Movatterモバイル変換

Shor’s AlgorithmUniversity Of CalcuttaMRINAL KANTI MONDAL

Introduction “Ithink I can safely say that nobodyunderstands quantum mechanics” - Feynman 1982 - Feynman proposed the idea of creatingmachines based on the laws of quantummechanics instead of the laws of classicalphysics. 1985 - David Deutsch developed the quantum Turing machine,showing that quantum circuits are universal[1]. 1994 - Peter Shor came up with a quantum algorithm to factorvery large numbers in polynomial time.1997 - Lov Grover develops a quantum search algorithm withO(√N) complexity.

Background• It is a Quantum algorithm, to find the prime factors ofany given integer N.• Formulated and named after mathematician Peter Shorin 1994.• It takes time O((log N)3). Which demonstrates that aninteger factorization can be solved in a quantumcomputer in polynomial time.• It makes RSA vulnerable to attack, as RSA is based onthe assumption that factoring large numbers iscomputationally infeasible.3

Quantum computation• In quantum computing single particle’s actual state is notrestricted to only one of the two observable states.• When observed or measured, the particle will manifestitself in one observable state or the other with respectiveprobability encoded in the superposition.• Any state of the system is modeled by a unit-length vectorin the Hilbert space C2n.4

Qubit (short of quantum bit)Computational basisState:Measurementnon-deterministiccollapseTwo possible outputs(constraint)5

Multi-qubit Systems2-qubit QC:N-qubitquantum computerstates6

Entangled states2-qubitsystemEntangled stateExample:7

Quantum gatesNOT Gate (Bit Flip)8

Significance of Shor’s algorithm• Fast factorization of any given no.• Finding a factor of a n-bit integer requiresexp( (n^1/3(log n)^2/3) operations using best classicalalgorithm.• Shor’s algorithm can accomplish this same task usingO(n^2(log n(log log n)) operations.• i.e. a quantum computer can factor a number inexponentially faster than the best known classicalalgorithm.11

Shor’s AlgorithmShor’s algorithm shows (in principle,) that a quantumcomputer is capable of factoring very large numbers inpolynomial time.The algorithm is dependant onModular ArithmeticQuantum ParallelismQuantum Fourier Transform

The algorithm• The problem is: given an odd composite number N,find and integer d, strictly between 1 and N, thatdivides N.• The Shor’s algorithm consists of two parts:1. Conversion of the problem of factoring to theproblem of finding the period.(this part can beimplemented classically).2. Finding the period (Quantum period finding) usingthe quantum Fourier transform, and is responsiblefor quantum speedup.13

Classical part1. A random number a<N picked.2. Compute gcd(a, N). This may be done using the Euclideanalgorithm.3. If gcd(a,N)≠1, then there is a nontrivial factor of N.4. f(x + r) = ax + r mod N = ax mod N=f(x).5. If r is odd, go to step 1.6. If a r/2 -1 (mod N), go back to step 1.7. gcd(a r/2 ± 1, N) is a non trivial factor of N.14

Shor’s Algorithm - Periodicity An important result from Number Theory:aF(a) = x mod N is a periodic function Choose N = 15 and x = 7 and we get the following:70 mod 15 = 171 mod 15 = 772 mod 15 = 437 mod 15 =1347 mod 15 = 1...

Shor’s Algorithm - In Depth AnalysisTo Factor an odd integer N (Let’s choose N=15) :21. Choose an integer q such that N < q < 2N2let’s pick 2562. Choose a random integer x such that GCD(x, N) = 1 let’s pick 73. Create two quantum registers (these registers must also beentangled so that the collapse of the input registercorresponds to the collapse of the output register)•Input register: must contain enough qubits to representnumbers as large as q-1. up to 255, so we need 8 qubits•Output register: must contain enough qubits to representnumbers as large as N-1. up to 14, so we need 4 qubits

Shor’s Algorithm - Preparing Data4. Load the input register with an equally weightedsuperposition of all integers from 0 to q-1. 0 to 2555. Load the output register with all zeros.The total state of the system at this point will be:1255∑ |a, 000>√256 a=0InputRegisterOutputRegisterNote: the comma heredenotes that the registersare entangled

Shor’s Algorithm - Modular Arithmetic6. Apply the transformation x a mod N to each number in theinput register, storing the result of each computation in theoutput register.Input Register7a Mod 15Output Register|0>70 Mod 151|1>71 Mod 157|2>7 Mod 154|3>73 Mod 1513|4>74 Mod 151|5>75 Mod 157|6>76 Mod 154|7>77 Mod 15132..

