1. Review from last lecture: A Simple Quantum (3,1) Repetition Code
Recovered state

2. Single Qubit errors
Bit flip error:
Do a bit flip using a operator.

3. Phase flip error: Do a phase flip using a operator.
Phase flip affects interference.
Interference: when parallel computation are carried out by a quantum computer, these parallel computation can reinforce or cancel out each other.

4. Bit and phase flip error:
Do a bit and phase flip using a operator.
X,Y,Z are pauli matrices. They are also called depolarisation errors

5. A review of a simple classical error correction encoding
3 bit repetition encoding:
0 encoded as 000
1 encoded as 111
Assuming only 1 bit error
Decoding: Take majority vote of the 3 bits
E.g.
This scheme can correct 1 bit error and detect up to 2 bit errors.
Classical linear error correcting codes involve encoding k bits to be protected into a n bit string, where n>k
Recall also hypercubes

6. Why using classical error correction for correcting Qubits is not trivial?
First reason
No cloning theorem
Unable to encode as
Measurement of qubits cause disturbance
Need to do error correction without measuring the value of each qubit.
Classical error correction takes for granted that bits can be measured as much as you want.
Second reason

7. Why using classical error correction for correcting Qubits is not trivial?
Third reason
Unable to correct phase errors
Unable to correct small errors
For , an error might change α and β by a small order.
These small errors can accumulate.
Classical methods only designed to correct large discrete errors (i.e. bit flips)
Fourth reason
Quantum errors are usually continuous.
But we will solve all these problems

8. Quantum Error correcting codes
Correcting single bit flip error using 3 qubits
Correcting single phase error using 3 qubits
9 qubits error correcting code
5 qubits error correcting code
Concatenated code

9. Simple (3,1) repetition code circuit
This circuit can correct single bit flip and detect double bit flip.

10. Error Correction for 1 Bit Flip
This shows what happened
If bit flip occurred in data bit than syndrome is 11, used for correction

11. Encoder for (3,1) Repetition Code
For encoding, use 2 extra qubits initially set to
Encoding circuit:
Calculated from in Dirac notation as xor of 1 and 0

12. We use slightly different notation to explain it even better+

13. How decoder works?
Assuming at most 1 bit will be flipped and the bit flip is just as likely to affect any qubit.
Decoding circuit:
Changes in second bit
As usually red bits show change in our pictures
Changes in third bit

14. The important idea of Syndrome
The last 2 qubits are called the syndrome and their values indicate the error type that occurred.
All possible states at the end of decoding circuit:
Syndrome
Error 00
No error 01
3rd qubit flipped 10
2nd qubit flipped 11
1st qubit flipped
Syndrom as a result of error that happened
Only this is wrong
good

15. In this case correction is trivial
Correction circuit:
If syndrome bits are not ’00’, discard them and re-encode using new qubits.

16. Let us analyze one more time the Decoder for (3,1) Repetition Code using another notation
This are all possible signals with no error or with error from transmission
This are all their counterpart final signals
Results of correction. As we see this is majority

17. Correcting single phase flip in (3,1) circuits
Use Hadamard to convert a phase flip to bit flip
Similarly:
Pauli X
Pauli Z
This is another fundamental trick – convert one type of error to another which is easier to manipulate

18. Proof of the first of the above convertions
Now we will see how this idea is used

19. Correcting single phase flip
Complete Circuit for correcting single bit flip:
Modified circuit to correct single phase flip.
1st one can correct bit flip but not phase flip. 2nd one can correct phase flip but not bit flip.
To detect phase flip we add Hadamards at the end of encoder and at beginning of decoder
If there is a phase flip, two hadamards will convert it to bit flip

20. Initial Problems Avoided
No cloning involved in encoding
Able to diagnose the error without damaging the quantum information.
Able to correct errors without knowing state of qubit.
Able to correct bit flip or phase flip error depending on the circuit used.
Few tricks solves many of problems listed earlier!!

21. Able to correct small errors
Few tricks solves many of problems listed earlier!!
Able to correct small errors
Example: Assume encoded qubit damaged such that:
0.7 probability of getting no errors
0.3 probability of getting 1st bit flipped

22. Step by step analysis of decoding and correction
After the circuit, 1st qubit will always be
The decoding circuit maps the state into either one with no error, or one with an error which we know how to correct.
Unique syndroms allow to correct if 11
Error correction is possible even if error is a superposition.
Quantum error correction will digitalize the errors.

23. Shor’s 9 qubits error correcting code
The 2 codes earlier corrects either bit flips or phase flips.
Shor’s 9 qubits error correcting code combines both codes.
It can correct any arbitrary single qubit error
9 qubits used to encode 1 qubit.

24. Basic Idea of Shor Code
Correction of bit & phase flip errors

25. Shor code:

26. Architecture of Shor Code
encoder
decoder

27. First we explain the principle of Encoding in Shor code
Use 9 qubits to encode 1 qubit (9,1):

28. Thus for general qubit we have
Encoding circuit:

29. Shor code –Encoding
Bell state |+> and |->
Entangled GHZ states

30. Now we show step by step how encoder works
Tensor product of results of Hadamards with zeros
Xoring in second and third bits with 1 from first bit

31. Now we show step by step how DECODER works
Assuming at most 1 qubit error and the error is just as likely to affect any qubit.
The decoding and correction circuit:
Appreciate please the mirror like symmetry

32. Detailed analysis of an error
Example: Assume encoded qubit damaged such that:
Send to line
Received from line
As we see the red error is in phase and bit flip of first qubit

33. Shor code –Decoding
We can explain it using Bell and GHZ states quickly
Or use full notation for analysis

34. From line
Before correction

35. After correction

36. Before Hadamards, from previous slide
After Hadamards
After bit flip decoding
Xoring in bits 2 and 3 only in alpha part

37. And finally after the entire bit flip and phase flip corrections we get:
So we got what we wanted to get!

38. The (9,1) circuit – put all together
This paper published recently started a furry of results and great ideas

39. Now we explain in a different way how Shor’s [9,1,3] code works
Pauli Z in bit one

40. All nine bits used

41. Thus there are no small errors, only large errors which we can fix