Fredkin gate

Circuit representation of Fredkin gate

The Fredkin gate (also CSWAP gate) is a computational circuit suitable for reversible computing, invented by Ed Fredkin. It is universal, which means that any logical or arithmetic operation can be constructed entirely of Fredkin gates. The Fredkin gate is the three-bit gate that swaps the last two bits if the first bit is 1.

Definition

The basic Fredkin gate^[1] is a controlled swap gate that maps three inputs (C, I₁, I₂) onto three outputs (C, O₁, O₂). The C input is mapped directly to the C output. If C = 0, no swap is performed; I₁ maps to O₁, and I₂ maps to O₂. Otherwise, the two outputs are swapped so that I₁ maps to O₂, and I₂ maps to O₁. It is easy to see that this circuit is reversible, i.e., "undoes" itself when run backwards. A generalized n×n Fredkin gate passes its first n-2 inputs unchanged to the corresponding outputs, and swaps its last two outputs if and only if the first n-2 inputs are all 1.

The Fredkin gate is the reversible three-bit gate that swaps the last two bits if the first bit is 1.

Truth table

Matrix form

INPUT			OUTPUT
C	I₁	I₂	C	O₁	O₂
0	0	0	0	0	0
0	0	1	0	0	1
0	1	0	0	1	0
0	1	1	0	1	1
1	0	0	1	0	0
1	0	1	1	1	0
1	1	0	1	0	1
1	1	1	1	1	1

$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}$

It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the billiard ball model means the same number of balls are output as input. This corresponds nicely to the conservation of mass in physics, and helps to show that the model is not wasteful.

Logic function with XOR and AND gates

O₁ = I₁ XOR S

O₂ = I₂ XOR S

with S = (I₁ XOR I₂) AND C

It can also be implemented by the following logic:

O₁ = (NOT C AND I₁) OR (C AND I₂) = CI₁+CI₂

O₂ = (C AND I₁) OR (NOT C AND I₂) = CI₁+CI₂

C_out= C_in

Completeness

One way to see that the Fredkin gate is universal is to observe that it can be used to implement AND and NOT:

I_2 = 0

, then

O_2 = C \operatorname{AND} I_1

I_1 = 0

and

I_2 = 1

, then

O_2 = \operatorname{NOT} C

Example

Here is a diagram of a three-bit adder implemented using Fredkin gates. The three inputs are A, B and C, supplemented by the constant T and F. In the diagram, the leftmost input (before the colon) swaps the two rightmost inputs if it is true.

References

↑ Brown, Julian, The Quest for the Quantum Computer, New York : Touchstone, 2000.