•Controlled-
Controlled NOT
=Toffoli Gate
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Controlled - Quantum Computing - Lecture Slides, Slides of Computer Science
These are the Lecture Slides of Quantum Computing which includes Classical Computers, Quantum Computers, Significantly Faster, Factorization Problems, Exponential, Classical Computers, Non Polynomial Problems, Unstructured Search, Circuit Level Representation etc. Key important points are: Controlled, Toffoli Gate, Circuit, Reversible Function, Earlier Argument, Boolean Functions, Relying, Scratch Space, Circuits, Karnaugh Maps
Typology: Slides
2012/2013
1 / 13
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• Controlled-
Controlled NOT
=Toffoli Gate
The Toffoli Gate
- The Toffoli gate Q(3)^ is universal in the sense that we can
build a circuit to compute any reversible function using
Toffoli gates alone (if we can x input bits and ignore output
bits).
- It will be instructive to show this directly, without relying
on our earlier argument that NAND/NOT is universal for
Boolean functions.
- In fact, we can show the following:
- From the NOT gate and the Toffoli gate Q(3)^ , we can construct any invertible function on n bits, provided we have one extra bit of scratchpad space available. Docsity.com
Simple Idea – Toffoli gate with any number of inputs
- If we generalize the Toffoli Gate, we can realize any binary
function in a very efficient way
One can build Toffoli gate with 3 inputs
Can one build Toffoli gate with n inputs??????
xn
z
xn
z@x1 x2 … xn-1 xn
xn-1 xn-
x1 x
Of course, from many gates, but directly???
Karnaugh Maps
- A 4-variable K-
map.
YZ WX 00 01
00
01
0 1
4 5
3 2
7 6
11 10
124
8 9 11 10
13 15 14 11
10
Simple Idea
ESOP = Positive RM cover
YZ WX 00 01
00
01
0 1
4 5
3 2
7 6
11 10
124
8 9 11 10
13 15 14 11
10
Simple Idea
F=wx @ yz
Simple Idea – build ESOP circuit from Toffoli gates
w x
y z
0
wx wx⊕yz
Realizations of binary logic with
Toffoli and reversible logic with
Toffoli-like circuits
- Kronecker functional Diagram (uses
Davio expansions and Shannon Expansions)
- Kronecker function-driven Diagram
- ESOP
- DSOP (disjunctive SOP = Disjunctive
ESOP)
A
B
B
g f^ A
X Y= C
C 1
¬ C
00 01 11 10
00 AB^ C^0 01 11 10
AB C 0 1
00 01 11 10
AB C 0 1
0 1 0 1 1 0 1 1
1 1 1 0
00 01 11 10
AB C 0 1 0 1 0 1
f (^) A’
X=f (^) A’⊕ f (^) A
00 01 11 10
AB C 0 1
1 0 ¬ C
f (^) B’
00 01 11 10
AB C 0 1
f (^) B’ ⊕ f (^) B 0 1
Graphical method to calculate decision diagram from Toffoli gates
We use Davio expansions
Use Toffoli gates to realize Davio expansions
What you have to remember
- Toffoli gate equation and symbol.
- Kmaps for up to 5 variables.
- The concept of ESOP circuits.
- How to synthesize ESOP circuit from equation
- How to synthesize ESOP equation from Kmap.
- The concept of Mirror Circuit
- The concept of ancilla bit
- How to create large Toffoli gates from small Toffoli gates
- The concept of Davio Expansion realized in Toffoli gate.
The concepts of Davio expansions and ESOP/DSOP circuits will be more elaborated in next lectures. Now you can just use their ideas for simple circuits.