# Qudit

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{{short description|Unit of information in a quantum computer}}
{{fundamental info units}}

In [quantum computing](/source/quantum_computing), a '''qudit''' (/ˈkjuː/dɪt/) or '''quantum dit''' is the generalized unit of [quantum information](/source/quantum_information) described by a superposition of states, where the number of states ''d'' is an integer equal to or greater than two. The term "dit" in qudit refers to ''d''-ary digit.<ref>{{Cite web |last=Ranade |first=Kedar S. |last2=Alber |first2=Gernot |date=2006-09-26 |title=Asymptotic correctability of Bell-diagonal qudit states and lower bounds on tolerable error probabilities in quantum cryptography |url=https://arxiv.org/abs/quant-ph/0609196v1 |access-date=2026-05-10 |website=arXiv.org |language=en}}</ref><ref>{{Cite journal |last=Johnson |first=Donald B. |date=1975-12-01 |title=Priority queues with update and finding minimum spanning trees |url=https://www.sciencedirect.com/science/article/pii/0020019075900010 |journal=Information Processing Letters |volume=4 |issue=3 |pages=53–57 |doi=10.1016/0020-0190(75)90001-0 |issn=0020-0190}}</ref><ref>{{Cite journal |last=Klein |first=Shmuel T. |last2=Serebro |first2=Tamar C. |last3=Shapira |first3=Dana |date=2022 |title=Generalization of Fibonacci Codes to the Non-Binary Case |url=https://ieeexplore.ieee.org/document/9919813/ |journal=IEEE Access |volume=10 |pages=112043–112052 |doi=10.1109/ACCESS.2022.3214820 |issn=2169-3536}}</ref><ref>{{Cite journal |last=Chi |first=Yulin |last2=Huang |first2=Jieshan |last3=Zhang |first3=Zhanchuan |last4=Mao |first4=Jun |last5=Zhou |first5=Zinan |last6=Chen |first6=Xiaojiong |last7=Zhai |first7=Chonghao |last8=Bao |first8=Jueming |last9=Dai |first9=Tianxiang |last10=Yuan |first10=Huihong |last11=Zhang |first11=Ming |last12=Dai |first12=Daoxin |last13=Tang |first13=Bo |last14=Yang |first14=Yan |last15=Li |first15=Zhihua |date=2022-03-04 |title=A programmable qudit-based quantum processor |url=https://www.nature.com/articles/s41467-022-28767-x |journal=Nature Communications |language=en |volume=13 |issue=1 |page=1166 |doi=10.1038/s41467-022-28767-x |issn=2041-1723}}</ref><ref>{{Cite journal |last=Daboul |first=Jamil |last2=Wang |first2=Xiaoguang |last3=Sanders |first3=Barry C. |date=2003-03-14 |title=Quantum gates on hybrid qudits |url=http://arxiv.org/abs/quant-ph/0211185 |journal=Journal of Physics A: Mathematical and General |volume=36 |issue=10 |pages=2525–2536 |doi=10.1088/0305-4470/36/10/312 |issn=0305-4470}}</ref><ref>{{Cite journal |last=Heinosaari |first=Teiko |last2=Hillery |first2=Mark |date=2024-01-02 |title=Can a qudit carry more information than a dit? |url=http://arxiv.org/abs/2406.16566 |journal=Contemporary Physics |volume=65 |issue=1 |pages=2–11 |doi=10.1080/00107514.2024.2390279 |issn=0010-7514}}</ref><ref>{{Cite journal |last=Mato |first=Kevin |last2=Hillmich |first2=Stefan |last3=Wille |first3=Robert |date=2024-06-23 |title=Mixed-Dimensional Qudit State Preparation Using Edge-Weighted Decision Diagrams |url=http://arxiv.org/abs/2406.03531 |journal=Proceedings of the 61st ACM/IEEE Design Automation Conference |pages=1–6 |doi=10.1145/3649329.3656260}}</ref>

