This document discusses quantum error correction. It begins by explaining the need for quantum error correction due to noise and imperfections in real-world quantum systems. It then discusses barriers to quantum error correction like the no-cloning theorem. Different types of quantum errors like bit flips, phase flips, and more complex errors are described. Classical error correction techniques are compared. Finally, specific quantum error correcting codes like the repetition code, phase flip code, and Shor's code are explained as ways to protect quantum information against noise by encoding quantum states.

1. Quantum Error Correction
Yamini Singh.

2. Contents
What is error correction
• Intention to study this
• Why do we need this (Quantum) Error
Correcting (QEC)
• Classical error correction
• Barriers to QEC
• Types of Errors
• Quantum Error Correcting Codes

3. What is error correction
• To correction the error in the system is known as
error correction
• In what frame of reference :-
to process the quantum information without
errors even in the noisy environment.

4. Intention to study…
• How to process the information in the presence
of noise correctly.
• We begin by developing the basic theory of qec
codes, which protect the quantum information
against noise.
• These codes work by encoding quantum states in
a special way that make them resilient against
the effect of noise and then decoding when it is
wished to recover the orginal state.

5. Why do we need (Quantum) Error Correction?
• Theoretically operators and states are pristine
and perfect.
• In a lab:
▫ Approximations must be made.
▫ Operators don’t always do as they should.
▫ Fidelity of the prepared states and the theoretical
states is not perfect .
• Quantum Error Correction (QEC) deals with the
imperfection of the real world.

6. Barriers to Quantum Error Correction
• Measurement of error destroys
superpositions. (Classically we can
observe all bits)

7. Barriers to Quantum Error Correction
• No-cloning theorem prevents repetition.
• Not the same state. => contradiction proves a
cloning operator cannot exist.
• Means you can’t just keep “back up” of states.
Must protect original.
• Also prevents some error correction codes.
0 + 1  00 + 11
 (0 + 1)(0 + 1)

8. Barriers to Quantum Error Correction
• Must correct multiple types of errors
(not just bit flips).
• Must correct continuous errors and
decoherence.

9. Classical Error
• Say, probability failure per gate = p
• Probability getting right answer with n gates:n
p)1( 

10. Classical Error Correction


0|1|)1(1|
1|0|)1(0|
pp
pp
If you have a possible bit-flip error like:
Can try an error correction code like:


01|11|)1(11|
10|00|)1(00|
pp
pp

11. Take a example
• Error detection or syndrome diagnosis
• Let us examine more closely the error syndrome
for the classical repetition code.
• We performed a measurement which tells us
what error,if any,occurred on quantum state.
• Mesurement result is called quantum syndrome
• For bit flip channel there are four error
syndrome.

12. Syndrome mesurement
• Does not cause any change to the state:it is
a100 +b011 both before and after syndrome
measurement.
• Contain only information about what error has
occurred
• Does not allow us to infer anything about the
value of aor b.i.e it contain no information about
the state being protected.
• recovery: value of error syndrome to tell us what
produre is used to recover initial state

13. Know thy enemy - Errors
Phase Flip Z:
Complete dephasing:
(depolarisation)
Rotation R: R0 = 0, R1 =ei1
At R->Z 
 1|0|1|0| 
2/)1( pIp  

14. Know thy enemy - Errors
• A general operator :
• A density matrix ρ describes the statistical
state of a system.
*
||




AA
A



15. Correcting Phase (Z) Errors
Hadamard transform H exchanges bit flip
and phase errors:
H (0 + 1) = + + -
X+ = +, X- = -- (acts like phase flip)
Z+ = -, Z- = + (acts like bit flip)
Repetition code corrects a bit flip error
+ + -  +++ + ---
The same code in a new basis
corrects a phase error!
2/)1|0(||
2/)1|0(||



16. The shor’s code:-
• This is the simple quantum code which can
protect against the effects of arbitrary error on a
single qubit!
• This code is known as shor code, after its
inventor.
• Combination of three qubit phase flip and bit flip
code.

17. Shor code
• First we encoded each of these qubit using the
phase flip code:|0> |+++>,|1>|--->
• Next we encoded each of these qubits using the
three qubit bit :|+> is encoded as
(|000>+|111>)sqrt2 and |-> is encoded as
(|000>-|111>)sqrt2 .
• The result is a nine qubit code,with codewords
given by

18. Shor’s Code
•This is simply a combination of the two codes
above.
•Had to stick to one basis, so it’s a little less
intuitive.
•Correct for both X (bit-flip) and Z(phase-flip)
•Also, get Y errors for free Y=iXY!

19. Summary
•Applied concepts of classical error
correction to QEC.
•Learnt about quantum errors.
•Circumvented the problem caused by no
cloning and superposition.
•Learnt codes to correct for multiple types
of errors that can occur in quantum
computing.

20. Questions?