1. New Approach to Quantum Calculation of Spectral Coefficients Marek Perkowski Department of Electrical Engineering, 2005

2. X Y 0 1 1 0 1 1 Data Function X Y 0 1 1 0 Standard Trivial function for XOR of input variables X Y 0 1 1 0 Standard Trivial Function for whole map X Y 0 1 1 0 Standard Trivial function for input variable X X Y 0 1 1 0 Standard Trivial function for input variable Y As you remember from last lecture, rows of Hadamard transform correspond to some Boolean functions which are called Standard Trivial Functions.

3. Important ideas Values of spectral coefficients of function F represent correlations of this function to each of the spectral coefficients. May be this correlation can be found differently than shown before.

4. New Notation = Symbols a -- the number of true minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 1; b -- the number of false minterms of Boolean function F, where the function F has the logical value 0 and the standard trivial function has the logical value 1; c -- the number of true minterms of Boolean function F, where the function F has the logical value 1 and the standard trivial function has the logical value 0; d -- the number of false minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 0, and e be the number of don't care minterms of Boolean function F.

5. Then, for completely specified Boolean functions having n variables, this formula holds: a + b + c + d = 2 n Accordingly, for incompletely specified Boolean functions, having n variables, holds: a + b + c + d + e = 2 n Obvious properties: We disregard normalization for simplification.

6. The spectral coefficients for completely specified Boolean function can be defined in the following way: s 0 = 2 n – 2 * a s i = 2 * (a + d) - 2 n, when i ≠ 0. X Y 0 1 1 0 1 1 a -- the number of true minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 1; d -- the number of false minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 0 a = 2, d = 2, n =2 s 0 = 2 n – 2 * a = 4 – 2 * 2 = 0 s 3 = 2 * (a + d) - 2 n = 2(2+2) – 4 = 4 best correlation

7. s i = 2 * (a + d) - 2 n, when i ≠ 0. X Y 0 1 1 0 1 1 a -- the number of true minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 1; d -- the number of false minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 0 a = 2, d = 2, n =2 s 3 = 2 * (a + d) - 2 n = 2(0+0) – 4 = - 4 worst correlation Negation of the previous function

8. X Y 0 1 1 0 1 1 a -- the number of true minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 1; Other functions 1 1 s 0 = 2 n – 2 * a = 4 – 2 * 4 = - 4 X Y 0 1 1 0 0 0 0 0 s 0 = 2 n – 2 * a = 4 – 2 * 0 = + 4

9. Butterflies + + + + - - - - 0 0 = 1 1 1 = -1 2 1 = -1 3 0 = 1 0 2 0 -2 0 0 0 4 S encoding in complex number XOR Number of minterm Boolean Value + - 2 This is kernel of butterfly. It corresponds to 2*2 unitary matrix of Hadamard (disregarding normalization coefficient). This in red illustrates how kernels are repeated.

10. Butterflies in S encoding + + + + - - - - 0 0 = 1 1 1 = -1 2 1 = -1 3 0 = 1 0 2 0 -2 0 0 0 4 S encoding XOR Boolean Value

11. Butterflies in S encoding + + + + - - - - 0 1 = -1 1 0 = 1 2 0 = 1 3 1 = -1 0 -2 0 2 0 0 0 -4 S encoding XNOR Boolean value

12. Butterflies in S encoding + + + + - - - - 0 1 = -1 1 0 = 1 2 0 = 1 3 0 = 1 0 -2 2 0 2 S encoding X’ Y’ Boolean value We cannot find anything from measurement since modules (probabilites) are equal

13. Butterflies in S encoding + + + + - - - - 0 0 = 1 1 1 = -1 2 0 = 1 3 0 = 1 0 2 2 0 2 2 -2 2 S encoding X’ Y Boolean value

14. Butterflies in S encoding + + + + - - - - 0 0 = 1 1 0 = 1 2 1 = -1 3 0 = 1 2 0 0 -2 2 2 2 S encoding X Y’ Boolean value

