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1. New Results in Reversible Logic. Marek Perkowski LDL seminar, January 18, 2002

2. Plan • 1. recent results in Reversible Logic obtained by our group. • 2. review the work of DeVos, Storme and others. • 3. new reversible gates in CMOS. • 4. reversible fuzzy logic. • 5. methods to create reversible gates of higher number of inputs/outputs. • 6. new regular structures and synthesis methods for them. • 7. an approach to synthesize arbitrary multi-output functions with no garbage. • 8. the system of EDA tools for reversible/quantum logic. • 9. binary, data mining, multiple-valued and quantum benchmarks

3. constraint constraint constraint A B A=X A XOR B=Y reversible Think about a gate as an input/output constraint A B X Y 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 R(input, output) = R(<A,B>,<X,Y>) R(<A,B>,<X,Y>) = Permute(2,3) <<1,0> , <1,1>> YES <0,0> ==> <0,0> <1,0> <== <1,1> <<1,1> , <0,1>> NO

4. Reversible computation. • Charles Bennett, IBM, 1973. • Logical reversibility of computation, • IBM J. Res. Dev. 17, 525 (1973). This principle applies also to combinational circuits that we build, but is this a best way, question for us to answer

5. Landauer's principle • Landauer's principle:logic computations that are not reversible, necessarily generate heat: • i.e. kTlog(2), for every bit of information that is lost. wherek is Boltzmann's constant and T the temperature. • For T equal room temperature, this package of heat is small, i.e. 2.9 x 10 -21 joule, but non-negligible. • In order to produce zero heat, a computer is only allowed to perform reversible computations. • Such a logically reversible computation can be `undone': the value of the output suffices to recover what the value of the input `has been'. • The hardware of a reversible computer cannot be constructed from the conventional gates • On the contrary, it consists exclusively of logically reversible building blocks.

6. Tomasso Toffoli, 1980: There exists a reversible gate which could play a role of a universal gate for reversible circuits. x x y y z z@xy Q(3) :(x,y,z)==>(x,y,z@xy) @ denotes EXOR Fredkin and Toffoli created the first (3,3) universal gate A B A A XOR B reversible Toffoli Gate

7. x x y x@y Quantum XOR • Of special interest is the controlled-NOT or reversible XOR gate: XOR : (x; y) ==> (x, x @ y); by @ we denote EXOR (modulo-2 sum) These notations were introduced by physicists and they are inconsistent with standard electrical engineering notations, however it will be convenient for us to use both notations.

8. Swapping bits using XOR cascade With the circuit Of importance in quantum, quantum dot, but not CMOS Constructed from three Quantum XORs, we can swap two bits: (x,y) ==> (x,x@y) ==> (y,x@y) ==> (y,x) Conclusion: in quantum logic you pay for crossing wires!!! cascade

9. Use of Toffoli Gate The first step is to show that from the three-bit Toffoli Gate Q (3) we can construct an n-bit Toffoli Gate Q (n). The n-bit gate works as follows: • From three-bit Toffoli-Gate Q(3) (x1,x2,…,x n-1, xn)==>(x1,x2,…,x n-1 y@x1 x 2…x n-1 ) The construction requires one extra bit of scratch space. For example, we construct Q (4) circuit from Q (3) circuits as follows: Q 4 Q 4 from three Q 3 Scratch space

10. Use of Toffoli Gate • The purpose of the last Q (3) gate is to reset the scratch bit back to its original value 0. • With one more gate we can obtain an implementation of Q (4) that works irrespective of the initial value of the scratch bit: We can eliminate the last gate if we don’t mind flipping value of the scratch bit

11. Use of Toffoli Gate • When we need to construct Q (k), any available extra bit will do, since the circuit returns the scratch bit to its original value. • The next step is to note that, by conjugating Q(n)with NOT gates, we can in effect modify the value of the control string that “triggers" the gate. • For example, the circuit flips the value of y if x1 x2 x3 = 010, and it acts trivially otherwise.

12. Use of Toffoli Gate • Flips the value of y if x1 x2 x3 = 010, and it acts trivially otherwise. • Thus this circuit transposes the two strings 0100 and 0101. • In like fashion, with Q(n) and NOT gates, we can devise a circuit that transposes any two n-bit strings that differ in only one bit. • (The location of the bit where they differ is chosen to be the target of the Q (n) gate.)

