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The most common quantum gates operate on spaces of one or two qubits. This means that as matrices, quantum gates can be described by 2 × 2 or 4 × 4 matrices with Orthonormal rows. Remark. "Quantum logic" can refer either to the performance of quantum logic gates or to a foundational formalism for Quantum Mechanics called Quantum Logic based on a modification of some of the rules of Propositional Logic . EXAMPLES Hadamard gate. This gate operates on a single qubit. It is represented by the Hadamard matrix: : Since the rows of the matrix are orthogonal, ''H'' is indeed a unitary matrix. Phase shifter gates. Gates in this class operate on a single qubit. They are represented by 2 × 2 matrices of the form : where θ is the ''phase shift''. Controlled gates. Suppose ''U'' is a gate that operates on single qubits with matrix representation : The ''controlled-U gate'' is a gate that operates on two qubits in such a way that the first qubit serves as a control. | ||
| :<math> 1 0 Angle \mapsto 1 Angle U 0 Angle | 1
angle \left(x_{00} 0
angle + x_{01} 1
angle
ight) </math> |
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| :<math> 1 1 Angle \mapsto 1 Angle U 1 Angle | 1
angle \left(x_{10} 0
angle + x_{11} 1
angle
ight) </math> |
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| :<math> \operatorname{D( Heta)}: i,j,k Angle Ightarrow \begin{cases} I \cos( Heta) i,j,k Angle + \sin( Heta) i,j,1-k Angle & \mbox{for }i | j=1 \ i,j,k
angle & \mbox{otherwise}\end{cases}</math> |
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