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April 25, 2011

April 25, 2011. A Constructive Group Theory based Algorithm for Reversible Logic Synthesis. Main Ideas . We know many representations of reversible circuits, such as reversible vectors, truth tables, Kmaps and permutative unitary matrices.

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April 25, 2011

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  1. April 25, 2011

  2. A Constructive Group Theory based Algorithm for Reversible LogicSynthesis

  3. Main Ideas • We know many representations of reversible circuits, such as reversible vectors, truth tables, Kmaps and permutative unitary matrices. • In this lecture we will learn one more useful representation – a set of cycles. • A cycle can be decomposed to transpositions. Set of transpositions is another representation. • This is the initial paper in “group-theory approach to quantum circuits synthesis”

  4. Background • Reversible logic plays an important role in quantum computing; • oracles are reversible (permutative) circuits. • Any computing system of irreversible logic gates leads inevitably to energy dissipation. • To avoid power dissipation, circuits must be constructed from reversible gates. • De Vos – CMOS • Optical • DNA • Nano-technologies • There are a lot of research on the construction of reversible logic gates and circuits: • De Vos 1998, group theory. • Kerntopf 2000, enumerative – decision diagrams. • Perkowski et al. 2001, exor logic, spectral, group theory, heuristic, genetic algorithm, exhaustive. • Markov et al 2002, exhaustive, group theory. • Yang et al, 2003 – group theory approach – GAP and exhaustive search • Miller, Dueck and Maslov 2003, spectral and heuristic non-optimal algorithm • Aggrawal and Jha 2004, heuristic RM-transformation-based.

  5. Definition 1 (Binary reversible gate): • Let B = {0, 1}. • A binary logic circuit fwith n inputs and outputs is denoted by a binary multiple-output function f : Bn Bn. • Let B1,…, Bn  Bn and P1,…, Pn  Bnbe the input and output vectors, • where B1,…, Bn are input variables • and P1,…, Pnare output variables. • There are 2n different assignments for the input vectors. • A binary logic circuit f is reversible if it is a one-to-one and onto function (bijection). • A binary reversible logic circuit with n inputs and n outputs is also called an n-qubit binary reversible gate. • There are a total of (2n)! different n-qubit binary reversible circuits.

  6. Permutations, Cycles and Transpositions

  7. Permutation groups and their relationshipwith reversible circuits • Definition 2 (Permutation):Let M = {d1, d2 , , dk}. • A bijection of M onto itself is called a permutation on M. • The set of all permutations on M forms a group under composition of mappings, called a symmetric group onM. • It is denoted by Sk. • A permutation group is simply a subgroup of a symmetric group.

  8. Mappings and cycles • A mapping s : M  M can be written as: ( d1 , d2 ,.., dk di1 , di2 ,.., di1(1) ( • Here we use a product of disjoint cycles (Definition 3) as an alternative notation for a mapping. • For example, • d1 , d2 , d3 ., d4 , d5 , d6 , d7 , d8 , d9 • d1, d4 , d7 , d2 ,d5 , d8 , d3 , d6 , d9(2) • can be written as (d2, d4) (d3, d7) (d6, d8). ( (

  9. F(a14) = a14 F(1101) = 1101 F(13) = 13 Identity minterm 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Identity function as a mapping Identity function as a circuit

  10. Simple function Simple function has many identity minterms, N is small The function is a single 2-cycle (6,7) 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Toffoli gate as a mapping Toffoli gate as a circuit

  11. Simple function Simple function has no identity minterms The function has four 2-cycles (0,1), (2,3), (4,5) and (6,7) N is high but function is simple 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Identity function as a mapping Identity function as a circuit

  12. Problem to be solved has simple formulation It is a kind of “Rubik Cube” game 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Toffoli Feynman Feynman Fredkin

  13. Patterns based on transpositions and costs based on transpositions

  14. How many minterms in distance k cycle Distance Histogram of a function 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 6 How many minterms in distane k cycle 0 1 2 3 Distance k cycle 8 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 Distance k cycle

  15. Notation • Denote “( )” as the identity mapping (i.e., direct wiring) and call this the unity element in a permutation group. • A product f * g of two permutations applies mapping fbeforeg.

