1. Error-Correcting Codes:
Classical to Quantum
Timothy S Woodworth and Kishor T. Kapale Department of Physics,
Western Illinois University, Macomb IL, 61455
2 more slides, titles, and cartoon.

2. Outline Classical Intro Quantum Intro Error-Correcting Codes
Binary Numbers
Universal Turing Machine
Quantum Intro
Stern-Gerlach Experiment
Vectors
Bell Inequality and Hidden Information
Quantum Computers
Error-Correcting Codes
Repeats
Matrices
Future Research
Outline

3. Binary Numbers
Binary is a base 2 representation of numbers. What could be thought of as Normal Numbers are in base 10
Base 2
2 4 | | | | 2 0
1 | 0 | 1 | 1 | 0
Base 10
10 2 | 10 1 | 10 0
0 | 2 | 2
10110=22 2 10
Adding
XOR
A | B | X
0 | 0 | 0
0 | 1 | 1
1 | 0 | 1
1 | 1 | 0
Multiplying
AND
A | B | X
0 | 0 | 0
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1
Classical-What is binary, binary math

4. Universal Turing Machine
1: ‹qs, ,q1, ,+1›
2: ‹q1,0,q1,b,+1›
3: ‹q1,1,q1,b,+1›
4: ‹q1,b,q2,b,-1›
5: ‹q2,b,q2,b,-1›
6: ‹q2, ,q3, ,+1›
7: ‹q3,b,qh, 0,1›.
F(x)=1
Program
Finite State
Control
Classical-universal Turing machine
Read/Write
Head
Tape 1 1 1 1 1 1 1
Micheal A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information.. Cambridge University Press, 2000.

5. Outline Classical Intro Quantum Intro Error-Correcting Codes
Binary Numbers
Universal Turing Machine
Quantum Intro
Stern-Gerlach Experiment
Vectors
Bell Inequality and Hidden Information
Quantum Computers
Error-Correcting Codes
Repeats
Matrices
Future Research
Outline

6. Stern-Gerlach Experiment^
Stern-Gerlach Experiment z ^ y
+ 1 2
Oven
Silver Ions
−1 2 ^ x ^ y
+ 1 2
Quantum-SG experiment
Oven
Silver Ions
−1 2

7. Vectors Sin θ Cosφ Sin θ Sinφ R= Cosθ N C Bloch Sphere Z= 𝑒 𝑖𝜑 𝑡𝑎𝑛 𝜃 2
Quantum-Bloch Sphere, poincare
Z= 𝑒 𝑖𝜑 𝑡𝑎𝑛 𝜃 2
θ 2
Poincaré Sphere S
Stig Stenholm and Kalle-Antti Souminem. Quantum Approach to Informatics.. John Wiley & Sons Inc., 2005.

8. Vectors2 |𝜓 =𝛼 | + 1 2 +𝛽 | − 1 2 𝜓|= 𝛼 ∗ + 1 2 | + 𝛽 ∗ − 1 2 | 𝜓 𝜓 =1
|𝜓 =𝛼 | 𝛽 | − 1 2
𝜓|= 𝛼 ∗ | + 𝛽 ∗ − |
𝜓 𝜓 =1
+ − =0
𝛼 𝛼 ∗ +𝛽 𝛽 ∗ =1
|𝜓 𝑧 = | | − 1 2
Quantum-Superposition, orthogonallity.
| + |𝜓
| −

9. Bell Inequality and Hidden Information
Alice
Q = ±1
R = ±1
Bob
S = ±1
T = ±1
1 Particle
QS+RS+RT-QT
= (Q+R)S+(R-Q)T
= ±2
1 Particle
R,Q = ±1
Q+R = 0 or Q-R = 0
QS+RS+RT-QT ≤ 2
𝑄𝑆 =
𝑅𝑆 =
𝑅𝑇 =
𝑄𝑇 = − 1 2
𝑄𝑆 + 𝑅𝑆 + 𝑅𝑇 − 𝑄𝑇 =2 2 >2
Quantum-Bell inequality, hidden information
You cannot know everything about a system at once

