Quantum ErrorCorrection:

• Andrew Landahl• David Gottesman• Dr. Wu• And others

Stabilizer measurement?? Stabilizer measurement??

0

This is implementing a measurement (interpreting 0 as +1, and 1 as –1)

1Z

Z1

0

This is implementing a measurement

21ZZ

Stabilizer measurement?? Stabilizer measurement??

Z1

Z2

Stabilizer measurement??

0

This is implementing a measurement

21XX

H H

Stabilizer measurement?? Stabilizer measurement??

Notation clarification Notation clarification

For an n-qubit system denotes

E.g. n=3, then

jZ

jnj

IIZIII

1

IZZIZIIIZZZ 21

What does it mean Z1 Z2?

The Shor The Shor Code as a Code as a Stabilizer Stabilizer

code code

Stabilizer codesStabilizer codes

Stabilizer Codes by Stabilizer Codes by Gottesman, 1997Gottesman, 1997

Classical (7,4,3) Hamming Classical (7,4,3) Hamming Code CCode C11

Stabilizer for Steane code

Steane code: based on Hamming code

Vector Space under Symplectic Inner Products

Weight and Distance

Theorem for Existence of Additive QECC

Stabilizer for 5-qubit [5,1,3] codeStabilizer for 5-qubit [5,1,3] code

Symplectic Form for Stabilizer

Commuting Group

Linear Block Codes

Standard Symplectic Form

Encoder for Stabilizer Codes

[5,1,3] Quantum Encoder (X part)

(5,1,3) code(5,1,3) code

Complete [5,1,3] Quantum Encoder

(5,1,3) code(5,1,3) code