Calculating with the square root of NOT

Transcription

1 Calculating with the square root of NOT Alexis De Vos Imec v.z.w. and Universiteit Gent Belgium Cryptasc workshop Université Libre de Bruxelles, 6 Oct 2009

2 This is about groups

3 This is about groups Groups exist in three different flavours:

4 This is about groups Groups exist in three different flavours: finite groups Order is an integer infinite, but still discrete groups Order = ℵ 0 infinite groups with a non-countable number of elements = Lie groups Order = m

5 But first another story...

6 A logically irreversible computer 3 1 adding computer 4

7 3 1 adding computer 4?? adding computer 4

8 A logically reversible computer : 3 adding & subtracting computer 1 2 4

9 3 adding & subtracting computer adding & subtracting computer 1 2 4

10 Truth table of three irreversible logic gates (a) XOR gate (b) NOR gate (c) AND gate. AB P AB P AB P (a) P = A B (b) P = A + B (c) P = A B Truth table of a reversible logic gate AB PQ P = A B Q = A

11 Truth table of three reversible logic gates of width 2 (a) an arbitrary reversible gate r (b) the identity gate i (c) the inverse r 1 of r AB PQ (a) AB PQ (b) AB PQ (c) The group of reversible gates of width w is isomorphic to the symmetric group S 2 w. Its order is (2 w )!. Here w = 2 Thus S 4 Its order is 4! = 24.

12 The number r of reversible gates w r , ,922,789,888,000 r(w) = (2 w )!

13 The CONTROLLED NOT gate: P = A Q = A B. is equivalent with P = A Q = if (A = 0) then B else B. The CONTROLLED CONTROLLED NOT gate: P = A Q = B R = AB C. is equivalent with P = A Q = B R = if (AB = 0) then C else C.

14 Schematic for (a) CONTROLLED NOT gate (b) CONTROLLED CONTROLLED NOT gate (c) CONTROLLED SWAP gate a B _ Q _ A Q A A A B c C R A Q A A A B C R A Q A A A B b R _ C _ R C A A B B A A B B

15 Transistor cost: The CONTROLLED NOT gate or FEYNMAN gate : 8 transistors TheCONTROLLED CONTROLLED NOT gate ortoffoli gate : 16 transistors The CONTROLLED SWAP gate or FREDKIN gate : 16 transistors.

16 Microscope photograph (140 µm 120 µm) of 2.4-µm 4-bit reversible ripple adder (192 transistors).

17 Microscope photograph (610 µm 290 µm) of 0.8-µm 4-bit reversible carry-look-ahead adder (320 transistors).

18 Microscope photograph (1,430 µm 300 µm) of 0.35-µm 8-bit reversible multiplier (2,504 transistors).

19 Oscilloscope view of 0.35 µm full adder.

20 The number r of reversible circuits and the number q of quantum circuits w r q , ,922,789,888, r(w) = (2 w )! q(w) = (2w ) 2

21 Example: w = 1 r(w) = r(1) = 2 The two 1-bit computers (group S 2 ) are represented by the 2 2 permutation matrices: and q(w) = q(1) = 4 The infinitely many 1-qubit computers (group U(2)) are represented by the 2 2 unitary matrices: cos(θ) exp(iα) sin(θ) exp(iβ) sin(θ) exp(iγ) cos(θ) exp( iα + iβ + iγ)

22 Example: w = 1 r(w) = r(1) = 2 The two 1-bit computers (group S 2 ) are represented by the 2 2 permutation matrices: and q(w) = q(1) = 4 The infinitely many 1-qubit computers (group U(2)) are represented by the 2 2 unitary matrices: cos(θ) exp(iα) sin(θ) exp(iβ) sin(θ) exp(iγ) cos(θ) exp( iα + iβ + iγ) S 2? U(2) 2 < Order(?) < 4

23 Take S 2 and introduce an extra generator: i 1 i 2 1 i 1 + i Obtain a group with four 2 2 matrices:, i 1 i 2 1 i 1 + i, , and i 1 + i 1 + i 1 i S 2 Z 4 U(2) 2 < 4 < 4

24 Next: w = 2 Take S 4 and introduce an extra generator: i 1 i i 1 + i i 1 i i 1 + i S 4? U(4) 24 < 192 < 16

25 Next: w = 2 Take S 4 and introduce an extra generator: i 1 i i 1 + i i 1 i i 1 + i S 4? U(4) 24 < 192 < 16 Four representative circuits: a b c d

26 Next: w = 2 Take S 4 and introduce an other extra generator: i 1 i i 1 + i S 4? U(4) 24 < ℵ 0 < 16

27 The number of reversible circuits, as a function of the circuit width w, if we allow no square roots of NOT, if we allow square roots of NOT, and if we allow controlled square roots of NOT. w classical classical plus NOT classical plus controlled NOT not applicable ℵ 0 3 4

28 The number of reversible circuits, as a function of the circuit width w, if we allow no square roots of NOT, if we allow square roots of NOT, and if we allow controlled square roots of NOT. w classical classical plus NOT classical plus controlled NOT not applicable ℵ ,320 ℵ 0 ℵ ,922,789,888,000 ℵ 0 ℵ 0

29 The number of reversible circuits, as a function of the circuit width w, if we allow no square roots of NOT, if we allow square roots of NOT, and if we allow controlled square roots of NOT. w classical classical plus NOT classical plus controlled NOT not applicable ℵ ,320 ℵ 0 ℵ ,922,789,888,000 ℵ 0 ℵ 0 Making two controlled NOT gates from a single NOT gate.

30 Future research: (1) Replace square root by other root θ/π = 1/2 = i 1 i 1 i 1 + i, cos(θ) exp(iθ) i sin(θ) exp(iθ) i sin(θ) exp(iθ) cos(θ) exp(iθ)

31 Future research: (1) Replace square root by other root θ/π = 1/2 = i 1 i 1 i 1 + i, cos(θ) exp(iθ) i sin(θ) exp(iθ) i sin(θ) exp(iθ) cos(θ) exp(iθ) S 2 U(1) U(2) 2 < 1 < 4

32 Future research: (1) Replace square root by other root θ/π = 1/2 = i 1 i 1 i 1 + i, cos(θ) exp(iθ) i sin(θ) exp(iθ) i sin(θ) exp(iθ) cos(θ) exp(iθ) (2) Take square root of other (hermitian) 2 2 matrix e.g. Pauli matrix = Hadamard matrix = i i

33