13. Q R P Q R P 0 1 0 1 A 0 1 A B C B C A circuit from two multiplexers Its schematics This is a reversible gate, one of many Notation for Fredkin Gates

14. Q R P 0 1 0 1 0 1 A A B C C A B Margolus Gate

15. Toffoli Gate P Q R • The 3 * 3 Toffoli gate is described by these equations: P = A, Q = B, R = AB  C, • Toffoli gate is an example of two-through gates, because two of its inputs are given to the output. + * A B C

16. P Q R P Q + + * A B C A B Feynman (b) Toffoli P Q R + + 1 0 * * C A B Kerntopf Gate

17. Kerntopf Gate • The Kerntopf gate is described by equations: P = 1 @ A @ B @ C @ AB, Q = 1 @ AB @ B @ C @ BC, R = 1 @ A @ B @ AC. • When C=1 then P = A + B, Q = A * B, R = !B, so AND/OR gate is realized on outputs P and Q with C as the controlling input value. • When C = 0 then P = !A * ! B, Q = A + !B, R = A @ B. • 18 different cofactors!

18. Kerntopf Gate • As we see, the 3*3 Kerntopf gate is not a one-through nor a two-through gate. • Despite theoretical advantages of Kerntopf gate over classical Fredkin and Toffoli gates, so far there are no published results on optical or CMOS realizations of this gate.

19. Fredkin D Feynman A Toffoli B C Feynman copy copy How to build garbage-less circuits Feynman D Fredkin A Toffoli B C Feynman inputs reconstructed F1 from spy F2 from spy 2 outputs no garbage width = 4 delay = 9 A,B,C,D are original inputs This process is informationally reversible It can be in addition thermodynamically reversible

20. Observations • We reduced garbage at the cost of delay and number of gates • We were not able to reduce the width of the scratchpad register

21. 0 1 1 0 1 1 1 0 1 0 Conservative circuit = the same number of ones in inputs and outputs X3 xor3 Variable Davio Majority X2 xor2 Shannon Examples of balanced functions = half of Kmap are ones

22. YZ X YZ - - - - 00 01 11 10 X - - - - 0 1 1 0 1 0 YZ 0 1 1 1 1 0 1 0 garbage g - - - - f X garbage garbage h gf Y garbage hfg fgh 1 Z garbage 1 0 1 0 YZ X X - - - - 0 1 1 1 garbage 0 1 i fg garbage 0 1 0 1

23. YZ X - - 1 - 1 0 - - - - - - 1 0 1 0 - - - - - - - - - - - 0 - - - - - - - - - - - - - - 1 0 1 0 1 0 - - - - - - - - - - - - 0 1 - - - - - - - - - - - - - - 0 1 1 1 - - - - - - - - 0 1 1 1 - - - - - - - - - - 1 1 = 1 - 0 - - 1 - - - - 1 - - 0 - - - =1 =0 =1 =0 X Y Z

24. Lattice Structure for Multiple-valued and Binary Logic • Realizes every binary symmetric function • Realizes every non-symmetric function by repeating variables • Realizes piece-wise linear multivalued functions Patented by Pierzchala and Perkowski 1994/1999

25. Lattice Structure for Binary Logic F = S 1,3 (A,B,C) A 0 1 B C 0 1 1 0 S0 S1 S2 S3

26. YZ X YZ - - - - 00 01 11 10 X - - - - 0 1 1 0 1 0 YZ 0 1 1 1 1 0 1 0 garbage g - - - - f X garbage garbage h gf Y garbage hfg fgh 1 Z garbage 1 0 1 0 YZ X X - - - - 0 1 1 1 garbage 0 1 i fg garbage 0 1 0 1

27. Control gates as building blocksfor reversible computers • A. De Vos 1 , B. Desoete 2 , F. Janiak 3 , and A. Nogawski 3 • 1 Universiteit Gent and Imec v.z.w., B-9000 Gent, Belgium 2 Universiteit Gent, B-9000 Gent, Belgium 3 Politechnika L odzka, PL-90-924 L odz, Poland

28. As an illustration, two reversible 4-bit carry-look-ahead adders in 0.8 m c-MOS have been built.

29. 2 Simple control gates • 2.1 Definition • A gate with k inputs (A1, A2 ,…, Ak) and k outputs (P1, P2,,... ,…, Pk), satisfying Pi = Ai for all i  { 1, 2,.., k } Pk = f(A1,A2,…, Ak-1)  Ak with fan arbitrary boolean function of kboolean arguments, is called a simple control gate. • The number k is called the width of the gate. • The logic inputs A1, A2, ..., Ak-1 are named the controlling bits, whereas the input Ak is the controlled bit. • Finally, the functionf is called the control function.

30. 2.2 Properties • Any simple control gate is reversible. • We cascade two identical simple control gates, yielding Pk = [ f(A1, A2,…, A k-1)  f(A1, A2,…, Ak-1) ] Ak and thus Pi = Ai for all i, because of the two boolean identities X X = 0 and 0 Y = Y . • The result is thus the k-bit follower. • In other words: any simple control gate is its own inverse, and thus is necessarily reversible.

