CNOT and Toffoli Gates in Multi-Qubit Setting

The most commonly used two-qubit gate is the controlled-not (CNOT) gate. In two-qubit space, CNOT is a matrix of the form . In general it is a matrix. The most important three-qubit gate is the universal Toffoli gate, or controlled-controlled-not (CCNOT) gate with two control bits. For the Toffoli gate, both control bits are operative; for CNOT, only the first control bit.
  • Contributed by: Rudolf Muradian
  • Based on a program by: Bruno Juliá-Díaz and Frank Tabakin


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The program is taken from the work of Bruno Juliá-Díaz and Frank Tabakin, QDENSITY: A Quantum Computer Simulation, available as the Mathematica package Qdensity.m and also available here.
The program generates the basic elements that make up a quantum computation. Especially instructive is a method for constructing operators (gates) acting within multi-qubit states.
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