nLab qudit

Idea

In quantum information theory and quantum computing, by a qdit (or qudit) one means a quantum state in a $d$-dimensional Hilbert space, for any natural number $d$.

Hence for fixed $d\in \mathbb{N}$, the quantum data type of qdits is the $d$-dimensional complex vector space equipped with a quantum measurement-basis

$$\mathbb{C}^d \;\simeq\; \underset{n \in \{1, \cdots, d\}}{\bigoplus} \mathbb{C} \cdot \vert n \rangle \,.$$

• For $d = 2$ one speaks of qbits, this is the original terminology;

• For $d = 3$ one also speaks of qtrits.

Examples

Anyon states

A key example of quantum gates which naturally act on qdits for $d \gt 2$ are anyon braid-gates in topological quantum computation:

For the potentially realistic case of Chern-Simons theory/WZW-model-controled anyons (such as su(2)-anyons), the elementary quantum logic gates act by the monodromy of the Knizhnik-Zamolodchikov connection on Hilbert spaces of conformal blocks, whose dimension $d$ is given by a Verlinde formula.

References which make this point explicit include Kolganov, Mironov & Morozov (2023).

Related concepts

• data type

• qbit

• quantum gate, quantum circuit

References

General:

• IEEE Spectrum Qudits: The Real Future of Quantum Computing? (June 2017)

• Yuchen Wang, Zixuan Hu, Barry C. Sanders, Sabre Kais, section 3.2.1 of: Qudits and high-dimensional quantum computing, Front. Phys. 8 479 (2020) [arXiv:2008.00959, doi:10.3389/fphy.2020.589504]

See also:

• Wikipedia, Qutrit

On the quantum Fourier transform with qdits:

• Ashok Muthukrishnan, C. R. Stroud Jr: Quantum fast Fourier transform using multilevel atoms, Journal of Modern Optics 49 13 (2002) [arXiv:quant-ph/0112017, doi:10.1080/09500340210123947]

• Zeljko Zilic, Katarzyna Radecka: Scaling and better approximating quantum Fourier transform by higher radices, IEEE Transactions on Computers 56 2 (2007) 202-207 [arXiv:quant-ph/0702195, doi:10.1109/TC.2007.35]

• Cao Ye et al.: Quantum Fourier Transform and Phase Estimation in Qudit System, Commun. Theor. Phys. 55 790 (2011) [doi:10.1088/0253-6102/55/5/11]

and its applications to arithmetic quantum algorithms:

• Archimedes Pavlidis, Emmanuel Floratos: Quantum-Fourier-transform-based quantum arithmetic with qudits, Phys. Rev. A 103 (2021) 032417 [doi:10.1103/PhysRevA.103.032417]

Explicit mentioning of the qdit-nature of the elementary gates in topological quantum computation:

• Nikita Kolganov, Sergey Mironov, Andrey Morozov, Large $k$ topological quantum computer, Nuclear Physics B

  987 (2023) 116072 [arXiv:2105.03980, doi:10.1016/j.nuclphysb.2023.116072]