Classical and Quantum Error Correction - Duke Universityreif/courses/randlectures/ALnotes/Landahl.quantum.errorcor.pdfDecoherence times in practice •Decoherence time refers to the

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