  16. Cascading gates and permutations • A n-qubit reversible circuit is a permutation in S2n, and vice versa. • Cascading two gates is equivalent to multiplying two permutations in S2n. • Thus, in what follows, we will not distinguish a n-qubit reversible circuit from a permutation in S2n.

  17. Definition 3 (‘j’-cycle): • Let Sk be a symmetric group of symbols {d1, d2,…,dk}, then (di1 , di2 ,…, dij ) is called a ‘j’-cycle, • where j  k, 1  i1 , i2 ,…, ij k. 01234567 01234567 Cycle (0,1,2,3)

  18. Realizations of the same cycle 0 1 00 Cost 1+2=3 012 01 11 (0,1,2) = (0,2) * (1,2) 10 We should select the set of 2-cycles with the minimum total distance 0 1 00 Cost 1+1=2 012 01 11 (0,1,2) = (0,1) * (0,2) This is better 10

  19. Definition 4 (NOT gate): • A NOT gate Nj connects an inverter to the j-th wire, i.e.: Pj = Bj 1, Pi = Bi , if i  j . 1  j  n: • An example NOT gate is shown in figure 1.

  20. Definition 5 (‘n-1’-CNOT gate): • A ‘n-1’-Controlled-NOT (CNOT) gate Cjis defined as follows: • If m  j, then Pm = Cj (Bm) = Bm . • If m = j, and Bi = 1 for all i  j, then Pj = Cj (Bj ) = Bj 1; else, Pj = Bj . • An example ‘4’-CNOT gate is shown in figure 2. • A ‘n-1’-CNOT gate is a generalized Toffoli gate where two inputs control an output of another input. These are “big Toffoli gates” that we want to avoid in general

  21. MMD and New methods • MMD is a specific algorithm that processes permutation vectors (truth table) in such a way that the set of self-mapping minterms grows from top. • This can be rewritten to our new notation. • But this is only one way of realizing permutation from transpositions. • We should investigate more of them.

  22. MMD method 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 We add identity minterms from top MMD method can start from function with small N and make it a new function with big N Our idea here is to decompose to 2-cycles of the minimum total distance

  23. How many minterms in distane k cycle Distance 3 function – Miller 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 6 0 1 2 3 Distance k cycle Distance 1 function – Toffoli Distance 2 function – Fredkin 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

  24. Programs for reversible logic Perform heuristic search with one ordering MMD Perform heuristic search with many orderings Function (permutation) with high N MMDS Template Greedy Distance Perform local optimization to reduce costs of gates Reduce N Decompose to smallest distance gates Function with small N This can be greedy or evolutionary algorithm

  25. III. THEORETICAL RESULTS on decomposing to cycles • The process to constructively synthesize any ‘n’-qubit reversible circuit by Not and ‘n-1’- CNOT gates without ancilla qubits. • Lemma 1:Every permutation can be decomposed to some ‘2’-cycles. 2-cycle = transposition

  26. Hamming Distance one Transpositions or “Neighboring 2-cycles” • Remark 1: • This lemma is a well-known result in group theory • Definition 6 (neighboring ‘2’-cycle): • If two n-dimension vectors u, s have only one bit difference, we call the permutation (u, s) a neighboring ‘2’-cycle.