31. Cascading two arbitrary simple control gates (one with control function f’ and one with control function f’’) results in a new simple control gate, with control function f’f’’. • Therefore the simple control gates of width k together with the operation cascading form a group. • The group has elements.

32. and thus can be built into a square geometry, provided we use dual line electronics, i.e. any signal X is accompanied by its counterpart NOT X. • Fig. 1 shows Pkis connected to Ak if f = 0 but is connected to A’k(short notation for NOT Ak) if f = 1.

33. Because a boolean function f(A1, A2,….,A k-1) can always be written • either as an `OR of ANDs' (often referred to as `sum of minterms') • or as an `AND of ORs' (often referred to as `product of maxterms'), we can implement Pk = f(A1, A2,….,A k-1) A k in the square, using • either four parallel connections of series connections of switches • or four series connectionsof parallel connections of switches • or a combination of both.

34. Each switch is composed of one n-MOS transistorin parallel with one p-MOS transistor (forming together a transmission gate). • This leads to a reversible electronic implementationin dual-line pass-transistor logic:so-called r-MOS technology . • Such logic is naturally suited for adiabatic addressing   . • All energy supplied to the outputs Pkand P’kcomes from the inputs Akand A’k, i.e. not from separate power lines. • 9. A. De Vos: Reversible computing. Progress in Quantum Electronics 23 (1999) 1-49 • 10. B. Desoete and A. De Vos: Optimal charging of capacitors. In: A. Trullemans and J. Spars½ (eds.): Proc. 8 th Int. Workshop Patmos, Lyngby (Oct. 1998) 335-344 • 11. A. De Vos and B. Desoete: Equipartition principles in finite-time thermodynamics. Journal of Non-Equilibrium Thermodynamics 25 (2000) 1-13 • 12. A. De Vos, B. Desoete, A. Adamski, P. Pietrzak, M. Sibi nski, and T. Widerski: Design of reversible logic circuits by means of control gates. In: D. Soudris, P. Pirsch, and E. Barke (eds.): Proc. 10 th Int. Workshop Patmos, G.ottingen (Sept. 2000) 255-264

35. Fig. 1. Implementation of the function f Ak, with the help of four switches f Ak

36. Example of extending functions to reversible functions

37. Fig. 2. Implementation of boolean Table 2 using 12 switches

38. Fig. 2. Implementation of boolean Table 2 using 28 switches • An alternative approach makes use of standard cells, where the particular function f(A, B, C), to be XORed with D, is hardwired by the vias between the Metal1 and Metal2 layers of the chip. • These vias are displayed as small black squares in Fig. 2b. • The programmable gate however needs 2 k+1 - 4 = 28 switches.

39. Fig. 3. Decomposition of a control gate into simple control gates 3. Control gates • Because each output is only one boolean function f away from the inputs, its logic depth is only 1. • We call such gates control gates, as each output Piis either equal to the controlled bit Aior to its inverse A’i, depending on the value of its k-1 controlling inputs A1, A2, .., Ai-1. 3.1 Definition When we cascade k simple control gates (one of width k, one of width k-1,..., and one of width 1), in the way of Fig. 3, we have a new gate of width k.

40. We thus come to the definition of a control gate: • a logic gate with k inputs (A1, A2,….,A k) and k outputs (P1, P2,….,P k), satisfying Pi= fi (A1, A2,….,A k) Ai for all i  {1,2,…,k} with fi arbitrary boolean functions of (i-1) arguments, is called a control gate. • Note that a control gate with width khas (k-1) controlling bits (A1, A2,….,A k) as well as (k-1) controlled bits (A2,….,A k) • We remark that the above definition is somewhat more general than the preliminary definition presented at Patmos 2000 .

41. 3.2 Properties • As any control gate is composed of simple control gates and any simple control gate is reversible, the control gate is thus also reversible. • The inverse of a simple control gate is equal to itself. • This is not the case with an arbitrary control gate. • The inverse of the control gate of Fig. 3 consists of first putting the simple control gate f1, then the simple control gate f2, etc. • Now the cascading of two simple control gates of different width is not commutative. • Thus an arbitrary control gate and its reverse are not necessarily equal, the simple building blocks appearing in opposite order.

42. Two control gates (one with control functions f’ and one with control functions f’’), when cascaded, form a new control gate, • with control functions fi (A1, A2,….,A i-1) =f0 •  Ai fi(A1; A2; :::; Ai-1) = f0 i • (A1, A2,….,A i-1)  f’’ • (A1, A2,….,A i-1)  f’’ • (A1; A2; :::; Ai -1) XOR f00 i ( f0 1 (:) XOR A1; f0 2 (A1) XOR A2; :::; f0-1 (A1; A2; :::; Ai-2) XOR Ai 1 ) :

43. Thus, the control gates of width k, together with the cascading operation, form a group. • This group has 2^(2 k-1) elements and is solvable, but not abelian.