15. Butterflies in S encoding + + + + - - - - 0 0 = 1 1 0 = 1 2 0 = 1 3 1 = -1 2 0 0 2 2 2 2 -2 S encoding X Y Boolean value

16. Butterflies in S encoding + + + + - - - - 0 1 = -1 1 1 = -1 2 0 = 1 3 0 = 1 -2 0 2 0 0 0 -4 0 S encoding X’ Boolean value

17. Butterflies in S encoding + + + + - - - - 0 0 = 1 1 0 = 1 2 1 = -1 3 1 = -1 2 0 -2 0 0 0 4 0 S encoding X Boolean value

18. Butterflies in S encoding + + + + - - - - 0 0 = 1 1 1 = -1 2 0 = 1 3 1 = -1 0 2 0 2 0 4 0 0 S encoding Y Boolean value

19. Butterflies in S encoding + + + + - - - - 0 1 = -1 1 0 = 1 2 1 = -1 3 0 = 1 0 -2 0 0 -4 0 0 S encoding Y’ Boolean value

20. Butterflies in S encoding + + + + - - - - 0 -1 1 -1 2 -1 3 -1 -2 0 0 -4 0 0 0 S encoding Constant 1 minterm

21. Butterflies in S encoding + + + + - - - - 0 0 = 1 1 0 = 1 2 0 = 1 3 0 = 1 2 0 2 0 4 0 0 0 S encoding Constant 0 Boolean value

22. Butterflies in R encoding + + + + - - - - 0 1 0 2 0 3 0 2 0 2 0 0 0 0 0 R encoding Constant 0 minterm

23. Butterflies in R encoding + + + + - - - - 0 1 1 2 1 3 1 2 0 2 0 4 0 0 0 R encoding Constant 1 minterm

24. Butterflies in R encoding + + + + - - - - 0 1 2 1 3 0 1 1 1 2 0 0 -2 R encoding XOR minterm

25. Butterflies in R encoding + + + + - - - - 0 1 1 0 2 0 3 1 1 1 1 2 0 0 2 R encoding XNOR minterm

26. Butterflies in R encoding + + + + - - - - 0 1 2 1 3 1 1 2 0 3 R encoding X OR Y minterm

27. Butterflies in S encoding + + + + - - - - 0 1 1 -1 2 -1 3 1 0 2 -2 0 2 2 2 S encoding X OR Y Boolean value

28. Butterflies in S encoding + + + + - - - - 0 -1 1 -1 2 1 3 -1 -2 0 0 2 2 S encoding X’ OR Y 11 1 X Y

29. s i = 2 * (a + d) - 2 n, when i ≠ 0. X Y 0 1 1 0 1 1 a -- the number of true minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 1; d -- the number of false minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 0 a = 2, d = 2, n =2 s 2 = 2 * (a + d) - 2 n = 2(1+1) – 4 = 0

30. s i = 2 * (a + d) - 2 n, when i ≠ 0. X Y 0 1 1 0 1 1 a -- the number of true minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 1; d -- the number of false minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 0 a = 2, d = 2, n =2 s 1 = 2 * (a + d) - 2 n = 2(1+1) – 4 = 0

31. Since each standard function has the same number of true and false minterms that is equal to 2 n-1, then we can have alternative definitions of spectral coefficients. Please note that only for the spectral coefficient s 0, is the above rule not valid and the appropriate standard function is a tautology, –i.e. the logical function that is true for all its minterms. Thus, we have: a + b = c + d = 2 n-1 and s i = 2 *(c + b) - 2 n

32. or s i = 2 * (a + d) - 2 n = 2 * (a + 2 n-1 - c) - 2 n or s i = 2 * (a + d) - 2 n = 2 * (a + d) - (a + b + c + d) = (a + d) - (b + c), when i ≠ 0. and s 0 = 2 n – 2 * a = a + b + c + d – 2 * a = b + c + d - a = b - a, since for s 0, c and d are always equal to 0.

