QUANTUM LOGIC GATE

Quantum Logic Gates :

Traditional computers are like microscopic cities. The roads of these cities are wires with electricity coursing through them. These roads have lots of gates, known as logic gates, which enable computers to do their job. Like physical gates that allow or block cars, logic gates allow or block electricity. Electricity that goes through the gates represents a “1” of digital data, and blocked electricity is a “0.”

Logic gates are building blocks for processing information. One kind of logic gate, known as the AND gate, could, for example, quickly determine whether two people agree to a business deal. It takes in two bits of information, and generates a 1 if both incoming bits are 1 s. So, if both business people say “yes” (1) to the deal, the AND gate will output 1. If one or both say “no” (0), the AND gate generates a 0 or a no.

By arranging gates in a circuit, engineers can create something akin to a flowchart that enables computers to carry out many kinds of logical operations, such as mathematical calculations and perform the kinds of tasks that computers can do.

In their quantum logic gate, BY controlle the energy levels in an individual ion so that a lower-energy state represented a 0 and a higher-energy state represented a 1. The ion’s internal energy was the first q bit. They created a second quantum bit with the atom’s external motion: 0 represented less motion and 1 represented a greater amount of motion.

The group entangled the ion’s internal energy state with its overall motion. In the process, they made a quantum version of a CONTROLLED NOT gate. In their gate, the ion’s energy of motion serves the “control” bit. If it is a 1, then it causes the ion’s internal energy state to flip.

Quantum gates are usually represented as matrices. A gate which acts on k qubits is represented by a 2^k x 2^k unitary matrix. The number of qubits in the input and output of the gate have to be equal. The action of the gate on a specific quantum state is found by multiplying the vector which represents the state by the matrix representing the gate. In the following, the vector representation of a single qubit is:

Hadamard (H) gate

The Hadamard gate acts on a single qubit. It maps the basis state  to  and  to , which means that a measurement will have equal probabilities to become 1 or 0 . It represents a rotation of  about the axis .s. It is represented by the Hadamard matrix.

${\displaystyle H={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}}$The hadamard gate is the one-qubit version of the quantum furier trasform..

Since ${\displaystyle HH^{*}=I}$ where I is the identity matrix, H is indeed a unity matrix.

Pauli-X gate

The Pauli-X gate acts on a single qubit. It is the quantum equivalent of the NOT gate for classical computers (with respect to the standard basis , 

 which privileges the Z-direction) . It equates to a rotation of the bloch spare around the X-axis by pai radians. I Due to this nature, it is sometimes called bit flip.It is represented by the pauli metrix

Pauli-Y gate

The Pauli-Y gate acts on a single qubit. It equates to a rotation around the Y-axis of the Bloch sphere by pai radians. 

${\displaystyle Y={\begin{bmatrix}0&-i\\i&0\end{bmatrix}}}$.

Pauli-Z gate

The Pauli-Z gate acts on a single qubit. It equates to a rotation around the Z-axis of the Bloch sphere by ${\displaystyle \pi }$ radians. Thus, it is a special case of a phase shift gate (which are described in a next subsection) with ${\displaystyle \phi =\pi }$. It leaves the basis state ${\displaystyle |0\rangle }$ unchanged and maps ${\displaystyle |1\rangle }$ to ${\displaystyle -|1\rangle }$. Due to this nature, it is sometimes called phase-flip. It is represented by the pauli Z matrix:

${\displaystyle Z={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$.

Square root of NOT gate (√NOT)

The NOT gate acts on a single qubit. so this gate is a square root of the NOT gate.

Similar squared root-gates can be constructed for all other gates by finding the unitary matrix that, multiplied by itself, yields the gate one wishes to construct the squared root gate of. All rational exponents of all gates can be created in this way. (Only approximations of irrational exponents are possible to synthesize from composite gates whose elements are not themselves irrational, since exact synthesis would result in infinite gate depth.)

Phase shift (${\displaystyle R_{\phi }}$) gates

This is a family of single-qubit gates that leave the basis state  unchanged and map  to  The probability of measuring a  or  is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle (a line of latitude) on the Bloch sphere by ${\displaystyle \phi }$ radians.

{\displaystyle R_{\phi }={\begin{bmatrix}1&0\\0&e^{i\phi }\end{bmatrix}}}

where the phase shift. Some common examples are the  gate (commonly written as T) where , the phase gate (written S, though S is sometimes used for SWAP gates) where and the Pauli-Z gate where.

Swap (SWAP) gate

.

The swap gate swaps two qubits.

 With respect to the basis, it is represented by the matrix:

Square root of Swap gate (√SWAP)

The sqrt(swap) gate performs half-way of a two-qubit swap. It is universal such that any quantum many qubit gate can be constructed from only sqrt(swap) and single qubit gates. 

The sqrt(swap) gate is not, however maximally entangling, more than one application of it is required to produce a Bell state from product states.

Controlled (cX cY cZ) gates

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation. For example, the controlled NOT gate (or CNOT or cX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is , and otherwise leaves it unchanged. With respect to the basis, it is represented by the matrix:

More generally if U is a gate that operates on single qubits with matrix representation

then the controlled-U gate is a gate that operates on two qubits in such a way that the first qubit serves as a control. It maps the basis states as follows.

${\displaystyle |00\rangle \mapsto |00\rangle }$The matrix representing the controlled U is

The CNOT gate is generally used in quantum computing to generate entangled states.When U is one of the pauli matrices, σ_x, σ_y, or σ_z, the respective terms "controlled-X", "controlled-Y", or "controlled-Z" are sometimes used.

Toffoli (CCNOT) gate

The Toffoli gate, also CCNOT gate or Deutsch ${\displaystyle D(\pi /2)}$ gate, is a 3-bit gate, which is universal for classical computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If the first two bits are in the state, it applies a Pauli-X (or NOT) on the third bit, else it does nothing. It is an example of a controlled gate. Since it is the quantum analog of a classical gate, it is completely specified by its truth table. The Toffoli gate is universal when combined with the single qubit Hadamard gate.

Fredkin (CSWAP) gate

The Fredkin gate (also CSWAP or cS gate) is a 3-bit gate that performs a controlled swap. It is universal for classical computation. 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.

Ising (XX) gate

The Ising gate (or XX gate) is a 2-qubit gate that is implemented natively in some trapped-ion quantum computers. It is defined as

{\displaystyle XX_{\phi }={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&0&-ie^{i\phi }\\0&1&-i&0\\0&-i&1&0\\-ie^{-i\phi }&0&0&1\\\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&0&e^{i(\phi -\pi /2)}\\0&1&-i&0\\0&-i&1&0\\e^{i(-\phi -\pi /2)}&0&0&1\\\end{bmatrix}}}

Deutsch gate

Deutsch  gate is a three-qubit gate. It is defined as

{\displaystyle |a,b,c\rangle \mapsto {\begin{cases}i\cos(\theta )|a,b,c\rangle +\sin(\theta )|a,b,1-c\rangle &{\mbox{for }}a=b=1\\|a,b,c\rangle &{\mbox{otherwise.}}\end{cases}}}

Unfortunately, a working Deutsch gate has remained out of reach, due to lack of a protocol. However, a method was proposed to realize such  with dipole-dipole interaction in neutral atoms.