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.

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

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 3 | 2 2 | 2 1 | 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

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.

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

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

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.

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

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 = 1 2 𝑅𝑆 = 1 2 𝑅𝑇 = 1 2 𝑄𝑇 = − 1 2 𝑄𝑆 + 𝑅𝑆 + 𝑅𝑇 − 𝑄𝑇 =2 2 >2 Quantum-Bell inequality, hidden information You cannot know everything about a system at once

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. http://www.ceid.upatras.gr/tech_news/papers/quantum_theory.pdf David Deutsch. Quantum theory, the Church-Turing pirinciple and the universal quantum computer. Proceedings of the Royal Society of London, 400 pp97-117, 1985.

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

Difference in classical and quantum models Bits are either a 1 or a 0 Qubits are in a superposition of and states. 101110111010 |𝟎 Bell States |𝟏

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 1010010 1010110 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 repeated But this requires a lot of space.

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

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.

Parity Check Matrix Every code for the all possible messages are: 000000 100011 001110 101101 010101 110110 011011 111000 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.

Minimum distance and dual code 000000 100011 001110 101101 010101 110110 011011 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 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 000000 110110 011100 011011 101010 101101 110001 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.

Current Study Dual code and superposition Example from MacWilliams Original Code 000000 100011 001110 101101 010101 110110 011011 111000 Dual Code 000000 110110 011100 011011 101010 101101 110001 010111 A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77:793, 1996.

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

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