Quantum Annealing for Prime Factorization

Transcription

1 Quantum Annealing for Prime Factorization Shuxian Jiang, Sabre Kais Department of Chemistry and Computer Science, Purdue University Keith A. Britt, Alexander J. McCaskey, Travis S. Humble Quantum Computing Institute, Oak Ridge National Laboratory Qubits 2018

2 Integer Factorization Problem Classical Factorization - Methods: 1. Trial Division(n 1 2 ): try 2,3,5,...,n 1 2 to divide n 2. Shanks, Pollard, Strassen (1970s)(n 1 4 ): complicated math 3. CFRAC(1970), QS(1980), NFS(1990) O(exp (ln(n))(ln(ln(n))) ): nonrigorous 4. Elliptic Curve Method(1985)O(exp (ln(n))(ln(ln(n))) ): probabilistic, nonrigorous - Limitations: EXP time if no randomness, no unproved hypotheses Quadratic Sieve Method(subexponential, not polynomial)

3 Integer Factorization Problem Quantum Factorization - Gate Model Shor s algo: order-finding Exponential speedup: polynomial time Hard to physically implement Largest Number: 21 - Adiabatic Computation Model Convert to optimization problem 1. NMR f = (N pq) 2 2. Ising Machine: D-wave,... Multiplication Table 3. Nitrogen-Vacancy (NV) center in diamond Multiplication Table

4 Integer Factorization Problem Quantum Factorization - Gate Model Shor s algo: order-finding Exponential speedup: polynomial time Hard to physically implement Largest Number: 21 - Adiabatic Computation Model Convert to optimization problem 1. NMR f = (N pq) 2 2. Ising Machine: D-wave,... Multiplication Table 3. Nitrogen-Vacancy (NV) center in diamond Multiplication Table

5 Quantum Adiabatic Computation Method 1 Define Hamiltonian: H(t) = (1 t/t )H 0 + (t/t )H P 2 Set system to ground state of initial H 0 (easy) 3* Define final H P s.t. its ground state encodes problem s solution 4 System slowly evolves to final state(eigenvector w.r.t. smallest eigenvalue),measure it to get soln

6 How to encode H p f p (s 1, s 2,..., s n ) = K n k=1 j 1,...,j k =1 J j1...j k s j1...s jk, s i = ±1 is the energy func.(eigenvalue func.) of Hamiltonian H p (σ z (1), σ z (2),..., σ z (n) ) = K n k=1 j 1,...,j k =1 J j1...j k σ (j 1) z...σ (j k) z i 1 n i Here σ z (i) {}}{{}}{ = I I... I σ z I... I with eigenvector x 1 x 2...x n, x i = {0, 1}, s i = ( 1) x i

7 Ising model Definition H p (σ z (1), σ z (2),..., σ z (n) ) = n i=1 h i σ (i) z + h i : external magnetic J ij : interaction between two adjacent sites i, j n i,j=1 Ising Model Quadratic Function J ij σ z (i) σ z (j)

8 Our Method for Factorization 1 Define Cost Function - Direct Method: f = (N pq) 2 - Modified Multiplication Table Method Order Reducing Method - x, y, z {0, 1} xy = z iff xy 2xz 2yz + 3z = 0, xy z iff xy 2xz 2yz + 3z > 0 x 1, x 2, x 3 {0, 1} min(x 1 x 2 x 3 ) = min (x 4 x 3 + 2(x 1 x 2 2x 1 x 4 2x 2 x 4 + 3x 4 )) x 4=x 1x 2 min( x 1 x 2 x 3 ) = min ( x 4 x 3 + 2(x 1 x 2 2x 1 x 4 2x 2 x 4 + 3x 4 )) x 4=x 1x 2 1 Shuxian Jiang et al. Quantum Annealing for Prime Factorization. In: arxiv preprint arxiv: (2018).

9 Cost Function: Direct Method For N = pq p = (p l 1 p l 2...p 1 1) 2, q = (q l 1q l 2...q 1 1) 2 Reduce Order: Define f = (N pq) 2 min(x 1 x 2 x 3 ) = l 1 l 1 = [N ( 2 i p i + 1)( 2 j q j + 1)] 2 i=1 i=1 j=1 l 1 l 1 = N 2 + ( 2 i p i + 1) 2 ( 2 j q j + 1) 2 i=1 j=1 l 1 l 1 2N( 2 i p i + 1)( 2 j q j + 1) j=1 min (x 4 x 3 + 2(x 1 x 2 2x 1 x 4 2x 2 x 4 + 3x 4 )) x 4 =x 1 x 2

