Quantum Inf Process (2013) 12:3625–3631
A SWAP gate for qudits
Juan Carlos Garcia-Escartin · Pedro
Received: 17 April 2013 / Accepted: 30 July 2013 / Published online: 14 August 2013
© Springer Science+Business Media New York 2013
Abstract We present a quantum SWAP gate valid for quantum systems of an arbitrary
dimension. The gate generalizes the CNOT implementation of the SWAP gate for
qubits and keeps its most important properties, like symmetry and simplicity. We
only use three copies of the same controlled qudit gate. This gate can be built with
two standard higher-dimensional operations, the quantum Fourier transform and the
d-dimensional version of the CZ gate.
Keywords Qudits · SWAP gate · CNOT
Quantum information protocols and quantum algorithms are usually designed for a
group of two level quantum systems (qubits). With n qubits, we can access Hilbert
spaces of dimension d = 2
. However, in some occasions it is a good idea to look into
systems with an arbitrary dimension d which is not necessarily a power of two .
Quantum information units with d possible states are called qudits.
The basic quantum operations performed by the quantum gates of the standard
toolbox are not always easy to generalize to the qudit setting. In many cases, there
is more than one possible generalization, each preserving some of the features of the
original qubit gate.
We present an implementation for a SWAP gate which interchanges two qudits. For
two d-dimensional states, |φ and |ψ, the SWAP gate will act as:
J. C. Garcia-Escartin (
) · P. Chamorro-Posada
ETSI Telecomunicación, Universidad de Valladolid, Paseo Belén n 15, 47011 Valladolid, Spain