33. The spectral coefficients for incompletely specified Boolean function, having n variables, can be defined in the following way: s 0 = 2 n – 2 * a - e and s i = 2 * (a + d) + e - 2 n, when i ≠ 0. As we can see, for the case when e = 0, i.e. for completely specified Boolean function, the above formulas reduce to the formulas presented previously.

34. All but S 0 coefficients And again, by easy mathematical transformations, we can define all but s 0 spectral coefficients in the following way: s i = 2 * (a +d) + e - 2 n = 2 * (a+d) + e - (a+b+c+d+e) = (a+d) - (b+c), when i≠ 0

35. s 0 spectral coefficient Simultaneously, the s 0 spectral coefficient can be rewritten in the following way: s 0 = 2 n - 2*a – e = a+b+c+d+e - 2*a – e = b+c+d-a = b - a, since for s 0, c and d are always equal to 0.

36. Thus, in the final formulas, describing all spectral coefficients, the number of don't care minterms e can be eliminated from them. Moreover, the final formulas are exactly the same as the ones for completely specified Boolean function. Of course, it does not mean, that the spectral coefficients for incompletely specified Boolean function do not depend on the number of don't care minterms. They do depend on those numbers, but the problem is already taken into account in the last two formulas themselves. Simply, the previously stated formula for the numbers a, b, c, d, and e bonds all these values together.

37. Conclusion If we have a method to calculate the coefficients a, b, c, d, e then we can calculate Hadamard spectral transform coefficients with full accuracy, not only some and not probabilistically, as shown b before. This can be done using quantum counting algorithm, which will be introduced in future.

38. Properties of Transform Matrices The transform matrix is complete and orthogonal, and therefore, there is no information lost in the spectrum S, concerning the minterms in Boolean function F. Only the Hadamard-Walsh matrix has the recursive Kronecker product structure – and for this reason is preferred over other possible variants of Walsh transform known in the literature as Walsh-Kaczmarz, Rademacher-Walsh, and Walsh-Paley transforms. Only the Rademacher-Walsh transform is not symmetric; –all other variants of Walsh transform are symmetric, –so that, disregarding a scaling factor, the same matrix can be used for both the forward and inverse transform operations.

39. When the classical matrix multiplication method is used to generate the spectral coefficients for different Walsh transforms, then the only difference is the order in which particular coefficients are created. –The values of all these coefficients are the same for every Walsh transform. Each spectral coefficient s I gives a correlation value between the function F and a standard trivial function e I corresponding to this coefficient. The standard trivial functions for the spectral coefficients are, respectively, –for the coefficient s 0 ( dc coefficient ) - the universe of the Boolean function denoted by e 0, –for the coefficient s i ( first order coefficient ) – the variable x i of the Boolean function denoted by e i, Properties of Transform Matrices

40. for the coefficient s ij ( second order coefficient ) - the exclusive-or function between variables x i and x j of the Boolean function denoted by e ij, for the coefficient s ijk ( third order coefficient ) - the exclusive-or function between variables x i, x j, and x k of the Boolean function denoted by e ijk, etc. Properties of Transform Matrices cont

41. The sum of all spectral coefficients s I of spectrum S for any completely specified Boolean function is  2 n. is notThe sum of all spectral coefficients s I of spectrum S for any incompletely specified Boolean function is not  2 n. Properties of Transform Matrices

42. The maximum value of any individual spectral coefficient s I in spectrum S is  2 n. –This happens when the Boolean function is equal to either a standard trivial function e I ( sign + ) or to its complement ( sign - ). –In either case, all the remaining spectral coefficients have zero values because of the orthogonality of the transform matrix T. Each but e 0 standard trivial function e I corresponding to n variable Boolean function has the same number of true and false minterms equal to 2 n-1. Properties of Transform Matrices

43. Probabilistic measurements If function is known to be affine, we can find which affine class it belongs to in one measurement with Hadamard gates after oracle. If function is unknown, we can find if it is affine or not in few measurements with high probability. (this result is new, although obvious).