10 Cost Function: Direct Method For N = 15 = 5 3 p = (x 1 1) 2 = x , q = (x 2 x 3 1) 2 = x x f (x 1, x 2, x 3 ) = (N pq) 2 = [15 (x )(x x )] 2 = (2x 1 + 1) 2 (4x 3 + 2x 2 + 1) 2 30(2x 1 + 1)(4x 3 + 2x 2 + 1) = 128x 1 x 2 x 3 56x 1 x 2 48x 1 x x 2 x 3 52x 1 52x 2 96x Reduce Order: f (x 1, x 2, x 3, x 4 ) = 128(x 4 x 3 + 2(x 1 x 2 2x 1 x 4 2x 2 x 4 + 3x 4 )) 56x 1 x 2 48x 1 x x 2 x 3 52x 1 52x 2 96x

11 Cost Function: Modified Multiplication Table Method Table: Multiplication table for = 143 in binary p 1 p 2 p 1 1 q 1 q 2 q p 2 p 1 1 q 1 p 2 q 1 p 1 q 1 q 1 q 2 p 2 q 2 p 1 q 2 q 2 1 p 2 p 1 1 carries c 4 c 3 c 2 c 1 p q = (p 2 + p 1 q 1 + q 2 ) + (p 1 + q 1 ) = 8c 2 + 4c

12 Cost Function: Modified Multiplication Table Methods From 2(p 2 + p 1 q 1 + q 2 ) + (p 1 + q 1 ) = 8c 2 + 4c We got f 1 = (2p 2 + 2p 1 q 1 + 2q 2 8c 2 4c 1 + p 1 + q 1 3) 2 Reduce order min(±x 1 x 2 x 3 ) = min x 4 =x 1 x 2 (±x 4 x 3 + 2(x 1 x 2 2x 1 x 4 2x 2 x 4 + 3x 4 ))

13 Embed to D-wave s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 10 s 11 s 12 s 5 s 6 s 7 s 8 s 9 s 10 s 11 s 12 s 9 s 10 s 11 s 12 Figure: Embed problem graph of factoring 143 using method one to Chimera graph

14 Embed to D-wave (a) Result (b) Result Figure: Final ground state of factoring 143. Red nodes:+1, Blue nodes:-1. (a) s 1 = 1, s 2 = 1, s 3 = 1, s 4 = 1 (p = 1101, q = 1011). (b) s 1 = 1, s 2 = 1, s 3 = 1, s 4 = 1 (p = 1011, q = 1101).

15 Cost Function: Modified Multiplication Table Method Table: Multiplication table for = in binary p 8 p 7 p 6 p 5 p 4 p 3 p 2 p q 8 q 7 q 6 q 5 q 4 q 3 q 2 q p 8 p 7 p 6 p 5 p 4 p 3 p 2 p 1 1 q 1 p 8q 1 p 7q 1 p 6q 1 p 5q 1 p 4q 1 p 3q 1 p 2q 1 p 1q 1 q 1 q 2 p 8q 2 p 7q 2 p 6q 2 p 5q 2 p 4q 2 p 3q 2 p 2q 2 p 1q 2 q 2 q 3 p 8q 3 p 7q 3 p 6q 3 p 5q 3 p 4q 3 p 3q 3 p 2q 3 p 1q 3 q 3 q 4 p 8q 4 p 7q 4 p 6q 4 p 5q 4 p 4q 4 p 3q 4 p 2q 4 p 1q 4 q 4 q 5 p 8q 5 p 7q 5 p 6q 5 p 5q 5 p 4q 5 p 3q 5 p 2q 5 p 1q 5 q 5 q 6 p 8q 6 p 7q 6 p 6q 6 p 5q 6 p 4q 6 p 3q 6 p 2q 6 p 1q 6 q 6 q 7 p 8q 7 p 7q 7 p 6q 7 p 5q 7 p 4q 7 p 3q 7 p 2q 7 p 1q 7 q 7 q 8 p 8q 8 p 7q 8 p 6q 8 p 5q 8 p 4q 8 p 3q 8 p 2q 8 p 1q 8 q 8 1 p 8 p 7 p 6 p 5 p 4 p 3 p 2 p 1 1 c 14 c 13 c 9 c 8 c 7 c 6 c 5 c 4 c 3 c 2 c 1 c 12 c 11 c

16 Experimental Results (a) N = 15, T = 200ms (b) N = 21, T = 2000ms Figure: D-wave Results for Method One: rates of getting different solutions.

17 Experimental Results Table: D-wave Results for Method Two N factors logic#qb phy#qb embed sol