  27. Realizations of the same cycle 0 1 00 Cost 1+2=3 012 01 11 (0,1,2) = (0,2) * (1,2) 10 We should select the set of 2-cycles with the minimum total distance 0 1 00 Cost 1+1=2 012 01 11 (0,1,2) = (0,1) * (0,2) This is better 10

  28. Lemma 2: • Suppose between u and s, there is only one bit Bj different, and i same bits are zeros. • These zero bits are Bi1 ,…, Bil . • Then, (u , s) = Ni1 * … * Nil * Cj * Nil * … * Ni1 (4) This is a distance 1 decomposition of a 2-cycle

  29. Lemma 3: Decomposition to distance-1 transpositions If two n-dimension vectors u , s have k bits different, then there is an ordered set M = {d1, d2 ,.., dk+1 } such that d1 = u , dk+1 = s and for any i, 1 i < k + 1, there is only one bit difference between di and di+1, and (u , s) = (d1, d2)(d2 , d3) … (dk , dk+1) (dk , dk-1) … (d2 , d1) (5) This is a decomposition of a 2-cycle to 1-distance cycles. Total distance cost should be minimized

  30. Decomposing to sequences of neighbor minterms in transpositions • In order to make the number of NOT gates as small as possible, we give two rules for constructing the ordered set M. • If the number of 1’s in the vector u is more than that in s, then d1 = u , dk+1 = s. Else, d1 = s , dk+1 = u. • In the different bits between u and s, change the zero bit to one first, then change one bit to zero bit. • For example, if u = (0 , 1 , 0 , 0 , 0 ), s = (0 , 0 , 1 , 1 , 1), then k = 4; d1 = s , d5 = u and, d2 , d3 , d4 are given in the following table.

  31. Hamming Distance and “Distance-Gates” Various Distance 3 gates 0 1 00 When F(a) = b, (or (a,b) is a 2-cycle) Hamming distance of a and b is 3 01 11 10 All Distance 1 gates 0 1 00 When F(a) = b, Hamming distance of a and b is 1 01 11 10

  32. REMOVAL OF NOT gates • can remove a pair of the adjacent NOT gates in the same quantum wire. • Lemma 4: (complexity of NOT gates) • In decomposing ‘2’-cycle (u , s) to NOT gates and ‘n-1’-CNOT gates, suppose there are j bits different between u and s, and there are j0 zero bits in these bits in d1 and j1 one bits in these bits in d1, where j0 j1, then the number of NOT gates is no more than 2(j - 2) + 2(j0 - 1) if j0 1 , or 2(j-1) if j0 = 0.

  33. Theorem 1: • For a given n-qubit reversible circuit f, if there are Ndistinct input patterns that are different from their corresponding outputs, where N  2n, and the other (2n - N) input patterns are the same as their outputs, then this circuit can be synthesized by at most 2n * N‘(n - 1)’- CNOT gates and 4n2 * N NOT gates without ancilla qubit.

  34. Theorem 2: All n-qubit reversible circuits can be constructed by: • less than 2n * 2nNOT gates • and less than (2n - 1) * 2n ‘n - 1’-CNOT gates • without ancilla qubit.

  35. ALGORITHM • Step 1. • Check the truth table of f to determine before using equation 3 and lemma 3, whether we need NOT gates or not. • Step 2. • After step 1, write the reversible circuit in a product of cycles form. • For every cycle (d1, d2 ,…, dk), calculate the number riof different bits between diand di+1, i = 1, 2 ,…, k where dk+1 = d1. • Let rjbe the maximal number. • The basic idea to decompose the reversible circuit by equation 3 is to break the mapping relation from dj to dj+1 without increasing the number of different bits between adjacent vectors. • (d1, d2 ,…, dk) = (dj , dj+1) (dj , dj+2 , dj+3 ,…, d1) (6)

  36. ALGORITHM (cont) • Recursively repeating this process, we decompose the reversible circuit to ‘2’-cycles. • Step 3. • Decompose every ‘2’-cycles by NOT and ‘n-1’-CNOT gates by Lemma 3, two rules in remark 2, lemma 2, and remove pairs of adjacent NOT gates when possible.

