Reversible Logic Synthesis: A Group-Theory Approach, Slides of Computer Science

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A Constructive Group

Theory based

Algorithm for

Reversible Logic

Synthesis

Main Ideas

1. We know many representations of reversible

circuits, such as reversible vectors, truth tables,

Kmaps and permutative unitary matrices.

2. In this lecture we will learn one more useful

representation – a set of cycles.

3. A cycle can be decomposed to transpositions.

Set of transpositions is another representation.

4. This is the initial paper in “group-theory

approach to quantum circuits synthesis”

Definition 1 (Binary reversible gate):

• Let B = {0, 1}.
• A binary logic circuit f with n inputs and outputs is denoted by

a binary multiple-output function f : B n^  B n.

• Let 〈B 1 ,…, B (^) n 〉 ∈ B n^ and 〈P 1 ,…, P (^) n 〉 ∈ B n^ be the input and

output vectors,

• where B 1 ,…, B (^) n are input variables
• and P 1 ,…, P (^) n are 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. Docsity.com

Permutations,

Cycles and

Transpositions

• A mapping s : M  M can be written as:

d 1 , d 2 ,.., d k

di1 , di2 ,.., d i1 (1)

• Here we use a product of disjoint cycles (Definition 3) as an

alternative notation for a mapping.

• For example,

d 1 , d 2 , d 3 ., d4 , d 5 , d 6 , d 7 , d 8 , d 9

d1, d 4 , d 7 , d 2 ,d 5 , d 8 , d 3 , d 6 , d 9 (2)

can be written as (d 2 , d 4 ) (d 3 , d 7 ) (d 6 , d 8 ).

(

(

(

(

Mappings and cycles

Identity

minterm

F(a 14 ) = a 14

F(1101) = 1101

F(13) = 13

Identity function as a mapping (^) Identity function as a circuit

Identity function as a mapping (^) Identity function as a circuit

Simple function

Simple function has

no identity minterms

The function has four 2-cycles N is high but function is simple (0,1), (2,3), (4,5) and (6,7)

Problem to be solved has simple formulation

It is a kind of “Rubik Cube” game

Feynman Toffoli Feynman Fredkin

Distance Histogram of a function

How many minterms in distance k cycle

Distance k cycle

0 1 2 3

How many minterms in distane k cycle

Distance k cycle

6

8

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 f before g.

Definition 3 (‘j’-cycle):

• Let Sk be a symmetric group

of symbols {d 1 , d 2 ,…,d (^) k},

then (d (^) i1 , d (^) i2 ,…, d (^) ij ) is

called

a ‘j’-cycle,

• where j ≤ k, 1 ≤ i 1 ,

i 2 ,…, ij ≤ k.

Cycle (0,1,2,3)

Realizations of the same cycle

00

0 1

01

11

10

Cost 1+2=

00

0 1

01 11

10

Cost 1+1=

(0,1,2) = (0,1) * (0,2)

(0,1,2) = (0,2) * (1,2)

We should select the set of 2-cycles with the minimum total distance

This is better

Definition 5 (‘ n-1 ’-CNOT gate):

• A ‘n-1’-Controlled-NOT (CNOT) gate Cj is defined as follows:
• If m ≠ j, then P (^) m = Cj (B (^) m ) = B (^) m.
• If m = j, and B (^) i = 1 for all i ≠ j, then P (^) j = Cj (B (^) j ) = B (^) j ⊕ 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

MMD and New

methods

1. 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.

2. This can be rewritten to our new notation.

3. But this is only one way of realizing

permutation from transpositions.

4. We should investigate more of them.