Algorithms in the Quantum Computing Framework

Transcription

1 Algorithms in the Quantum Computing Framework Juanjo Rué, Sebastià Xambó Laboratoire d Informatique, École Polytechnique (Palaiseau, Paris), ExploreMaps ERC Project Seminar on Quantum Processing: Mathematics, Physics and Technology, IEC, 25th February 2010

2 Summary of the talk Explain the phase estimate framework. Explain the order finding q-algorithm. Explain the factoring q-algorithm. Scope of these techniques Open problems and conclusions

3 Shor s algorithm: order finding and factoring

4 The work of Peter Shor Combinatorics, bin packing, quantum computing,... Nevalinna prize (1998), Gödel prize (1999) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer SIAM Journal of Computing 26, pp (1997) Laudatio ICM 1998: To read our , how mean of the spies and their quantum machine; Be comforted though, they do not yet know how to factorize twelve or fifteen.

5 Quantum Fourier Transform revisited Appereance of periodic phenomenon: change of basis The quantum context provides optimal realizations of the Fourier Transform: F k = 1 2 n 1 kj 2πi e 2 n j 2 n j=0 Exponential order in the classical framework (FFT) vs quadratic in the quantum framework (QFT).

6 The phase estimate: a general algorithm The quantum Fourier transform can be used to get eigenvalues of an unitary operator: Uu = e 2πiϕ u The general procedure starts in the following way: z = 1 2 t ( 0 +e2πi2t 1ϕ 1 ) t)... ( 0 +e 2πi21ϕ 1 )( 0 +e 2πi20ϕ 1 ) u

7 Phase estimate (2) z can be written in the form 1 2 t 2 t 1 j=0 e2πiϕj j u. We proceed with F I : 2 1 t 1 2 t j=0 Two cases: e 2πiϕj j u = 1 2 t 1 2 t j=0 ϕ 2πi e 2 t j j u F I = ϕ u. If ϕ could be written with exactly t bits, then ϕ = ϕ. In the other case, ϕ is a superposition. What do we do in the second case?

8 Phase estimate (and 3) Let ε > 0, 0 < ϕ = 1 2 ϕ < 1 and k a natural integer. Let t be t the number of bits used. Then, if ( t = k + log ) 2ε with probability 1 ε the value ϕ measured after applying F I satisfies ϕ ϕ < 2 k. 2 t Roughly speaking, taking a large value for the number of bits the algorithm gives, with high probability, a value which is closed to the real one. Get good implementations for U How do we construct u? Can we recover the real ϕ from a good estimate ϕ?

9 Searching the order of an element We solve a different problem: given an integer a such that (a, n) = 1, which is the order of a in (Z/nZ)? This problem can be stated in terms of an spectral one: Definition of a convenient unitary operator: U(a, n)u j = u ja (mod n), 0 j n Construction of a family of eigenvectors (reminiscent of representation theory): v s = A nice eigenvalue: U(a, n)v s = ord(a) 1 k=0 ord(a) 1 k=0 e 2πi e 2πi sk ord(a) u a k (mod n) sk ord(a) u a k+1 (mod n) = e 2πi s ord(a) v s

10 Analysis: looking at U(a, n), v s Operator U(a, n): classical modular exponenciation Complexity O ( log(n) 3). Eigenvectors v s ord(a) 1 1 ord(a) s=0 e 2πij s ord(a) v s = a j (mod n) 2 1 t 1 j t 1 j a j (mod n) 2 t 2 t j=0 j=0 1 2 t ord(a) ord(a) 1 s=0 2 t 1 j=0 e 2πij s ord(a) j v s

11 Last step: what we do which a bad estimate? Applying F I we get ord(a) 1 1 s ord(a) v s ord(a) s=0 At the end, we measure the first component: s ord(a) Can we extract information from the real value using this estimate?