44. The spectrum S of each true minterm of n variable Boolean function is given by s 0 = 2 n - 2, and all remaining 2 n - 1 spectral coefficients s I are equal to  2. Properties of Transform Matrices + + + + - - - - m0= -1 m1= -1 m2= 1 m3= 1 -2 0 2 0 0 0 -4 0 S encoding + + + + - - - - m4= -1 m5= -1 m6= 1 m7= 1 -2 0 2 0 0 0 -4 0 + + + + - - - - 0 0 - 8 0 0 0 0 0 EXAMPLE

45. The spectrum S of each true minterm of n variable Boolean function is given by s 0 = 2 n - 2, and all remaining 2 n - 1 spectral coefficients s I are equal to  2. Properties of Transform Matrices + + + + - - - - m0= -1 m1= 1 m2= 1 m3= 1 0 -2 2 0 2 S encoding + + + + - - - - m4= 1 m5= 1 m6= 1 m7= 1 2 0 2 0 4 0 0 0 + + + + - - - - 6 = 2 3 - 2 - 2 EXAMPLE

46. In Hilbert Space +6 -2 -2 -2 -2 -2 -2 -2 |000> |001> |010> |011> |100> |101> |110> |111> state Complex amplitude (not normalized) measurement 9/16 1/16 probabilities EXAMPLE

47. Probabilities + + + + - - - - m0= -1 m1= 1 m2= 1 m3= 1 0 -2 2 0 2 S encoding + + + + - - - - m4= 1 m5= 1 m6= 1 m7= 1 2 0 2 0 4 0 0 0 + + + + - - - - 6=2 3 - 2 - 2 p=36/64 Sum p= 36 + 7*4 = 64 p=4/64 p=4/64=1/16 p=9/16 Selected coefficient, in this case S 0 for SAT testing

48. What is the probability that there is a single minterm being “one”? 9/16 0 New state, other than previous 7/16 9/16 0 7/16 New state, other than previous 0 9/16 0 7/16 New state, other than previous 9/16 0 7/16 How many times I have to measure and obtain first spectral coefficient to be sure 99,99% that my function is constant? Is it possible? I measure always first coefficient If I measure K times, and every time get the selected coefficient then the probability that my function is not the standard trivial function of this coefficient is (9/16) K which I can make arbitrarily close to 0 by increasing K.

49. measurement Probability(satisfied) = (7/16) + (9/16)*(7/16) + (9/16) 2 * (7/16) +… 7/16=0.4375 0.246 0.138 0.77 0.04 0.02 SUM = 0.902 after 5 measurement. Conclusions: Measuring five times and not getting other coefficient than 0 we have more than 90 probability that function is constant

50. The spectrum S of each true minterm of n variable Boolean function is given by s 0 = 2 n - 2, and all remaining 2 n - 1 spectral coefficients s I are equal to  2. The spectrum S of each don't care minterm of n variable Boolean function is given by s 0 = 2 n - 1, –and all remaining 2 n - 1 spectral coefficients s I are equal to  1. The spectrum S of each false minterm of n variable Boolean function is given by s I = 0. Properties of Transform Matrices

51. What we achieved? 1. If the function is known to be linear or affine, by measuring once we can distinguish which one is the linear function in the box. We cannot distinguish a linear function from its negation. They differ by sign that is lost in measurement. 2. If function is a constant (zero for satisfiability and one for tautology) we can find with high probability that it is constant. –Thus we can solve SAT with high probability but without knowing which input minterm satisfies. (a single one in a Kmap of zeros) –Thus we can solve Tautology with high probability but without knowing which input minterm fails (a single zero in kmap of ones).