  37. COMPLETE SYNTHESIS EXAMPLE WITH NON-MINIMIZED TOFFOLI GATES

  38. SYNTHESIS EXAMPLE • Given a binary reversible circuit f which has a truth table as shown in Table II. • Therefore, f = (a1, a3, a4 , a16) (a2 , a6 , a14). • Step 1. • The total number of changed vectors is 7, less than 24-1 = 8, thus, we deal with the input reversible circuit f without pre-cascading NOT gates. • Step 2. • Decompose each cycle into the product of 2-cycles by using equation 6.

  39. a4=1100 a16=1111 d2 N2 * N2= () a4=1100 a3=0100 (a1,a3,a4,a16) (a2,a6,a14) d1 a6=1010 a6=1010 (a4,a16) (a4,a3) (a1, a3) (a6,a14) (a6,a2) a2=1000 a14=1011 d2 d1 d1 d1 d1 d1 d1 inverters

  40. C3

  41. (a4,a16) (a4,a3) (a1, a3) (a6,a14) (a6,a2) d2 d1 d1 d1 d1 • The synthesis process is finished, and f is decomposed into the product of 12 NOT gates and 7 ‘n-1’-CNOT gates, shown in figure 3. C3 N3 C4

  42. IMPROVED SYNTHESIS METHOD BASED ON “DISTANCE GATE”

  43. New idea = a better way to implement transpositions (13,14) a b c d a a4=1101 b c a16=1110 d d2 Realization of transposition (1101, 1110) = (13,14) HD =2

  44. New idea = a better way to implement transpositions (12,15) a b c d a b a4=1100 1 0 1 c c a16=1111 0 1 d d d2 d2 Realization of transposition (1100, 1111) = (12,15) a b 0 1 HD =2 0 c c 1 0 d d

  45. New idea = a better way to implement transpositions (12,15) a b c d 1 a 1 a4=1100 b 1 1 0 0 c c a16=1111 1 1 d d d2 Realization of transposition (1100, 1111) = (12,15) 1 a 1 b 0 0 1 1 HD =2 c c 0 0 d d

  46. New idea = a better way to implement transpositions (12,11) a b c d 1 a 1 1 1 1 a4=1100 b 0 0 0 c c 11001100 no change a16=1101 0 0 0 d d d2 Realization of transposition (1001, 1110) = (9,14) 1 a 1 1 0 1 b 0 0 HD =2 0 c c 0 11011101 no change 1 1 1 1 d d

  47. New idea = a better way to implement transpositions a b c d 1 a 1 0 0 0 a4=1001 b 1 1 1 c c 1 a16=1110 0 0 0 0 d d d2 10101010 Realization of transposition (1001, 1110) = (9,14) 1 a 1 1 0 1 1 b 1 0 HD =2 0 c c 1 10011110 0 1 1 0 d d Here we see the transposition in action

  48. Distance gates • The gates shown in last few slides are called “Distance gates”. • They are all inexpensive and realize transpositions. • By adding inverters on inputs and outputs of these gates a wider category (called family) of these gates is created. • We should consider the widest family and find synthesis methods (algorithms) for them.

  49. CONCLUSIONS • The concept of decomposition of permutation to cycles • The concept of decomposition of cycle to 2-cycles • The concept of distance-k-gates and decomposition with smallest total distance • A constructive algorithm for synthesizing reversible circuits by NOT and ‘n-1’-CNOT gates • Synthesis examples based on this algorithm, which show that even by hand, synthesizing any ‘4’-qubit reversible circuit is not difficult. Especially for functions with small N. • The computational complexity of our synthesis algorithm is exponentially lower than the complexity of breadth-first search based synthesis algorithm. • To be used as part of a comprehensive system of programs

  50. CONCLUSIONS • Knowledge reduces search – group theory • Knowledge is not sufficient to find minimal solution – search is necessary • Role for evolutionary algorithms, PSO, immune, etc • We found some heuristics, more are needed. • Based on last section, the method can be improved, which is a good project.

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