Boolean State Transformation

Transcription

1 Quantum Computing Boolean State Transformation

2 Quantum Computing 2 Boolean Gates Boolean gates transform the state of a Boolean system. Conventionally a Boolean gate is a map f : {,} n {,}, where n is the number of input lines and there is only one output line. For a large n, an n-input gate may be decomposed in smaller realisable gates.

3 Quantum Computing 3 Boolean Gates But if the system state is represented by n-bits then a state transition map is g : {,} n {,} n. The map g may be viewed as an n input and n output gate. This also may be realised using smaller gates. Following diagram is an example.

4 Quantum Computing 4 Boolean Gate Array s i s s 2 s 3 s o G G 3 G 5 I I 2 I 3 I 4 O O 2 O 3 O 4 G 2 G 4

5 Quantum Computing 5 Note s i is the input state, s o is the output state. s,s 2,s 3 are intermediate states. Transition from s to s is through the gates G,I,I,G 2, where I may be viewed as identity map. This transition may be viewed as a 4-bit transformation G I I G 2. Other transitions are similar.

6 Quantum Computing 6 -bit Boolean Gates There are four one variable Boolean functions. Two constant functions: c :,, c :,, the identity map i :,, and the not gate :,.

7 Quantum Computing 7 -bit Linear Algebra If we encode Boolean as [ and as [ the following transformation matrices represent the four gates. [ [ c =,c =,i = [, : [, The i and gate are invertible, but other two are not.

8 Quantum Computing 8 -bit Linear Algebra c () = c () = = = [ [ [ [ [ [ [ [ = = = = [ [ [ [ =, =, =, =.

9 Quantum Computing 9 -Bit Boolean Transformation Matrix A 2 2 matrix over F2 is a valid transformation matrix for a single-bit if every column has exactly one. This restriction is due to our encoding of and. We get 2 2 = 4 valid transformation matrices corresponding to c,c,i and. Only two of them are reversible.

10 Quantum Computing Reversibility Reversibility of computation or invertibility of an operator is an issue even in classical computation. It is known from thermodynamics that there is no increase in entropy in a reversible process. But completely isentropic circuits are impossible to design.

11 Quantum Computing Reversibility There are adiabatic circuits that use reversible logic to reduce power consumption during switching. Landauer s principle claims that any logically irreversible change in information causes more change in entropy. A physical reversibility demands logical reversibility.

12 Quantum Computing 2 Reversibility We shall see that state transition in a quantum mechanical system is reversible. Only the reversible classical logical gates may have quantum mechanical counterpart. So we explore the classical Boolean gates that are reversible as well as universal.

13 Quantum Computing 3 Tensor Product We define the tensor product of two 2-dimensional vectors x = [ a and [ y = c over some field F as [ a b [ c d = a c d b c d b = ac ad bc bd. d This can be generalised to higher dimensions.

14 Quantum Computing 4 2-Bit Boolean Data -bit Boolean data is encoded as = [ and = [. Two bit Boolean data may be viewed as the tensor product of two -bit vectors.

15 Quantum Computing 5 2-Bit Boolean Data [ [ = =. [ [ = =.

16 Quantum Computing 6 2-Bit Boolean Data [ [ = =. [ [ = =.

17 Quantum Computing 7 2-bit Boolean Data Four possible combinations of inputs,,,, are encode as four 4-vectors. :, :, : This can be generalised for n-bits., :.

18 Quantum Computing 8 2-bit Boolean Gates There are sixteen 2-variable conventional Boolean functions e.g. and, nand, or, nor etc. They correspond to maps from {,} 2 {,}. But we are interested about state transition maps from {,} 2 {,} 2 that are reversible. There are total 4 4 = 256 such maps.

19 Quantum Computing 9 2-Bit Boolean Transformation Matrix Similar to -bit transformations, a 2-bit transformation is a 4 4 matrix over F2. It is a valid if every column has exactly one. There are 4 4 = 256 valid transformations corresponding to 256 maps {,} 2 {,} 2. But only 4! among them are reversible, where every column has exactly one.

20 Quantum Computing 2 Universality and Reversibility But is there any universal gate out these 24 reversible gates? One possibility is to consider the tensor products of -bit reversible gates e.g. i or - but they cannot be universal. Another temptation is to try with a pair of conventional gates and see what happens.

21 Quantum Computing 2 Xor-And Transformation Here is an example of a 2-bit transformation matrix of a pair of conventional gates. () = () Xor And

22 Quantum Computing 22 Note The transformation matrix of Xor-And pair is not invertible - the 4 th -row has all zeros. An invertible transformation matrix correspond to a permutation of rows of 4 4 identity matrix.

