From classical circuits to quantum circuits. Alexis De Vos and Stijn De Baerdemacker. Waterloo, 9 June 2015

Transcription

1 From classical circuits to quantum circuits Alexis De Vos and Stijn De Baerdemacker Waterloo, 9 June 2015

2 Un bonjour de Waterloo, Belgique pour Waterloo, Ontario

3 From classical circuits to quantum circuits Alexis De Vos and Stijn De Baerdemacker

4 U(n) = the unitary group = the group of n n unitary matrices

5 U(n) = the unitary group = the group of n n unitary matrices U(2 w ) = the quantum circuits acting on w qubits

6 U(n) = the unitary group = the group of n n unitary matrices U(2 w ) = the quantum circuits acting on w qubits Thus: U(2) = the quantum circuits acting on 1 qubit = the group of 2 2 unitary matrices

7 Single-qubit circuits ( ) 1 0 I = 0 1

8 Single-qubit circuits ( ) 1 0 I = 0 1 Two square roots: ( ) 0 1 X = 1 0 and ( ) 1 0 Z = 0 1

9 Single-qubit circuits NEGATOR = N(θ) = ( ) cos(θ/2) exp( iθ/2) i sin(θ/2) exp( iθ/2) i sin(θ/2) exp( iθ/2) cos(θ/2) exp( iθ/2)

10 Single-qubit circuits NEGATOR = N(θ) = ( cos(θ/2) exp( iθ/2) i sin(θ/2) exp( iθ/2) ) i sin(θ/2) exp( iθ/2) cos(θ/2) exp( iθ/2) with special values N(0) = N(π) = N(π/2) = 1 2 ( ) ( ) = I = I = X ( 1 i 1 + i 1 + i 1 i ) = X = V N(π/4) = V = W

11 Single-qubit circuits NEGATOR = N(θ) = ( cos(θ/2) exp( iθ/2) i sin(θ/2) exp( iθ/2) ) i sin(θ/2) exp( iθ/2) = cos(θ/2) exp( iθ/2) exp(iθ) exp(iθ) exp(iθ) exp(iθ)

12 Single-qubit circuits NEGATOR = N(θ) = ( cos(θ/2) exp( iθ/2) i sin(θ/2) exp( iθ/2) ) i sin(θ/2) exp( iθ/2) cos(θ/2) exp( iθ/2) N(θ)

13 Single-qubit circuits ( ) 1 0 I = 0 1 Two square roots: ( ) 0 1 X = 1 0 and ( ) 1 0 Z = 0 1

14 Single-qubit circuits PHASOR = Φ(θ) = ( exp(iθ) )

15 Single-qubit circuits PHASOR = Φ(θ) = ( exp(iθ) ) with special values Φ(0) = Φ(π) = Φ(π/2) = Φ(π/4) = S = T ( ) 1 0 = I 0 1 ( ) 1 0 = I = Z 0 1 ( ) 1 0 = Z = S 0 i

16 Single-qubit circuits PHASOR = Φ(θ) = ( exp(iθ) ) Φ(θ)

17 Single-qubit circuits PHASOR = Φ(θ) = ( exp(iθ) ) Φ(θ) NEGATOR = N(θ) = ( cos(θ/2) exp( iθ/2) i sin(θ/2) exp( iθ/2) ) i sin(θ/2) exp( iθ/2) cos(θ/2) exp( iθ/2) N(θ)

18 Two-qubit circuits N(θ)

19 Two-qubit circuits N(θ) Φ(θ)

20 Multiple-qubit circuits N(θ) cos(θ/2)e iθ/2 i sin(θ/2)e iθ/ i sin(θ/2)e iθ/2 cos(θ/2)e iθ/

21 Multiple-qubit circuits N(θ) cos(θ/2)e iθ/2 i sin(θ/2)e iθ/ i sin(θ/2)e iθ/2 cos(θ/2)e iθ/ XU(n) U(n)

22 Multiple-qubit circuits N(θ) cos(θ/2)e iθ/2 i sin(θ/2)e iθ/ i sin(θ/2)e iθ/2 cos(θ/2)e iθ/ XU(n) U(n)

23 Multiple-qubit circuits Φ(θ) exp(iθ)

24 Multiple-qubit circuits Φ(θ) exp(iθ) ZU(n) U(n)

25 Multiple-qubit circuits Φ(θ) exp(iθ) ZU(n) U(n)

26 Within the unitary group U(n), two subgroups : the group XU(n) of all n n unitary matrices with all line sums equal to 1 the group ZU(n) of all n n unitary diagonal matrices with first entry equal to 1.

27 ZU(n) U(n) XU(n) Whereas U(n) has n 2 dimensions, XU(n) has (n 1) 2 dimensions and ZU(n) has (n 1) dimensions.

28 An arbitrary member of U(n) can be decomposed U = exp(iβ) Z 1 X Z 2

29 An arbitrary member of U(n) can be decomposed U = exp(iβ) Z 1 X Z 2 where Z 1 ZU(n) X XU(n) Z 2 ZU(n)

30 An arbitrary member of U(n) can be decomposed U = exp(iβ) Z 1 X Z 2

31 An arbitrary member of U(n) can be decomposed U = exp(iβ) Z 1 X Z (n 1) + (n 1) 2 + (n 1)

32 An arbitrary member of U(n) can be decomposed U = exp(iβ) Z 1 X Z 2 n 2 = 1 + (n 1) + (n 1) 2 + (n 1)

33 We conjectured (arxiv:math-ph ) on 30 Jan 2014.

34 We conjectured (arxiv:math-ph ) on 30 Jan Idel and Wolf proved (arxiv:math-ph ) on 25 Aug 2014

35 Thus U = exp(iβ) Z 1 X Z 2 e iβ Z 1 X Z 2

36 The two ZU(n) parts : Z =

37 The two ZU(n) parts : Z =... Φ(α 7 ) Φ(α 5 ) Φ(α 3 )... Φ(α 8 ) Φ(α 6 ) Φ(α 4 ) Φ(α 2 )

38 The XU(n) part : X =

39 The XU(n) part : X = C 1 C 2 C 3... C 2n 2

40 The XU(n) part : X = C 1 C 2 C 3... C 2n 2 where C j is a block-circulant XU(n) matrix

41 E.g. C = /3 2/3 2/3 0 2/3 1/3 2/3 0 2/3 2/3 1/3

42 E.g. C = /3 2/3 2/3 0 2/3 1/3 2/3 0 2/3 2/3 1/3 N 2 N 3 N 1 N 1 N 1 N 4... N 1 N 3 N 2... N 5 N 1 N 1

43 E.g. C = /3 2/3 2/3 0 2/3 1/3 2/3 0 2/3 2/3 1/3 N 2 N 3 N 1 N 1 N 1 N 4... N 1 N 3 N 2... N 5 N 1 N 1 where N 1 = N(π/2), N 2 = N(π/4) N 3 = N(7π/4) N 4 = N(π + Arccos(1/3)) N 5 = N( π Arccos(1/3)).

44 Conclusion : U = exp(iβ) Z 1 X Z 2 e iβ Z 1 X Z 2

45 Conclusion : U = exp(iβ) Z 1 X Z 2 e iβ Z 1 X Z 2 Z 1 with PHASORs X with NEGATORs Z 2 with PHASORs

46 The end / La fin Waterloo (Belgique), 18 June 1815