QUANTUM COMPUTING: SHORS FACTORING ALGORITHM A MATHEMATICAL - PowerPoint PPT Presentation

QUANTUM COMPUTING: SHOR’S FACTORING ALGORITHM A MATHEMATICAL DESCRIPTION OF THE FUNCTION OF SHOR’S ALGORITHM AND THE BASIC PRINCIPLES OF QUANTUM COMPUTATION A HIGH-LEVEL SUMMARY Eero Jääskeläinen Mathematics

MOTIVATION Quantum computation is at the heart of the intersection of physics and mathematics. • Abstract yet functional. • Striking elegance in the function algorithms. Quantum computation provides solutions to real problems • Optimization • Simulation • Cryptography

AIMS AND SCOPE Not… But instead… The development of a new method or invention.  A complete introduction to a revolutionary  algorithm in a way that provides new intuitions. Including original proofs and discussion.  A light or popularized introduction to quantum Detailed description including proofs, but largely   computing. self-contained. A focus on the practical implementation of the A mathematical description of the function of the   algorithm. quantum process. Highlights the novelty, peculiarity and elegance of  QC.

MATHEMATICAL BACKGROUND Quantum computation is described in the language of complex linear algebra  Dirac bra-ket notation useful in representing quantum states  Unitary operations represent quantum state evolution, i.e. quantum gates    v   0   1 1 1   =  = = v   U  w w w − 0 n   1 1   2   v n Modular arithmetic is also utilized in the discussion. 

MATHEMATICAL BACKGROUND The Euclidean algorithm is a method of computing the greatest common divisor of two integers.  = + A BQ R 1 1 = + B R Q R 1 2 2 = + R R Q R 1 2 3 3 = R R Q + + k k 1 k 2 = = It is shown that, for the sequence above, gcd( , A B ) gcd( R R , Q ) Q  + + + k k 1 k 2 k 2 This algorithm is efficient : It can be shown that the value of R at least halves in two iterations. Hence the number of  B + iterations necessary is at most log 2 2

REFORMULATION OF FACTORING N To take advantage of the properties of quantum computation, a corollary problem is developed:  Define Find r such that ( )  ( ) f r 1(mod N ) = x f x a (mod N ) a N , a N ,  = + f ( ) x f ( x r ) a N , a N , Then If r is even Obtain Factor of N:        r r r   N a − − + r  ∣      1 ∣ 2 2   N a 1 a 1 2 gcd N , a 1          

THE QUANTUM FRAMEWORK Qubits are the basic units of information.  States 1, 0, and superpositions of these         1 0 = =  =       0 , 1 ,        0 1 Quantum registers are systems of multiple qubits.  Numbers are stored in binary, like in classical computation.  Quantum gates are interactions that transform the states of qubits  Drawn with Qiskit -library in python. Represented by linear matrices (operators), satisfying requirements  based on the principles of quantum physics.

THE INVERSE QUANTUM FOURIER TRANSFORM AND MEASUREMENT After some manipulation of qubits, we obtain the superposition state  − o 1 1    = + A Periodic Superposition of t jr 0 States with Period r o = j 0 Inverse Quantum Fourier Transform:  ‘A sequence of gates that changes the state  to relate to the frequency of the states.’  Measure to get an integer z.

OBTAINING A RESULT: CONTINUED FRACTIONS 4  = It is shown that, with probability  P 0.40828  2 Some Measured integer. value. z k 1 −   m m 2 r 2 2 Number Required of qubits. period. ( 2 m >N 2 ) z k  takes the nearest for some integer k . possible value to 2 m r

OBTAINING A RESULT: CONTINUED FRACTIONS Resulting from the Euclidean algorithm, we can construct continued fraction representations of rationals:  A 1   =  +   =  a a a , , , , a a a , a for j 1,2, , m 0 1 2 m 0 0 j 1 B + a 1 1 + a 2 1 + a + 3 a m 𝐵 For all and C , 1 𝐶 = 𝑏 0 , 𝑏 1 , 𝑏 2 , … , 𝑏 𝑛 can take on any  x   = c c  non-zero value for C a a a , , , , a , , 0 1 2 m some real x . Continued fraction if c  1

OBTAINING A RESULT: CONTINUED FRACTIONS Given the condition z k 1 −   m m 2 r 2 2  Period r found in a way k   =  it is proven that if a a a , , , , a that is known to be  0 1 2 j r efficient computationally. z   =  then a a a , , , , a c ,  0 1 2 j m 2 𝑨 𝑙 where c>1. This implies that is one convergent of 2 𝑛 𝑠 (Assuming k and r are coprime so that the fraction is in lowest terms.)

 Self-contained description of a complex and valuable algorithm.  Including introduction to the mathematical framework, justification of the IQFT, and details of continued CONTRIBUTIONS fractions.  Insight into the Elegance of Quantum Computing and its mathematics.  Inspiration to the future from understanding a non-trivial process.

It is magic until you understand it, and mathematics thereafter. -Bharati Krishna

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REFERENCES (2) Lynn, Ben. “Continued Fractions – Convergence”. PBC Library ,  https://crypto.stanford.edu/pbc/notes/contfrac/converge.html. Accessed 5 September 2019. Lyons, James. “An Intuitive Discrete Fourier Transform Tutorial”. Practical Cryptography , 2019,  http://practicalcryptography.com/miscellaneous/machine-learning/intuitive-guide-discrete-fourier-transform/. Accessed 23 May 2019. Nielsen, Michael A., and Chuang, Isaac L.. Quantum Computation and Quantum Information . Cambridge University Press,  2010. Qiskit, https://qiskit.org/. Accessed 3 September 2019.  Shor, Peter W. “Polynomial -time algorithms for prime factorization and discrete logarithms on a quantum  computer.” SIAM review 41.2 (1996): 303-332. Stenholm, Stig, and Suominen, Kalle-Antti. Quantum Approach to Informatics . Wiley-Interscience, 2005.  Sysoev , Sergey. “The Introduction To Quantum Computing”. Saint Petersburg State University , 2019,  https://www.coursera.org/learn/quantum-computing-algorithms/home/welcome. Accessed 6 August 2019. Weisstein , Eric W. “Euclidean Algorithm”. Wolfram Mathworld , 2019,  http://mathworld.wolfram.com/EuclideanAlgorithm.html. Accessed 2 June 2019.

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