Quantum Logic Gates

Quantum Gates:

In Quantum Circuits, gates are mathematically represented as transformation matrices or linear operators that must be unitary. This is a required condition because the norm of a system wave function must be equal to 1 at all times. Unitary transformations preserve the norm. Each unitary transformation U has an inverse transformation U-1, where U+ is the conjugate of U. Thus, quantum computations are reversible.

All computations are performed by applying a succession of these unitary matrices on single or multi-qubit systems. A single qubit is represented as a 2 x 1 matrix, 2 qubits as a 4 x 1 matrix, and 3 qubits as an 8 x1 matrix. If a gate acts on a single qubit then it is called a single-qubit gate. A 2-qubit and 3-qubit code can be defined similarly. Every unitary operator (U) is represented by a 2×2 matrix every unitary operator is a valid single-qubit gate.

Types of Quantum Gates:

X Gate:

It is the quantum equivalent of a classical NOT gate, that is if |k> is the input to an X gate, the output of the gate is |k’>. Since the states of a qubit |0> and |1> are represented by the column vectors.

Y Gate:

This gate is represented by the Pauli matrix σy. It maps |0> to i|1> and |1> to -i|0>.

Z Gate:

This gate maps input |k> to

(-1)k |k>

Thus for an input |0> the output of the Z gate isn’t changed, that is also |0> and for an input |1> the output is -|1>.

√Not Gate:

A square-root gate is a 1-qubit gate that is designed to implement the expression:

√Not . √Not = NOT

There is no such operation in classical logic. The √Not gate is a good example of how a gate can exist even though Boolean algebra can’t be used to describe its operation.

Hadamard Gate:

The Hadmard gate is a truly quantum gate. It is one of the most important in quantum computing. It has some similar characteristics to the √Not gate. However, the Handmard gate, unlike the √Not gate is self-inverse. It maps input |m> to

Phase Gate:

This gate turns a |0> into |0>, and a |1>. It is represented by the following matrix:

T Gate:

The matrix for the T gate also known as the π/8 gate is –

CNOT Gate:

A CNOT gate basically implements a reversible EX-OR. It can be used to generate entanglement. The CNOT gate can be graphically represented as in fig: The control and the target inputs are shown as two horizontal lines. The dependence of output y on control a is shown by a vertical line from a to one of the inputs of the EX-OR gate, the other input of the gate is driven by target input b.

Controlled-U Gate:

A CNOT gate can be extended in a way that it can work on two qubits based upon a single control qubit. Assume U is a single qubit gate that can be represented with a unitary matrix:

Reversible Logic Gate:

Landauer showed that three-input/output reversible logic gates are extremely useful for classical reversible computation. Two such gates are the Fredkin gate and Toffoli gates. These have been proven to be universal gates for irreversible computation. A universal gate allows the building of a circuit corresponding to any Boolean function. Both NAND and NOR gates have been used as universal gates in classical digital circuit design. Fredkin gates and Toffoli gates have been shown to function as NAND gates.