Measuring progress in Shor s factoring algorithm

Transcription

1 Measuring progress in Shor s factoring algorithm Thomas Lawson Télécom ParisTech, Paris, France 1 / 58

2 Shor s factoring algorithm What do small factoring experiments show? Do they represent progress? 2 / 58

3 Overview Picking the calculation Shortcuts A measure of success 3 / 58

4 Shor s factoring algorithm Shor s factoring algorithm 4 / 58

5 Shor s factoring algorithm Factoring To factor N 1 coprime x 2 order r, (x r mod N = 1) 3 factors are gcd(x r 2 ± 1, N). N=21 3 Pick coprime x Find order r gcd 7 5 / 58

6 Shor s factoring algorithm Order finding Factoring N = 21 using coprime x = 4. The order is given by 4 r mod 21 = = = = = = = = / 58

7 Shor s factoring algorithm Order finding Factoring N = 21 using coprime x = 4. The order is given by 4 r mod 21 = mod 21 = mod 21 = mod 21 = mod 21 = mod 21 = mod 21 = mod 21 = Here r = 3. 7 / 58

8 Shor s factoring algorithm Factoring: an example Factoring N = 21 using coprime x = 4. The order is given by 4 r mod 21 = mod 21= mod 21= mod 21= mod 21 = mod 21 = mod 21 = mod 21 = Here r = 3. 8 / 58

9 Shor s factoring algorithm Factoring: an example The factors of N = 21 are gcd(4 3 2 ± 1, 21) = gcd(8 ± 1, 21) = gcd(7, 21) and gcd(9, 21) =7 and 3. 9 / 58

10 Shor s factoring algorithm Quantum order finding Quantum operators speed it up. 1 = = = 1 10 / 58

11 Shor s factoring algorithm Quantum order finding contains the whole order, r. φ k = α k φ k r φ k = φ k α r k = 1 11 / 58

12 Shor s factoring algorithm Quantum order finding contains the whole order, r. φ k = α k φ k r φ k = φ k α k = e 2πi k r 12 / 58

13 Shor s factoring algorithm Quantum order finding The QOFA quickly finds α k. n Mn... 2 M2 1 M > 2 n... ² k r = 0.M 1M 2 M 3... M n 13 / 58

14 Shor s factoring algorithm Quantum order finding The output of the quantum order finding algorithm. Pr... 1/r 2/r 3/r 4/r 5/r Output (for n ). 14 / 58

15 Picking the calculation Picking the calculation 15 / 58

16 Picking the calculation Trivial calculations Normally the distribution becomes fuzzy as n is reduced. Pr... 1/r 2/r 3/r 4/r 5/r Output 16 / 58

17 Picking the calculation Trivial calculations Normally the distribution becomes fuzzy as n is reduced. Pr... 1/r 2/r 3/r 4/r 5/r Output 17 / 58

18 Picking the calculation Trivial calculations But for trivial calculations this does not happen. For order r = 4, Pr 0/4 1/4 2/4 3/4 Output 18 / 58

19 Picking the calculation Trivial calculations But for trivial calculations this does not happen. For order r = 4, Pr Output 19 / 58

20 Picking the calculation Trivial calculations A circuit that does this 3 M3 2 M2 1 M > 4 ² Pr Output 20 / 58

21 Picking the calculation Trivial calculations A circuit that does this H 0 2 M2 1 M > I ² Pr Output 21 / 58

22 Picking the calculation Trivial calculations A circuit that does this H 0 I 2 0/1 I 1 0/ > I ² Pr Output 22 / 58

23 Picking the calculation Trivial calculations This happens if r = 2 p, because 1/r represented in binary. N = 15 gives r = 2 or r = 4. This is always trivial. 23 / 58

