Quantum Gate

   { Currently studying this, there may be errors. ~drummyfish }

   Quantum (logic) gate is a [1]quantum computing equivalent of a traditional
   [2]logic gate. A quantum gate takes as an input N [3]qubits and transforms
   their states to new states (this is different from classical logical gates
   that may potentially have a different number of input and output values).

   Quantum gates are represented by [4]complex [5]matrices that transform the
   qubit states (which can be seen as points in multidimensional space, see
   Bloch sphere). A gate operating on N qubits is represented by a 2^Nx2^N
   matrix. These matrices have to be unitary. Operations performed by quantum
   gates may be reversed, unlike those of classical logic gates.

   We normally represent a single qubit state with a column [6]vector |a> =
   a0 * |0> + a1 * |1> => [a0, a1] (look up bra-ket notation). Multiple qubit
   states are represented as a [7]tensor product of the individual state,
   e.g. |a,b> = [a0 * b0, a0 * b1, a1 * b0, a1 * b1]. Applying a quantum gate
   G to such a qubit vector q is performed by simple matrix multiplication: G
   * v.

Basic gates

   Here are some of the most common quantum gates.

  Identity

   Acts on 1 qubit, leaves the qubit state unchanged.

 1 0
 0 1

Pauli Gates

   Act on 1 qubit. There are three types of Pauli gates: X, Y and Z, each one
   rotates the qubit about the respective axis by [8]pi radians.

   The X gate is:

 0 1
 1 0

   The Y gate is:

 0 -i
 i  0

   The Z gate is:

 1  0
 0 -1

NOT

   The not gate is identical to the Pauli X gate. It acts on 1 qubit and
   switches the probabilities of measuring 0 vs 1.

CNOT

   Controlled NOT, acts on 2 qubits. Performs NOT on the second qubit if the
   first qubit is |1>.

 1 0 0 0
 0 1 0 0
 0 0 0 1
 0 0 1 0

   TODO

Links:
1. quantum.md
2. logic_gate.md
3. qubit.md
4. complex_number.md
5. matrix.md
6. vector.md
7. tensor_product.md
8. pi.md