes for interpretation
Draw yourselfhypercube pictures for
these, illustrate our(3,1,1) code fromprevious lecture
(3,1,1)
t=1, correct one error
d=2t+1, t=1,2t+1=3=d
w=3
n k d
0
1
transpose
identity
Role of Parity Check Matrix PRole of Parity Check Matrix PExplanation that P returnsonly error syndrome since itanihilates codewords v
Classical Linear Error Control CodesClassical Linear Error Control Codes
General idea of block linearGeneral idea of block linearcodescodes
Matrix vectormultiplication
Galois Fieldhypercube
Smallerspace
generator
We denote it by
Bigspace
distanceSmallerspace
n= length of vector
Error Error DetectionDetection and and CorrectionCorrectionCapabilityCapability
As in general case
3 in ourcase
1 in ourcase
Detection Capability of Linear Block Codes
If codewordis changedto anothercodeword itcannot bedetected
Detection & Correction of (n,k)Linear Block Codes
23-21=6
2 3-1 = 4-1=3
0
1
Linear (Linear (nn,,kk)) Cyclic Cyclic Codes over GF(2)GF(2)
Easy hardware tooperate on thesepolynomials
Encoding a Encoding a CyclicCyclic Code Code
From slide withgeneral diagramof linear codes
Cyclic ShiftsCyclic Shifts in Cyclic Codes
Cyclic propertyCyclic property
Thus we can talk abouta group
Cyclic Group Gc in Code Subspace
Redarrowsrepresentshifts
QuantumQuantumErrorError
CorrectionCorrection
Outline
• Sources and types of errors• Differences between classical and
quantum error correction• Quantum error correcting codes
Introduction: why quantum errorcorrection?
• Quantum states of superposition (which storesquantum information) extremely fragile.
• Quantum error correction more tricky thanclassical error correction.
• In the field of quantum computation, what ispossible in theory is very far off from what canbe implemented.
• Complex quantum computation impossiblewithout the ability to recover from errors
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 willaccumulate.
• External:– Dissipation
• A qubit loses energy to the environment.
– Decoherence
DecoherenceDecoherence• Decoherence is the loss of quantum
information of a quantum system due to itsinteraction with the environment.
• Almost impossible to isolate a quantumsystem from the environment.
• Over time, our quantum system will beentangled with the environment.
Detrimental role of environmentDetrimental role of environment• Information encoded in our quantum system
will be encoded instead in the correlationsbetween the quantum system and theenvironment.
• The environment can be seen as measuringthe quantum system, collapsing itssuperposition state.
• Hence quantum information (encoded in thesuperposition) is irreversibly lost from thequbit.
How to Deal With Decoherence?
Design quantum algorithms to finishbefore decoherence ruins the quantuminformation.
– Can be difficult as• Decoherence occurs very quickly.• Quantum algorithms may be very complex and
long.
First method to deal with First method to deal with decoherencedecoherence
Dealing With Decoherence
Try to lower the rate at whichdecoherence occurs.
– Accomplished by using a right combinationof:• Quantum particle type• Quantum computer size• Environment
Second method to deal with Second method to deal with decoherencedecoherence
DecoherenceDecoherence times in practice times in practice• 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 factorslike temperature and amount of surrounding particlesin the environment
Approximate decoherence time (in seconds) for various system sizes and environment
System size(cm)
Cosmic Radiation
RoomTemperature
SunlightVacuum
(106 particles/cm
3)
Air
10-3
10-7
10-14
10-16
10-18
10-35
10-5
1015
10-3
10-8
10-10
10-23
10-6
1025
105
10-2
10-6
10-19
Gate completion timeGate completion time• Time needed for a quantum gate
operation is as important as decoherencetime.
• Different types of quantum systems havedifferent decoherence time and per gateoperation time.
operation gate quantumper time
timeedecoherenc
edecoherenc before performed becan that operations of noMax
=
In next time we will compare thesecoefficients for real technologies
Maximum number of operations before Maximum number of operations before decoherencedecoherencefor various quantum systemsfor various quantum systems
• The better the decoherence time, theslower the quantum gate operations.
Quantum system
Decoherence
time
(sec)
Time per
Gate Operation
(sec)
Max number of
operations
before
decoherence
Electrons from gold atom 10-8
10-14
106
Trapped indium atoms 10-1
10-14
1013
Optical microcavity 10-5
10-14
109
Electron spin 10-3
10-7
104
Electron quantum dot 10-3
10-6
103
Nuclear spin 104
10-3
107
Decoherence time versus time required for a quantum gate operation
for various quantum systems
Dealing With Decoherence andother sources of errors
Use Quantum Error correcting codes• Encode qubits (together with extra ancillary
qubits) in a state where subsequent errors canbe corrected.
• Allows long algorithms requiring many operationsto run, as errors can be corrected after theyoccur.
Third method to deal with Third method to deal with decoherencedecoherence
History of Quantum ErrorQuantum ErrorCorrectionCorrection Codes (QECC)
Quantum Error Correcting Codes
QuantumErrors
General representation of single qubit
Cloning (copying) operator U doesnot exist
Assume thatsuch U exists
So we apply it togeneralsuperposed state
And we obtain this Which is not what wewanted
But this is still useful. Although not copying , this is a redundancy introducing operator soit may be used for error correcting codes. This was one of main ideas
Commuting and Anti-CommutingCommuting and Anti-CommutingQuantum OperatorsQuantum Operators
Commutator of A and B
Anti-commutator of Aand B
(1-qubit) Pauli Operators
We express Y interms of X and Z
Properties of Pauli Operators
Adjointoperator
commutative
PauliPauli operators are self- operators are self-inverses and anti-inverses and anti-commutecommute
anticommutative
1-qubit Pauli Group G1
4 * 2 = 8 elementsin this group
PauliPauli operators are a group operators are a groupPlease remember, this is important
Now we extend to group Gn
We model faultsin channels byGn
Example: error operator in GG55Tensor product
This will be ourerror model fromnow
Quantum network forQuantum network forcorrecting errorscorrecting errors
0
01s
2s
3eb!
2eb!
1eb!
• Assume thatb
b
b
111000
e1e1e1eee123123
!+"
#$$$!+"
1123!++ eee }1,0{!
ie
Input signal with error
Input signal after error correcting
Decoder and corrector
EquivalentlyEquivalently
0
01s
2s
3eb!
2eb!
1eb!
b
b
b
1s
2s
Perform operations on logicalPerform operations on logicalbitsbits
b H
• e.g. Hadamard gate
b
b bbb2
)1(
bbb2
1
b!
+
Quantum Error CorrectingQuantum Error Correctingby Peter by Peter ShorShor
• In 1995, Peter Shor developed animproved procedure using 9 qubits toencode a single qubit of information
• His algorithm was a majority vote typeof system that allowed all single qubiterrors to be detected and corrected
This was a starting point to great research area,although his paper had many bugs