== Etymology ==
Early papers in the 1990s exploring Quantum d-ary systems or non-binary quantum codes began using the term "qudit" to simplify the description of higher-dimensional [Hilbert spaces](/source/Hilbert_space). The term became standardized in the late 1990s and early 2000s with works by researchers like [Holevo](/source/Alexander_Holevo), Knill, and [Gottesman](/source/Daniel_Gottesman) who were developing the foundations for higher-dimensional systems.<ref>{{Cite journal |last=Cortese |first=John |date=2004-02-04 |title=Holevo-Schumacher-Westmoreland channel capacity for a class of qudit unital channels |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.69.022302 |journal=Physical Review A |language=en-US |volume=69 |issue=2 |article-number=022302 |doi=10.1103/PhysRevA.69.022302 |arxiv=quant-ph/0211093 |bibcode=2004PhRvA..69b2302C |issn=1050-2947 |archive-url=http://web.archive.org/web/20240419212417/https://journals.aps.org/pra/abstract/10.1103/PhysRevA.69.022302 |archive-date=2024-04-19}}</ref><ref>{{Cite journal |last=Gottesman |first=Daniel |date=1998-02-02 |title=Full length article |journal=Chaos, Solitons & Fractals |volume=10 |issue=10 |pages=1749–1758 |doi=10.1016/S0960-0779(98)00218-5 |arxiv=quant-ph/9802007 |language=en}}</ref><ref>{{Cite journal |last=Thew |first=R. T. |last2=Nemoto |first2=K. |last3=White |first3=A. G. |last4=Munro |first4=W. J. |date=2002-07-16 |title=Qudit Quantum State Tomography |url=http://arxiv.org/abs/quant-ph/0201052 |journal=Physical Review A |volume=66 |issue=1 |article-number=012303 |doi=10.1103/PhysRevA.66.012303 |issn=1050-2947}}</ref><ref>{{Cite web |last=Grassl |first=Markus |last2=Roetteler |first2=Martin |last3=Beth |first3=Thomas |date=2002-11-04 |title=Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes |url=https://arxiv.org/abs/quant-ph/0211014v1 |access-date=2026-05-08 |website=arXiv.org |language=en}}</ref><ref>{{Cite web |last=Knill |first=E. |date=2004-10-25 |title=Quantum Computing with Very Noisy Devices |url=https://arxiv.org/abs/quant-ph/0410199v2 |access-date=2026-05-08 |website=arXiv.org |language=en}}</ref>

== Qudit versus qubit ==
A qudit, characterized by {{nowrap|1=''d'' = 2}} states is a [qubit](/source/qubit).<ref>{{cite web |title=What is a Qudit? Advantages & Use Cases |url=https://www.quera.com/glossary/qudit |access-date=2025-09-21 |website=www.quera.com}}</ref><ref>{{Cite web |title=Qudits {{!}} Cirq |url=https://quantumai.google/cirq/build/qudits |access-date=2026-05-05 |website=Google Quantum AI |language=en}}</ref>

Qudits with ''d'' states greater than 2 can provide a larger [Hilbert space](/source/Hilbert_space), providing more ways to store and process quantum information.<ref>{{cite journal |last1=Meth |first1=Michael |last2=Zhang |first2=Jinglei |last3=Haase |first3=Jan F. |last4=Edmunds |first4=Claire |last5=Postler |first5=Lukas |last6=Jena |first6=Andrew J. |last7=Steiner |first7=Alex |last8=Dellantonio |first8=Luca |last9=Blatt |first9=Rainer |last10=Zoller |first10=Peter |last11=Monz |first11=Thomas |last12=Schindler |first12=Philipp |last13=Muschik |first13=Christine|author13-link=Christine Muschik |last14=Ringbauer |first14=Martin |date=2025-03-25 |title=Simulating two-dimensional lattice gauge theories on a qudit quantum computer |journal=Nature Physics |language=en |volume=21 |issue=4 |pages=570–576 |doi=10.1038/s41567-025-02797-w |issn=1745-2473 |pmc=11999872 |pmid=40248572 |arxiv=2310.12110 |bibcode=2025NatPh..21..570M }}</ref><ref>{{cite journal |last1=Meng |first1=Zhe |last2=Liu |first2=Wen-Qiang |last3=Song |first3=Bo-Wen |last4=Wang |first4=Xiao-Yun |last5=Zhang |first5=An-Ning |last6=Yin |first6=Zhang-Qi |date=2024-02-20 |title=Experimental realization of high-dimensional quantum gates with ultrahigh fidelity and efficiency |url=https://link.aps.org/doi/10.1103/PhysRevA.109.022612 |journal=Physical Review A |volume=109 |issue=2 |article-number=022612 |doi=10.1103/PhysRevA.109.022612 |arxiv=2311.18179 |bibcode=2024PhRvA.109b2612M }}</ref>

