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Note on Exchange Gate on the Qudit Space

Kazuyuki Fujii

Published Jun 29, 2002• 1 citation
OpenAlex

Abstract

We construct the exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three "reverse" gates. This is a natural extension of the qubit case. We also consider a similar subject on the Fock space, but in this case it is not clear what elementary gates are. We present some related problems.

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We construct an exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three 'reverse' gates. This is a natural extension of the qubit case. We also consider a similar situation in Fock space, but in this case we find some differences. However, we can construct the exchange gate by making use of a generalized coherent operator based on the Lie algebra su (2), which is a well-known method in quantum optics. We also make a brief comment on 'imperfect clones'.

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Guang Song, Andreas Klappenecker

The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases. realizations of controlled unitary gates (pp139-155) G. Song and A. Klappenecker The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases.

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Colloquium : Qudits for decomposing multiqubit gates and realizing quantum algorithms

Evgeniy O. Kiktenko, Anastasiia S. Nikolaeva, Aleksey K. Fedorov

Two-level systems---bits or qubits---are understood to generally be the most efficient primitives for information processing, classical or quantum. But this is not to say that there are no roles to be played by multilevel systems. This Colloquium surveys these possible roles for the quantum case. Here we speak of qudits: $d$-level quantum systems. In one interesting example, the use of just one three-level system permits a drastic simplification of the ``Toffoli gate,'' the basic three-qubit primitive of reversible logic. A survey is given of various qudit-qubit embeddings, and the current state of quantum computing experiments using qudits is reviewed.

Reviews of Modern Physics•Jun 3, 2025•27 citations
We construct the exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three "reverse" gates. This is a natural extension of the qubit case. We also consider a similar subject on the Fock space, but in this case it is not clear what elementary gates are. We present some related problems.
We construct an exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three 'reverse' gates. This is a natural extension of the qubit case. We also consider a similar situation in Fock space, but in this case we find some differences. However, we can construct the exchange gate by making use of a generalized coherent operator based on the Lie algebra su (2), which is a well-known method in quantum optics. We also make a brief comment on 'imperfect clones'.
The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases. realizations of controlled unitary gates (pp139-155) G. Song and A. Klappenecker The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases.
The SWAP gate has become an integral feature of quantum circuit architectures and is designed to permute the states of two qubits through the use of the well-known controlled-NOT gate. We consider the question of whether a two-qudit quantum circuit composed entirely from instances of the generalised controlled-NOT gate can be constructed to permute the states of two qudits. Arguing via the signature of a permutation, we demonstrate the impossibility of such circuits for dimensions d≡3d \equiv 3d≡3 (mod 4).
Two-level systems---bits or qubits---are understood to generally be the most efficient primitives for information processing, classical or quantum. But this is not to say that there are no roles to be played by multilevel systems. This Colloquium surveys these possible roles for the quantum case. Here we speak of qudits: ddd-level quantum systems. In one interesting example, the use of just one three-level system permits a drastic simplification of the ``Toffoli gate,'' the basic three-qubit primitive of reversible logic. A survey is given of various qudit-qubit embeddings, and the current state of quantum computing experiments using qudits is reviewed.