24 Picking the calculation Nontrivial calculations Nontrivial: N = 21 with x = 4 gives r = = / 58

25 Picking the calculation Nontrivial calculations 2 a" Factoring 0 N = 21 with x = 4 (giving r = 3). 1 b" Probability" H" H" H" {I,}" H" 2" " (n;1)" 2" " (n;2)" 0" 2" " Further"iteraBons" to..." 0.35 c" 2 " n=2" " Increasing"precision" FIG. 1: The iterative order finding algorithm for factoring 21. a, Measurement of the control qubit after each controlled unitary gives the next most significant bit in the output and the outcome is fed forward to the iterated (semi-classical) Fourier transform, which applies either the identity operation I or the appropriate phase gate, prior to the Hadamard H. b, As the number of iterations increases the precision increases. c, For two bits of precision the controlled unitary operations can be constructed with this arrangement of controlled-swap gates. Fourier transform is constructive for states contributing to the 00 term and boosts its probability of observation to three times that of the probability for observing the 10 term, which experiences destructive quantum interference among its contributory states. Decoherence in the two qubit control register, the single swap of 2 is implemented with a controlled-not (CNOT) gate; 1 is realised with two swaps, the first of which is a CNOT gate, while it is su cient for the second swap to be uncontrolled. (See Appendix for details). 25 / 58

26 Picking the calculation Second cause of triviality Lack of precision. 3 M3 2 M2 1 M > 4 ² If r > 2 n no interference happens. 26 / 58

27 Picking the calculation Second cause of triviality Lack of precision. 3 M3 2 M2 1 M > If r > 2 n no interference happens. 27 / 58

28 Picking the calculation Second cause of triviality For n steps, only 2 n states can be accessed. 3 M3 2 M2 1 M > If r > 2 n no interference happens. 28 / 58

29 Picking the calculation Second cause of triviality Lack of precision. 3 M3 2 M2 1 M > If r > 2 n no interference happens. 29 / 58

30 Picking the calculation Idée reçue 1) Short r are easy. In fact r = 2 12 is easier than r = 3, r must be small. 30 / 58

31 Experimental shortcuts Experimental shortcuts 31 / 58

32 Experimental shortcuts Simplifying the circuit Demonstrations use shortcuts: Compiling Circuit simplifications Substitutions 32 / 58

33 Experimental shortcuts Compiling Compiling removes the hardest part of the algorithm - making the unitary operators, 2n. n Mn... 2 M2 1 M > 2 n... ² 33 / 58

34 Experimental shortcuts Compiling 7 n Mn... 2 M2 1 M > 2 n... ² Figure: Niskanen et al / 58

35 Experimental shortcuts Compiling All demonstrations have used compiling. n Mn... 2 M2 1 M > 2 n... ² It is not scalable. It needs knowledge of the calculation. A part of the algorithm is missing. 35 / 58

36 Experimental shortcuts Compiling All demonstrations have used compiling. n Mn... 2 M2 1 M > 2 n... ² It is not scalable. It needs knowledge of the calculation. A part of the algorithm is missing. 36 / 58

37 Experimental shortcuts Simplifying the circuit Demonstrations use shortcuts: Compiling Circuit simplifications Substitutions 37 / 58

38 Experimental shortcuts Circuit simplifications nused qubits are removed. 3 M3 2 M2 1 M > This is fine (even if it needs knowledge of the calculation). 38 / 58

39 Experimental shortcuts Circuit simplifications nused qubits are removed. 3 M3 2 M2 1 M > This is fine (even if it needs knowledge of the calculation). 39 / 58