Shor’s Algorithm - SuperpositionCollapse7. Now take a measurement on the output register. This willcollapse the superposition to represent just one of theresults of the transformation, let’s call this value c.Our output register will collapse to represent one of thefollowing:|1>, |4>, |7>, or |13>For sake of example, lets choose |1>

Shor’s Algorithm - Entanglement8. Since the two registers are entangled, measuring theoutput register will have the effect of partially collapsingthe input register into an equal superposition of eachstate between 0 and q-1 that yielded c (the value of thecollapsed output register.)Since the output register collapsed to |1>, the input register will partially collapse to:1√64|0> +1|4> +√641√641|8> +1|12>, . . .√64The probabilities in this case are 64 since our register is now in an equal superpositionof 64 values (0, 4, 8, . . . 252)

Shor’s Algorithm - QFTWe now apply the Quantum Fourier transform on thepartially collapsed input register. The Fourier transformhas the effect of taking a state |a> and transforming itinto a state given by: 1 q-12iac / q√q1√64∑ |c> *ec=0∑ |a> ,|1>aA1√256255∑ |c> *e2iac / 256c=0Note: A is the set of all values that 7 a mod 15 yielded 1. In our case A = {0,4, 8, …, 252}So the final state of the input register after the QFT is:2552iac / 25611∑∑ |c> *e√64 a  A √256 c=0, |1>

Shor’s Algorithm - QFTThe QFT will essentially peak the probability amplitudes atinteger multiples of q/4 in our case 256/4, or 64.|0>, |64>, |128>, |192>, …So we no longer have an equal superposition of states, theprobability amplitudes of the above states are now higherthan the other states in our register. We measure theregister, and it will collapse with high probability to one ofthese multiples of 64, let’s call this value p.With our knowledge of q, and p, there are methods ofcalculating the period (one method is the continuousfraction expansion of the ratio between q and p.)

Shor’s Algorithm - The Factors10. Now that we have the period, the factors of N can bedetermined by taking the greatest common divisor ofN with respect to x ^ (P/2) + 1 andx ^ (P/2) - 1. The idea here is that this computationwill be done on a classical computer.We compute:Gcd (7Gcd (74/2+ 1, 15) = 54/2- 1, 15) = 3Successfully factored 15!

Shor’s Algorithm - Problems The QFT comes up short and reveals the wrong period.This probability is actually dependant on your choice of q.The larger the q, the higher the probability of finding thecorrect probability. The period of the series ends up being odd.If either of these cases occur, we go back to thebeginning and pick a new x.Quantum modular exponentiation, much slower than thequantum Fourier transform.

Applications• Factoring – RSA encryption.• Quantum simulation.• Spin-off technology – spintronics, quantum cryptography.• Spin-off theory – complexity theory, DMRG theory, Nrepresent ability theory.25

Recent works on Shor’s algorithm• In 2001, a 7 qubit machine was built andprogrammed to run Shor’s algorithm to successfullyfactor 15, but no entanglement is observed.• In 2012, the factorization of 21 was achieved.• In April 2012, the factorization of 143 was achieved.26

Reference:• Peter W. Shor “Polynomial-Time Algorithms for Prime Factorizationand Discrete Logarithms on a Quantum Computer “, SIAM Journalon Computing (1997).• Michael A. Nielsen & Isaac Chuang, ‘Quantum computation andquantum information’, Cambridge university press.• Quantum Computing Explained - D. McMahon , (Wiley, 2007) .• An introduction to Quantum Computing - Oxford University Press,Jan,2007.• www.eecis.udel.edu/~saunders/courses/879-03s/• http://www.cs.berkeley.edu/~vazirani/27

Movatterモバイル変換

Change Language

Shor’s algorithm the ppt

Embed presentation

Recommended

More Related Content

What's hot

Viewers also liked

Similar to Shor’s algorithm the ppt

Recently uploaded

Shor’s algorithm the ppt