10. Quantum Computers Keynote speech at MIT
“And therefore, the problem is, how can we simulate the quantum mechanics? There are two ways that we can go about it. We can give up on our rule about what the computer was, we can say: Let the computer itself be built of quantum mechanical elements which obey quantum mechanical laws. Or we can turn the other way and say: Let the computer still be the same kind that we thought of before--a logical, universal automaton; can we imitate this situation?”
Richard P. Feynman. Simulating Physics with Computers. International Journal of theoretical Physics, 21:6/7, 1982.
Quantum-Feynman Quantum Computer.
Quantum-Universal Quantum Turing machine.
David Deutsch. Quantum theory, the Church-Turing pirinciple and the universal quantum computer. Proceedings of the Royal Society of London, 400 pp97-117, 1985.

11. Outline Classical Intro Quantum Intro Error-Correcting Codes
Binary Numbers
Universal Turing Machine
Quantum Intro
Stern-Gerlach Experiment
Vectors
Bell Inequality and Hidden Information
Quantum Computers
Error-Correcting Codes
Repeats
Matrices
Future Research
Outline

12. Difference in classical and quantum models
Bits are either a 1 or a 0
Qubits are in a superposition of and states. |𝟎
Bell States |𝟏

13. Sending information
When information is sent from the Sender to the Receiver, there exist a probability that
Some error will occur due to noise in the channel.
Noise
Sender
Receiver
To help find and fix these errors, we attach a coded message to the end of the message.
A simple code would just be the message repeated
But this requires a lot of space.

14. Simple Quantum Code
P. Shor. Scheme for reducing decoherence in quantum memory. Phys. Rev. A, 52:2493, 1995.

15. The Generator Matrix
For ‘k’ symbols in a message (u), you would want a ‘n’ (where n>k) length code (x) that would check that the message was sent correctly and possibly be able to fix any errors.
We could use a Generator matrix (G) to create the code (x).
u.G=x
G can be found by [Ik|A], where Ik is the identity matrix size (k) and A is a matrix size k X (n-k).
So, if A was, Then G would be,
If I had a message (101), I would get the code (x) from:
F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-correcting Codes. North-Holland, Amsterdam, 1977.

16. Parity Check Matrix Every code for the all possible messages are:
At the receiving end, we would check the code with parity check matrix (H), where:
H.x =0 T
(H) is created by:
[A |In-k] T
So in our example, H=
If given the correct code (101101)
If given the wrong code (101111)
F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-correcting Codes. North-Holland, Amsterdam, 1977.

17. Minimum distance and dual code
The min distance(d) of this code
(the minimum difference between any
2 code words) is 3.
If d is odd, a code can correct (d-1)/2 errors.
If d is even, it can correct (d-2)/2 errors and detect d/2.
The dual code
The dual codes generator matrix (G) is the parity check matrix (H) of the original code
-and-
The dual codes parity check matrix (H) is the generator matrix (G) of the original code
Example from MacWilliams
The dual code for our example is:
F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-correcting Codes. North-Holland, Amsterdam, 1977.

18. Current Study Dual code and superposition Example from MacWilliams
Original Code
Dual Code
A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77:793, 1996.

19. Acknowledgments Dr. Kapale Research Class Dr. Babu Dr. McQuillan
Classical knowledge
Acknowledgments

20. Outline Classical Intro Quantum Intro Error-Correcting Codes
Binary Numbers 3
Universal Turing Machine 4
Quantum Intro
Stern-Gerlach Experiment 6
Vectors 7-8
Bell Inequality and Hidden Information 9
Quantum Computers 10
Error-Correcting Codes
Repeats 13-14
Matrices 15-16
Future Research 18
Outline