40 see that qubit 1 is in 0i, and qubits 2 and 3 are in a mixture of 0i and Experimental shortcuts 1i. The register is thus in a mixture of 000i, 010i, 100i and 110i, or 0i, 2i, 4i and 6i. The periodicity in the amplitude of yi is now 2, so Circuit simplifications r ˆ 8=2 ˆ 4 and g:c:d: 7 4=2 6 1; 15 ˆ3; 5. Thus, even after the long and complex pulse sequence of the dif cult case (Fig. 4), the experimental data conclusively indicate the successful execution Doing otherwise is a waste of resources. of Shor's algorithm to factor 15. Factor N = 15 giving Nevertheless, r = there 4. are obvious discrepancies between the measured and ideal spectra, most notably for the dif cult case. sing a numerical model, we have investigated whether these deviations H 0 2 M2 1 M1 and phase damping (PD p E 0 ˆ 1 l 0 with g ˆ 1 2 e 2t=T1, p ˆ following observations single spin descriptions (1) GAD (and PD) e commute; (2) the E k for applied to arbitrary r; an > a b 1: 2: 3: 4: 5: 6: 7: n m (0) I 0 1 T e m p or a l a v e r a g i n g (1) (2) ² (3) (4) H n H H H A x 1 B C D E x a x mod N F G H Inverse QFT H 90 H H Figure 1 Quantum circuit for Shor's algorithm. a, Outline of the quantum circuit. Wires represent qubits, Figure: and boxes represent Vandersypen operations. Timetgoes al. from2001. left to right. (0) Initialize a rst register of n ˆ 2dlog 2 Ne qubits to j0i # ¼ # j0i (for short 0i) and a second i ω i /2π T 1,i T 2, Figure 2 Structure and propert per uorobutadienyl iron comple measured J 13C 19 F values, we co different from that derived in ref. (in Hz) at 11.7 T, relative to a re 40 / 58 13

41 Experimental shortcuts Simplifying the circuit Demonstrations use shortcuts: Compiling Circuit simplifications Substitutions 41 / 58

42 Experimental shortcuts Substitutions Technology which is not ready is replaced. 2 M2 1 M1 ² The global properties can be tested. 42 / 58

43 Testing Shor s algorithm Testing Shor s algorithm 43 / 58

44 Testing Shor s algorithm Idée reçue 2) The number factored is a good measure This does not represent progress. 44 / 58

45 Testing Shor s algorithm Idée reçue 2) The number factored is a good measure This does not represent progress. 45 / 58

46 Testing Shor s algorithm Idée reçue 3) Factoring (like love) should be blind. Avoid trivial cases. Factoring is physics. 46 / 58

47 Testing Shor s algorithm Idée reçue 4) Factoring a large number is a good test. 3 7 = 21 This shows the computer works. But it doesn t say how well. 47 / 58

48 Testing Shor s algorithm Idée reçue 4) Factoring a large number is a good test. 3 7 = 21 Bad circuits occasionally get the factors Perfect circuits often fail 48 / 58

49 Testing Shor s algorithm A test of progress We should test the order finding algorithm. Pr... 1/r 2/r 3/r 4/r 5/r Output 49 / 58

50 Testing Shor s algorithm A test of progress The overlap improves with n. M M M M M M M M Pr... 1/r 2/r 3/r 4/r 5/r Output 50 / 58

51 Testing Shor s algorithm A test of progress The overlap improves with n. M M M M M M M M Pr... 1/r 2/r 3/r 4/r 5/r Output 51 / 58

52 Testing Shor s algorithm A test of progress The overlap improves with n. M M M M M M M M Pr... 1/r 2/r 3/r 4/r 5/r Output 52 / 58

53 Testing Shor s algorithm A test of progress The overlap improves with n. M M M M M M M M Pr... 1/r 2/r 3/r 4/r 5/r Output 53 / 58

54 Testing Shor s algorithm Conclusion To design a factoring experiment 1 choose a nontrivial case, 2 only leave out uninteresting parts of the circuit, 3 judge the quantum order finding algorithm. 54 / 58

55 Thank you for your attention. Thank you for your attention. 55 / 58

56 Extras Extras 56 / 58

57 Extras QFT vs iterative PEA a) b) 0 H... H. 0 H... n H 0 H n 1 H n 2. 0 H H... 0 H φ1 n 1... φn 2 1 H 0 H φ1 n... φn 1 1 H 1 2n / 58

58 Extras Odd r The factors are gcd(x r 2 ± 1, N) N = 21, x = 4 gives r = 3 gcd(4 3 2 ± 1, 21) = gcd(2 3 ± 1, 21) since x is a perfect square. 58 / 58