== Qudit states ==
* {{nowrap|1=''d'' = 1}} state corresponds to a one-dimensional Hilbert space and is therefore a trivial quantum system.<ref>See [two-state quantum system](/source/Two-state_quantum_system), which describes two-dimensional systems as the simplest non-trivial case.</ref>
* [Qubit](/source/Qubit) – Qudit with {{nowrap|1=''d'' = 2}} states
* [Qutrit](/source/Qutrit) – Qudit with {{nowrap|1=''d'' = 3}} states
* [Ququart](/source/Ququart) or [Ququat](/source/Ququat) – Qudit with {{nowrap|1=''d'' = 4}} states
* [Ququint](/source/Ququint)– Qudit with {{nowrap|1=''d'' = 5}} states
* [Quhexit](/source/Quhexit) – Qudit with {{nowrap|1=''d'' = 6}} states <ref>{{Citation |last=Kürkçüoglu |first=Doga Murat |title=Qudit Gate Decomposition Dependence for Lattice Gauge Theories |date=2024-10-21 |url=http://arxiv.org/abs/2410.16414 |access-date=2026-05-24 |publisher=arXiv |doi=10.48550/arXiv.2410.16414 |id=arXiv:2410.16414 |last2=Lamm |first2=Henry |last3=Maestri |first3=Andrea}}</ref>
* [Quoctit](/source/Quoctit) – Qudit with {{nowrap|1=''d'' = 8}} states
* [Qupit](/source/Qupit) – Qudit with prime number dimensions (commonly used in quantum error correction)

== Teleportation ==
Qudit teleportation is the transfer of qudit information from one particle to another at a distant location without moving the physical particle itself. Using [entanglement](/source/Quantum_entanglement) and classical communication, it allows qudit information to be transmitted, with higher dimensional qudit teleportation offering larger data capacity and better noise resilience than lower dimensional qudit teleportation.<ref>{{Cite journal |last1=Dey |first1=Indrakshi |last2=Marchetti |first2=Nicola |date=17 October 2017 |title=Quantum teleportation in higher dimension and entanglement distribution via quantum switches |url=https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/qtc2.12122 |journal=IET Quantum Communication |language=en |volume=6 |issue=1 |article-number=e12122 |doi=10.1049/qtc2.12122 |issn=2632-8925}}</ref>

In a paper published February 2026 researchers from Jiangxi University of Science and Technology and Gannan Normal University introduced a new resource efficient high dimensional protocol that dramatically reduces the resources needed to transmit information via high-dimensional quantum states. Their research demonstrates a scale of measurement complexity from ''O''(d<sup>2</sup>) to ''O'' (d) therefore reducing the communication overhead resolution and circumventing the previously assumed limits of measurement due to the quadratic growth of measurement (d<sup>2</sup> Bell states and 2 log<sub>2</sub> d of classical bits).  A quantitative robustness analysis reveals that the protocol remains highly resilient to operational errors, maintaining an efficiency above 99.6\% even under a 0.1 rad phase deviation for d=16, highlighting the expanding advantages of utilizing higher dimensional quantum systems for secure and efficient communication.<ref>{{Citation |last1=Huang |first1=Long |title=Resource-efficient teleportation of high-dimensional quantum coherence via initial phase engineering |date=2026-02-12 |url=http://arxiv.org/abs/2602.11869 |access-date=2026-03-15 |arxiv=2602.11869 |last2=Liao |first2=Cai-Hong |last3=Li |first3=Yan-Ling |last4=Xiao |first4=Xing |journal=Physical Review A |volume=113 |issue=3 |article-number=032403 |doi=10.1103/91z1-ykmj |bibcode=2026PhRvA.113c2403H }}</ref>

== Error correction ==
[Quantum decoherence](/source/Quantum_decoherence) is the natural process where quantum information is lost due to environmental interaction and [quantum error correction](/source/quantum_error_correction) is a technique that actively combats decoherence.

In a paper published by Nature in May 2025 researchers at Yale first demonstrate quantum error correction past the break-even point for higher dimensional qudit systems. The team used GKP bosonic codes to encode qudits with {{nowrap|1=''d'' = 3}} and {{nowrap|1=''d'' = 4}} in superconducting cavities and optimized the protocol using reinforcement learning.<ref>{{cite journal |title=Quantum error correction of qudits beyond break-even |journal=Nature |date=May 2025 |issn=1476-4687 |pages=612–618 |volume=641 |issue=8063 |doi=10.1038/s41586-025-08899-y |language=en |first1=Benjamin L. |last1=Brock |first2=Shraddha |last2=Singh |first3=Alec |last3=Eickbusch |first4=Volodymyr V. |last4=Sivak |first5=Andy Z. |last5=Ding |first6=Luigi |last6=Frunzio |first7=Steven M.  |last7=Girvin |first8=Michel H. |last8=Devoret |pmid=40369140 |pmc=12078179 |arxiv=2409.15065 |bibcode=2025Natur.641..612B }}</ref> These findings are regarded as a significant step in the creation of more efficient quantum computers and opens new paths for hardware-lean quantum architectures, fault tolerant computation, and compact error protected memories.<ref>{{cite web |title=Researchers Demonstrate Error-Corrected Qudits That Beat Break-Even |url=https://thequantuminsider.com/2025/05/15/google-and-yale-team-demonstrates-error-corrected-qudits-that-beat-break-even/ |website=The Quantum Insider |date=2025-05-15 |access-date=2025-11-29 |language=en-US |first=Matt |last=Swayne }}</ref><ref>{{Cite web |last=Dai |first=Lynn |title=Qudit Connection: Bringing Quantum Computers Beyond Binary – Yale Scientific Magazine |url=https://www.yalescientific.org/2025/10/qudit-connection/ |access-date=2026-05-08 |website=www.yalescientific.org |language=en-US}}</ref>