52. What we achieved? 3. We can find the highest spectral coefficients by generating them using quantum oracles. Those coefficients that have high values will be generated with high probability. –In terms of signals and images it gives the basic harmonics or patterns in signal, such as textures. –This can have applications to image processing and speech processing. It is known that animals’ hearing apparatus does spectral transform. 4. Instead of using Hadamard gates in the transform we can use V to find a separation of some Boolean functions. But we were not able to extend this to more variables.

53. measure Here, after oracle we have all information about the function (Kmap) but we cannot access it as is Common to many quantum algorithms I promised to discuss how can be generalized! V V Now we insert gates V instead of gates H V = 1/2 1+i1-i 1+i

54. What we achieved? 4. Instead of using Hadamard gates in the transform we can use V to find a separation of some Boolean functions. But we were not able to extend this to more variables. 01 11 Group 1- symmetric not affine Group 2-non- symmetric not affine 00 01 11 10 10 00 00 10 10 11 11 01 01 00 We can separate these two classes in single measurement

55. What we achieved? 5. Knowing some high coefficients, we can calculate their values exactly, one evaluation (called also one run or one measurement) for each spectral coefficient.

56. Symbols a -- the number of true minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 1; b -- the number of false minterms of Boolean function F, where the function F has the logical value 0 and the standard trivial function has the logical value 1; c -- the number of true minterms of Boolean function F, where the function F has the logical value 1 and the standard trivial function has the logical value 0; d -- the number of false minterms of Boolean function F, where both the function F and the standard trivial function have the logical values 0, and e be the number of don't care minterms of Boolean function F. 1 0 a=8b=0 c=0 d=8

57. s i = 2 * (a + d) - 2 n = 2 * (a + d) - (a + b + c + d) = (a + d) - (b + c), when i ≠ 0. 1 0 a=8 c=0 d=8 b=0 s i = (a + d) - (b + c) = 16

58. s i = 2 * (a + d) - 2 n = 2 * (a + d) - (a + b + c + d) = (a + d) - (b + c), when i ≠ 0. 1 0 1 0 a=7 c=0 d=8 b=1 s i = (a + d) - (b + c) = 16 Ones inside Zeros outside zeros inside Ones outside (7+8 ) - (1+0) = 14

59. s i = 2 * (a + d) - 2 n = 2 * (a + d) - (a + b + c + d) = (a + d) - (b + c), when i ≠ 0. 1 0 a=0 c=8 d=0 b=8 s i = (a + d) - (b + c) = -16

60. s i = 2 * (a + d) - 2 n = 2 * (a + d) - (a + b + c + d) = (a + d) - (b + c), when i ≠ 0. 1 0 a=4 c=4 d=4 b=4 s i = (a + d) - (b + c) = 0

61. s i = 2 * (a + d) - 2 n = 2 * (a + d) - (a + b + c + d) = (a + d) - (b + c), when i ≠ 0. 1 0 a=4 c=4 d=4 b=4 s i = (a + d) - (b + c) = 0

62. s i = 2 * (a + d) - 2 n = 2 * (a + d) - (a + b + c + d) = (a + d) - (b + c), when i ≠ 0. 1 0 1 0 a=3 c=4 d=4 b=4 s i = (a + d) - (b + c) = (3+4) – (4+4) = -1

63. Architecture for arbitrary coefficients Using the formulas developed above we can measure the correlation to every standard trivial function. Moreover, we can measure the number of minterms in arbitrary function that we create as an additional oracle XORed with the original oracle. This way we can calculate all cofactors and use methods of synthesis and analysis of Boolean functions based on cofactors.

64.  a b f f  (a  b)  Quantum circuit For calculation of numbers of ones We calculate the number of ones (true minterms) in the XOR of function f and standard trivial function (a  b). If this number is zero, then there is a maximum correlation and function f is (a  b). If there are 2 n true minterms, then the function is (a  b). Example to calculate the value of the coefficient S a  b