Error-Correcting Codes: Classical to Quantum

Timothy S Woodworth and Kishor T. KapaleDepartment of Physics,

Western Illinois University, Macomb IL, 61455

1

2

OutlineClassical Intro

o Binary Numberso Universal Turing Machine

Quantum Introo Stern-Gerlach Experimento Vectorso Bell Inequality and Hidden Informationo Quantum Computers

Error-Correcting Codeso Repeatso Matriceso Future Research

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 | | |

1 | 0 | 1 | 1 | 0

Base 10 ||

0 | 2 | 2

10110=22

AddingXOR

A | B | X 0 | 0 | 0 0 | 1 | 1 1 | 0 | 1 1 | 1 | 0

MultiplyingAND

A | B | X 0 | 0 | 0 0 | 1 | 0 1 | 0 | 0 1 | 1 | 1

2 10

0

1

2

3

4

5

6Voltage

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 StateControl

1 1 1 1 1 1 10 0 0 0

Read/WriteHead Tape

Micheal A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information.. Cambridge University Press, 2000.

5

Classical Intro Binary Numbers Universal Turing Machine

Quantum Introo Stern-Gerlach Experimento Vectorso Bell Inequality and Hidden Informationo Quantum Computers

Error-Correcting Codeso Repeatso Matriceso Future Research

Outline

6

OvenSilver Ions

z

y

^ Stern-Gerlach Experiment

+12

−12

OvenSilver Ions

+12

−12

x

y

^

7

Vectors

R=Sin θ CosφSin θ Sinφ

Cosθ

θ2

C

N

S

Z=

Stig Stenholm and Kalle-Antti Souminem. Quantum Approach to Informatics.. John Wiley & Sons Inc., 2005.

Bloch Sphere

Poincaré Sphere

8

Vectors2¿𝜓 ⟩=𝛼 ¿+ 1

2 ⟩+𝛽 ¿− 12 ⟩

⟨𝜓|𝜓 ⟩=1¿

¿𝜓 ⟩𝑧=1√2

¿+ 12 ⟩+ 1

√2¿− 1

2 ⟩

⟨ +¿− ⟩=0𝛼𝛼∗+𝛽 𝛽∗=1

¿𝜓 ⟩¿¿

¿ − ⟩

9

Bell Inequality and Hidden Information

AliceQ = ±1R = ±1

BobS = ±1T = ±1

1 Particle

1 ParticleQS+RS+RT-QT = (Q+R)S+(R-Q)T

Q+R = 0 or Q-R = 0

= ±2

R,Q = ±1

QS+RS+RT-QT ≤ 2

⟨𝑄𝑆 ⟩ +⟨ 𝑅𝑆 ⟩+ ⟨𝑅𝑇 ⟩ − ⟨𝑄𝑇 ⟩=2√2>2

You cannot know everything about a system at once

10

Quantum ComputersKeynote 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.

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

Classical Intro Binary Numbers Universal Turing Machine

Quantum Intro Stern-Gerlach Experiment Vectors Bell Inequality and Hidden Information Quantum Computers

Error-Correcting Codeso Repeatso Matriceso Future Research

Outline

12

Classical Quantum

Bits are either a 1 or a 0

101110111010

Difference in classical and quantum models

¿𝟎 ⟩

¿𝟏 ⟩

Bell States

Qubits are in a superposition of and states.

13

Sender Receiver1010010

1010110

Noise

When information is sent from the Sender to the Receiver, there exist a probability thatSome error will occur due to noise in the channel.

To help find and fix these errors, we attach a coded message to the end of the message.

101001010100101010010 A simple code would just be the message repeatedBut this requires a lot of space.

Sending information

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

Simple Quantum Code

15

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.

So, if A was, Then G would be,

If I had a message (101), I would get the code (x) from:

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).

F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-correcting Codes. North-Holland, Amsterdam, 1977.

The Generator Matrix

16

000000 100011001110 101101010101 110110011011 111000

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 =0T

(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.

Parity Check Matrix

17

000000 100011001110 101101010101 110110011011 111000

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 codeThe 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

000000 110110011100 011011101010 101101110001 010111

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.

Minimum distance and dual code

Example from MacWilliams

18

Current Study

Dual code and superposition

Original Code000000 100011001110 101101010101 110110011011 111000

Dual Code000000 110110011100 011011101010 101101110001 010111

A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77:793, 1996.

Example from MacWilliams

19

Acknowledgments

• Dr. Kapale• Research• Class

• Dr. Babu• Class

• Dr. McQuillan• Classical knowledge

20

Classical Intro 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