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[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 ...
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.
First, that NOT is a function map of the computational basis (it maps basis states to basis states), and second that when a NOT is commuted through a Z-rotation gate, that this rotation flips. Euler decomposition A Hadamard-gate can be expanded into three rotations around the Bloch sphere (corresponding to its Euler angles). Sometimes this rule ...
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.
The Hadamard test produces a random variable whose image is in {} and whose expected value is exactly | | . It is possible to modify the circuit to produce a random variable whose expected value is I m ψ | U | ψ {\displaystyle \mathrm {Im} \langle \psi |U|\psi \rangle } by applying an S † {\displaystyle S^{\dagger }} gate after the first ...
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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 .