23 Quantum Computing 23 CNOT Gate A controlled not (CNOT) gate, has invertible transition matrix. Its inputs are (c,d), where c is the control input and d is the data input. The mapping is (c,d) (c,c d). (,d) if c =, CNOT(c,d) = (, d) if c =.

24 Quantum Computing 24 CNOT Transition Matrix CNOT(,) = = = (,) CNOT is inverse of itself. CNOT(CNOT(c,d)) = CNOT(c,c d) = (c,c (c d)) = (c,d).

25 Quantum Computing 25 Diagram c c d c d CNOT

26 Quantum Computing 26 CNOT can Copy a Bit We can copy (clone) a bit using CNOT: (a,) (a,a ) = (a,a). a a a = a CNOT This is actually creating a FANOUT.

27 Quantum Computing 27 Note Tensor products of -bit reversible gate are i i, i, i,. CNOT cannot be realised using them. [ [ [ [ It is + [ [ where = [ and [ [ = [.,

28 Quantum Computing 28 2-bit Universal Gate is Impossible The truth-table for a 2-bit reversible gate is as follows: i i o o x y x y x 2 y 2 x 3 y 3

29 Quantum Computing 29 2-bit Universal Gate is Impossible The output of a reversible gate is a permutation of {,,, }. It has exactly two s and two s per output column of the truth table. But i i has three s and i i has three s in their output. None of them can be produced by a two bit reversible gate.

30 Quantum Computing 3 Reversible and Universal Boolean Gates The 2-bit CNOT gate is reversible. There are 24 reversible 2-Bit Boolean transformations. But that set cannot generate all logic function. 3-bit transformations are map from {,} 3 {,} 3. There are 8 8 = such transformations. But out of which 8! = 432 are reversible.

31 Quantum Computing 3 Universal Reversible Gates One 3-bit reversible transformation is Toffoli gate or CCNOT gate. It was invented by Tommaso Toffoli from Italy. The Toffoli gate or CCNOT gate is known to be universal.

32 Quantum Computing 32 Toffoli or CCNOT Gate Following is the map of CCNOT gate: (x,y,c) (x,y,c (x y)), where x,y remains unchanged, c = c if x = = y, c otherwise. If c =, the 3 rd output is (x y) = x y.

33 Quantum Computing 33 CCNOT Gate Transition Matrix CCNOT transition matrix is its own inverse.

34 Quantum Computing 34 Diagram a b c a b c (a b) CCNOT

35 Quantum Computing 35 NOT, NAND and FANOUT from CCNOT CCNOT gate can implement NOT, NAND and COPY/FANOUT operations if logic and are available. (,,c) (,, c), (a,b,) (a,b, (a b)) = (a,b, (a b)), (,a,) (,a, ( a)) = (,a,a).

36 Quantum Computing 36 OR using only CCNOT We know that ( a b) = a b.. a and b are actual input. We use several ancilla input. 2. Use two CCNOT gates to create a copies of a and b. 3. Use two CCNOT gates to compute a and b.

37 Quantum Computing 37 OR using only CCNOT 4. Finally another CCNOT gate to compute ( a b). 5. We are not taking care of crossovers.

38 Quantum Computing 38 OR using CCNOT a b a a a b b b a b a b a b

39 Quantum Computing 39 Note Note that we have not taken care of cross-over of bits in the diagram. There are several ancilla inputs in state and. Also there are several useless outputs.

40 Quantum Computing 4 Note It is not very difficult to create a 3-bit reversible transformation that will compute a 2-bit function e.g. or when the 3 rd -bit has a fixed value. Look at the following truth table.

41 Quantum Computing 4 Truth Table a b c o o 2 o 3 When c =, o 3 = a b. Other bits are filled to maintain reversibility.

42 Quantum Computing 42 Transition Matrix There are three permutations (2,3),(4,5),(6,7).

43 Quantum Computing 43 OR using CCNOT and NOT Another way of getting OR using CCNOT and NOT are as follows. We negate the input a,b and keep c unchanged. This can be done by applying the transformation [ [ [ to a 3-bit vector.

44 Quantum Computing 44 Tensor Product of Matrices a a 2 a 2 a 22 b b 2 b 2 b 22 = a a 2 b b 2 b 2 b 22 b b 2 b 2 b 22 a 2 b b 2 b 2 b 22 a 22 b b 2 b 2 b 22 = a b a b 2 a 2 b a 2 b 2 a b 2 a b 22 a 2 b 2 a 2 b 22 a 2 b a 2 b 2 a 22 b a 22 b 2 a 2 b 2 a 2 b 22 a 22 b 2 a 22 b 22.

45 Quantum Computing 45 Tensor Product I = =

46 Quantum Computing 46 Tensor Product I We encode (x,y,c) : (,,),,(,,) as an 8-dimensional Boolean vectors (,,,),, (,,,). Operator I gives the desired result. = : (,,) (,,).