12 Intermezzo: continued fractions Let p q be a rational number. Then, p q representation of the form admits a finite We write p q = [p 0, p 1,..., p k ]. p q = p p p The numbers [p 0,..., p r ], r < k are the convergents of p q. Main result: Let p q a rational number. If x satisfies the property p q x < 1 2q 2, then p q is a convergent of x.

13 Continued fractions and estimates Taking a number of bits equal to 2 log 2 (n) log ( ε ) we assure that, with probability 1 ε 1 s 2 t ord(a) s ord(a) < log 2 (n) +1 < 1 2 ord(a) 2 Hence, Two cases: s ord(a) is a convergent of s ord(a). (s, ord(a)) = 1, all works fine! (s, ord(a)) = r 1, we get ord(a) r. Repeat ALL with a ord(a) r

14 ...And now, Factoring! Let N be an integer. We want to find a proper divisor of N. The strategy is the following: let x (Z/NZ) x ord(x) 1 0(N) ord(x) 0(2) ) ) (x ord(x) 2 1 (x ord(x) (N) Not possible that ) ) ) ) ((x ord(x) 2 1, N = ((x ord(x) 2 + 1, N = 1 Hence, we need to find and element x in (Z/NZ) with even order, such that x ord(x) 2 is not 1(N). Can the choice be done uniformly at random?

15 ...And now, Factoring! (2) Let N = m i=1 pα i i its decomposition in prime factors. Let x be a uniformly distributed random element in (Z/NZ). Then ) P (ord(x) 0(2), x ord(x) 2 1(N) 1 2 m. Hence, for N = p 1 p 2 we have a probability greater than 0.75 to get a nice generator!

16 Shor s algorithm INPUT: an integer N which can be written with L bits. OUTPUT: a proper divisor of N. 1. If N 0(2), we return 2, and that s all! 2. We determine if N = a b. This can be done classically with L 3 operations. If this is the case, return a, and that s all! 3. Choose x in {1, 2,..., N 1} uniformly at random. If (x, N) > 1, return (x, N) and that s all! 4. If (x, N) = 1 ord(x) using the order searching algorithm with complexity O(L 3 ). 5. If ord(x) 1(2) or x ord(x) 2 1 (mod N), we return to point 3. Good cases have a high probability to happen! 6. Compute (x ord(x) 2 1, N), (x ord(x) 2 + 1, N), return the bigger, and that s all! A more detailed analysis of point 5 could be done O(log(L)).

17 Scope and open problems

18 Beyond the spectral techniques Order-finding is a problem in which periodicities appear: discrete logarithm problem, period finding,... All these problems can be generalized (in some sense) to finite Abelian groups...but more difficult is to do this for non-abelian groups. (Hallner, 2003) Connections with computacional and algebraic number theory: structure of the units of Z[ D] Pell s Equation: ( x ) ( Dy x ) Dy = 1 x 2 Dy 2 = 1 Deal with other algebraic parameters?

19 (Quantum) Random walks and problems on graphs Grover s search can be generalized to more complex structures: graphs Classically, to search on a graph we can perform a random walk. Do the same, but in the quantum framework Could be this ideas taken to deal with more general graph (and hypergraphs, and oriented graphs, and weighted graphs) problems? Strong connections with Complexity and P=NP in theoretical Computer Science.

20 Some philosophical questions The output of the problems treated so far can be tested The results for testing primality, finding optimal parameters on large graphs cannot be tested! Can we create frameworks to assure categorical algorithms or there is an unavoidable limitation on the language? Are problems that cannot be codified using the rules of the quantum formalism? Are there unavoidable limitations in order to simulate quantum phenomena? (Inherent to nature)

21 Conclusions

22 Conclusions We have presented the phase estimate general framework, and its connection with the QFT. Reduction of order-finding to a linear algebra problem We have reduced the problem of factoring to the problem of order-finding. Spectral techniques can be applied to more difficult problems.

23 Thank you 15 = 3 5