A Matrix Representation of Quantum Circuits over Non-Adjacent Qudits


Within the general context of the architecture in quantum computer design, this paper aims is to provide a general strategy to obtain a block-matrix representation of quantum gates applied to qubits placed in arbitrary positions over an arbitrary dimensional input state. The model is also extended to the framework of quantum computation with qudits. An application in the context of the quantum computational logic is provided.

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Fig. 1


  1. 1.

    As an example, an introductory result was developed by Wilmott [32] that showed how to represent an arbitrary SWAP gate between two qudits by involving CNot gates only and by following combinatorial considerations. In this paper we follow a different approach, providing a block-matrix representation of arbitrary unitary operators without involving the composition of other control gates.

  2. 2.

    From now on, let us assume that any qubit is written in the canonical basis \(\mathcal B=\{\vert 0\rangle =\left (\begin {array}{cc} 1 \\ 0 \end {array}\right ), \vert 1\rangle =\left (\begin {array}{cc} 0 \\ 1 \end {array}\right )\}\).

  3. 3.

    Let us give a slight abuse of the terms target and control according with the convention that the control position is related to the qubit that is not affected by the gate, otherwise we speak about target position.


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This work has been partially supported by Regione Autonoma della Sardegna in the framework of the project “Time-logical evolution of correlated microscopic systems” (CRP 55, L.R. 7/2007, 2015) and by Fondazione Banco di Sardegna in the framework of the project “Strategies and Technologies for Scientific Education and Dissemination”, cup: F71I17000330002.

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Correspondence to Giuseppe Sergioli.

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Sergioli, G. A Matrix Representation of Quantum Circuits over Non-Adjacent Qudits. Int J Theor Phys (2019). https://doi.org/10.1007/s10773-019-04051-5

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  • Quantum circuit
  • Block-matrix representation
  • Quantum computational logic