 Quantum Gates, Logic & Matrix 
Toffoli Gate (CCNOT)
In computer science, the Toffoli gate (also CCNOT gate),
invented by Tommaso Toffoli, is a universal reversible logic gate, which
means that any reversible circuit can be constructed from Toffoli
gates. It is also known as the "controlledcontrollednot" gate, which
describes its action.
A logic gate L is reversible if, for any output y, there is a unique input x such that applying L(x) = y. If a gate L is reversible, there is an inverse gate L′ which maps y to x for which L′(y) = x. From common logic gates, NOT is reversible, as can be seen from its truthtable below.
INPUT  OUTPUT 

0  1 
1  0 
The common AND gate is not reversible however. The inputs 00, 01 and 10 are all mapped to the output 0.
Reversible gates have been studied since the 1960s. The original
motivation was that reversible gates dissipate less heat (or, in
principle, no heat). In a normal gate, input states are lost, since less
information is present in the output than was present at the input.
This loss of information loses energy to the surrounding area as heat,
because ofthermodynamic entropy. Another way to understand this is that
charges on a circuit are grounded and thus flow away, taking a small
quantity of energy with them when they change state. A reversible gate
only moves the states around, and since no information is lost, energy
is conserved.
More recent motivation comes from quantum computing. Quantum
mechanics requires the transformations to be reversible but allows more
general states of the computation (superpositions). Thus, the reversible
gates form a subset of gates allowed by quantum mechanics and, if we
can compute something reversibly, we can also compute it on a quantum
computer.
Universal Gate
Any reversible gate must have the same
number of input and output bits, by the pigeonhole principle. For one
input bit, there are two possible reversible gates. One of them is NOT.
The other is the identity gate which maps its input to the output
unchanged. For two input bits, the only nontrivial gate is
the controlled NOT gate which XORs the first bit to the second bit and
leaves the first bit unchanged.
Universal Gate
Truth table  Matrix form  


Unfortunately, there are reversible functions that cannot be computed
using just those gates. In other words, the set consisting of NOT and
XOR gates is not universal (see Functional completeness). If we want to
compute an arbitrary function using reversible gates, we need another
gate. One possibility is the Toffoli gate, proposed in 1980 by Toffoli.
This gate has 3bit inputs and outputs. If the first two bits are set,
it flips the third bit. The following is a table of the input and output
bits:
Truth table  Matrix form  


It can be also described as mapping bits a, b and c to a, b and c XOR (a AND b).
The Toffoli gate is universal; this means that for any Boolean function f(x_{1}, x_{2}, ..., x_{m}), there is a circuit consisting of Toffoli gates which takes x_{1}, x_{2}, ..., x_{m} and some extra bits set to 0 or 1 and outputs x_{1}, x_{2}, ..., x_{m}, f(x_{1}, x_{2}, ..., x_{m}),
and some extra bits (called garbage). Essentially, this means that one
can use Toffoli gates to build systems that will perform any desired
Boolean function computation in a reversible manner.
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 threebit gate that
swaps the last two bits if the first bit is 1.
The basic Fredkin gate is a controlled swap gate that maps three inputs (C, I_{1}, I_{2}) onto three outputs (C, O_{1}, O_{2}). The C input is mapped directly to the C output. If C = 0, no swap is performed; I_{1} maps to O_{1}, and I_{2} maps to O_{2}. Otherwise, the two outputs are swapped so that I_{1} maps to O_{2}, and I_{2} maps to O_{1}. 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 n2 inputs unchanged to the corresponding outputs, and swaps its last two outputs if and only if the first n2 inputs are all 1.
The Fredkin gate is the reversible threebit gate that swaps the last two bits if the first bit is 1.
Truth table  Matrix form  


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.
CNOT Gate
In computing science, the controlled NOT gate (also CNOT or CNOT) is a quantum gate that is an essential component in the construction of a quantum computer. It can be used to disentangle EPR states.
Specifically, any quantum circuit can be simulated to an arbitrary
degree of accuracy using a combination of CNOT gates and single qubit rotations.
The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is 1.
Before  After  

Control  Target  Control  Target 
0  0  0  0 
0  1  0  1 
1  0  1  1 
1  1  1  0 
The resulting value of the second qubit corresponds to the result of a classical XOR gate.
The CNOT gate can be represented by the matrix:
The first experimental realization of a CNOT gate was accomplished in
1995. Here, a single Beryllium ion in a trap was used. The two qubits
were encoded into an optical state and into the vibrational state of the
ion within the trap. At the time of the experiment, the reliability of
the CNOToperation was measured to be on the order of 90%.
Transpose Matrix
Unitary Identity Matrix
In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr or At) created by any one of the following equivalent actions:
 reflect A over its main diagonal (which runs topleft to bottomright) to obtain AT
 write the rows of A as the columns of AT
 write the columns of A as the rows of AT
Unitary Identity Matrix
In linear algebra, the identity matrix or unit matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by I_{n}, or simply by I if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I.)
Unitary Matrix
In mathematics, a complex square matrix U is unitary if,
Where I is the identity matrix and U* is the conjugate transpose of U.