The Group Zoo of Classical Reversible Computing and Quantum Computing

  • Alexis De Vos
  • Stijn De Baerdemacker
Part of the Emergence, Complexity and Computation book series (ECC, volume 22)


By systematically inflating the group of \(n \times n\) permutation matrices to the group of \(n \times n\) unitary matrices, we can see how classical computing is embedded in quantum computing. In this process, an important role is played by two subgroups of the unitary group U(n), i.e. XU(n) and ZU(n). Here, XU(n) consists of all \(n \times n\) unitary matrices with all line sums (i.e. the n row sums and the n column sums) equal to 1, whereas ZU(n) consists of all \(n \times n\) diagonal unitary matrices with upper-left entry equal to 1. As a consequence, quantum computers can be built from NEGATOR gates and PHASOR gates. The NEGATOR is a 1-qubit circuit that is a natural generalization of the 1-bit NOT gate of classical computing. In contrast, the PHASOR is a 1-qubit circuit not related to classical computing.


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© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Vakgroep elektronika en informatiesystemenUniversiteit GentGentBelgium
  2. 2.Vakgroep anorganische en fysische chemieUniversiteit GentGentBelgium

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