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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.
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.
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.
[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 ...
Arbitrary Clifford group element can be generated as a circuit with no more than (/ ()) gates. [6] [7] Here, reference [6] reports an 11-stage decomposition -H-C-P-C-P-C-H-P-C-P-C-, where H, C, and P stand for computational stages using Hadamard, CNOT, and Phase gates, respectively, and reference [7] shows that the CNOT stage can be implemented using (/ ()) gates (stages -H- and -P ...
A CNOT gate may appear to only act the control on the target, but surrounding it with Hadamard gates reveals it also acts the target on the control. A CNOT appears asymmetric, but can be transformed into a symmetric operation by Hadamard gates. Symmetric operations don't distinguish target and control, resulting in effects like phase kickback.
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