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The Hadamard transform H m is a 2 m × 2 m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2 m real numbers x n into 2 m real numbers X k. The Hadamard transform can be defined in two ways: recursively, or by using the binary (base-2) representation of the indices n and k.
Common quantum logic gates by name (including abbreviation), circuit form(s) and the corresponding unitary matrices. In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits.
Under this definition, is infinite, since it contains all unitaries of the form for a real number and the identity matrix . [2] Any unitary in C n {\displaystyle \mathbf {C} _{n}} is equivalent (up to a global phase factor) to a circuit generated using Hadamard , Phase , and CNOT gates, [ 3 ] so the Clifford group is sometimes defined as the ...
The figures below are examples of making an equivalent Hadamard-gate and CNOT-gate using beam splitters (illustrated as rectangles connecting two sets of crossing lines with parameters and ) and phase shifters (illustrated as rectangles on a line with parameter ).
The Fredkin gate (also CSWAP or CS gate), named after Edward Fredkin, 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.
[2] [3] [4] This set of gates is minimal in the sense that discarding any one gate results in the inability to implement some Clifford operations; removing the Hadamard gate disallows powers of / in the unitary matrix representation, removing the phase gate S disallows in the unitary matrix, and removing the CNOT gate reduces the set of ...
Let H be a Hadamard matrix of order n.The transpose of H is closely related to its inverse.In fact: = where I n is the n × n identity matrix and H T is the transpose of H.To see that this is true, notice that the rows of H are all orthogonal vectors over the field of real numbers and each have length .
A Hadamard-gate can be expanded into three rotations around the Bloch sphere (corresponding to its Euler angles). Sometimes this rule is taken as the definition of the Hadamard generator, in which case the only generators of ZX-diagrams are the Z- and X-spider.