In a paper published September 2025, researchers demonstrate a new hybrid method that encodes information in both light and matter using a [cat state](/source/cat_state) qudit with {{nowrap|''d'' > 2}}, which allows for the detection of photon loss through the parity syndrome by entangling a light pulse with ancillary qubits. This method achieves parallel Bell-pair generation by leveraging the multi-level nature of the qudit.<ref>{{cite journal |last1=McIntyre |first1=Z. M. |title=Loss-tolerant parallelized Bell-state generation with a hybrid cat qudit |date=2025-09-10 |arxiv=2509.08577 |last2=Coish |first2=W. A. |journal=Physical Review A |volume=112 |issue=6 |article-number=062609 |doi=10.1103/x56x-vld7 |bibcode=2025PhRvA.112f2609M }}</ref>

The first open source qudit stabilizer simulator named "Sdim" was announced November 2025 in a pre-print paper on [arXiv](/source/arXiv).<ref>{{cite arXiv |last1=Kabir |first1=Adeeb |title=Sdim: A Qudit Stabilizer Simulator |date=2025-11-16 |eprint=2511.12777 |last2=Nguyen |first2=Steven |last3=Ghosh |first3=Sohan |last4=Kiran |first4=Tijil |last5=Kim |first5=Isaac H. |last6=Huang |first6=Yipeng |class=quant-ph }}</ref>

== Qudit logic gates ==
A '''qudit logic gate''' (or simply '''qudit gate''') is a basic [quantum circuit](/source/quantum_circuit) that acts on a qudit.

To achieve a universal qudit gate, (a gate that can be used to approximate any unitary transformation on a quantum computer to an arbitrary degree of accuracy) a set of gates must include a finite set of single qudit gates and at least one two qudit entangling gate that can create entanglement between qudits.

== Qudit control ==
Qudit control is the precise navigation of a qudit's quantum state through engineered signals to perform quantum computations.

In a paper published December 2025 a team of researchers achieved a breakthrough in qudit control by engineering five level qudits through individually addressable transitions between Zeeman sublevels (''see also'' [Zeeman Effect](/source/Zeeman_effect)), achieved by combining a large linear Zeeman shift with a state-dependent light shift. Simulations predict state-preparation fidelities of {{nowrap|''F'' ≃ 0.99}} within ∽1&nbsp;μs, single-qudit gate fidelities of {{nowrap|1=''F'' ≃ 0.99}} with ''π'' pulse durations of ∽2.5&nbsp;μs, and fast destructive imaging with durations below 10&nbsp;μs. These results establish a broadly applicable framework for high-fidelity control of Zeeman sublevel-encoded qudits and a promising platform for scalable qudit-based quantum technologies.<ref>{{cite arXiv |last1=Heizenreder |first1=Benedikt |title=Engineering Zeeman-manifold quintets using state-dependent light shifts in neutral atoms |date=2025-12-16 |eprint=2512.14611 |last2=Gerritsen |first2=Bas |last3=Fouka |first3=Katya |last4=Spreeuw |first4=Robert J. C. |last5=Schreck |first5=Florian |last6=Naini |first6=Arghavan Safavi |last7=Urech |first7=Alexander |class=physics.atom-ph }}</ref>

== Use in measurement ==
Quantum information is traditionally used in [Ramsey interferometry](/source/Ramsey_interferometry), a technique used for precise measurement across various areas of science and technology.

Qudits with {{nowrap|''d'' > 2}} have shown to increase precision and resolution of quantum measurements. Qutrits, for example, have shown to achieve a twofold increase in resolution compared to qubits without any reduction in measurement contrast.<ref>{{cite arXiv |last1=Ilikj |first1=Branislav |title=Ramsey Interferometry with Qudits |date=2025-09-08 |eprint=2509.06290 |last2=Vitanov |first2=Nikolay V. |class=quant-ph }}</ref>

== References ==
{{reflist}}

Category:Quantum computing
Category:Units of information
Category:Quantum states

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Adapted from the Wikipedia article [Qudit](https://en.wikipedia.org/wiki/Qudit) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Qudit?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.