47 Quantum Computing 47 Tensor Product CCNOT ( I) The transformation ( I) is clearly reversible (universal property). CCNOT ( I) transforms (x,y,c) ( x, y,c ( x y)). We can apply on the result. This gives us x y when c =.

48 Quantum Computing 48 Tensor Product (I I ) ( ) = = ( ) : (,,) (,,)

49 Quantum Computing 49 Fredkin Gate Another important 3-input reversible gate is the Fredkin gate. The input-output relation is (x,y,c) (cy +cx,cx+cy,c) i.e. (x,y,) (x,y,) and (x,y,) (y,x,). If c =, x,y remains unchanged. If c =, the outputs are interchanged.

50 Quantum Computing 5 Fredkin Gate is Universal CROSSOVER: (a,b,) (b,a,), NOT and FANOUT: (,,a) (a,a,a), AND: (,b,a) (a b,a b,a),

51 Quantum Computing 5 Fredkin Gate Transition Matrix Clearly Fredkin gate is its own inverse.

52 Quantum Computing 52 Diagram a b c c α = cb+ca β = ca+cb Fredkin

53 Quantum Computing 53 Note Both in case of Toffoli gate and in Fredkin gate we need some ancilla input bits in state or to make them universal. They also produce useless outputs not used in subsequent stages.

54 Quantum Computing 54 Fredkin using CNOT & CCNOT c c b a b a b (a b) (a ((a b) c))) = ca+cb a a a (c (ab)) = cb+ca

55 Quantum Computing 55 NOT and FANOUT a b c Fredkin c α = cb+ca β = ca+cb x +x = x x x NOT and FANOUT x +x = x

56 Quantum Computing 56 AND and CCNOT y x AND x xy xy z z xy CCNOT z (x y) z (x y) xy

57 Quantum Computing 57 CCNOT using Fredkin. c,x and y are actual input. We use several ancilla input. 2. Use a Fredkin gate to create a copy of y. 3. Use the second Fredkin gate to compute x y.

58 Quantum Computing 58 CCNOT using Fredkin 4. Use the third Fredkin gate to copy c. 5. Finally the fourth Fredkin gate computes c (x y). 6. There are several output containing useless data.

59 Quantum Computing 59 CCNOT using Fredkin x x y y c y x y c c c (x y)

60 Quantum Computing 6 Note We have several reversible and universal classical gates. Their quantum mechanical counterpart can be used to simulate classical computation.

61 Quantum Computing 6 Probabilistic Circuit The linear algebra formalism of classical bits and gates can be generalised to probabilistic circuits. Suppose a single-bit is at state with probability p and at state with a probability p. It is represented as a 2-dimensional real vector [ p p, where p [,.

62 Quantum Computing 62 -bit Probabilistic Circuit Applying our old transformation matrices we get, c [ [ p p p p = = [ [ [ [ p p p p = = [ [ p Transformation matrix for probabilistic bits is left stochastic (each column adds to ). p,

63 Quantum Computing 63 2-bit Probabilistic Circuit If we consider two bits so that the probability is p for the th -bit to be and the probability is p for the st -bit to be. The probabilities of,,, are p p,p ( p ),( p )q,( p )( p ) respectively.

64 Quantum Computing 64 2-bit Probabilistic Circuit The joint probabilities of two bits may be represented as the following tensor product - [ [ p p p ( p ) =. p p p p ( p )p ( p )( p )

65 Quantum Computing 65 Xor-And on Probabilistic Bits If we apply Xor-And transformation on two probabilistic bits, we get p p p ( p ) ( p )p ( p )( p ) = p p ( p )( p ) p ( p )+( p )p The interpretation of transformation makes sense..

66 Quantum Computing 66 CNOT on Probabilistic Bits p p p p p ( p ) p ( p ) = ( p )p ( p )( p ) ( p )( p ) ( p )p Again it has meaningful interpretation..

67 Quantum Computing 67 References [ERWP Quantum Computing: A Gentle Introduction by Eleanor Rieffel & Wolfgang Polak, Pub. MIT Press, 2, ISBN [MNIC Quantum Computation and Quantum Information by Michael A Nielsen & Isaac L Chuang, Pub. Cambridge University Press, 22, ISBN [SA Quantum Computing Since Democritus by Scott Aaronson, Pub. Cambridge University Press, 23, ISBN [AAM Classical and Quantum Computation by A Yu Kitaev, A H Shen & M N Vyalyi, Pub. American Mathematical Society (GSM vol 47) 22, ISBN

68 Quantum Computing 68 References [PRM An Introduction to Quantum Computing by Phillip Kaye, Raymond Laflamme & Michele Mosca, Pub. Oxford University Press, 27, ISBN