View source for Qutrit

{{short description|Unit of quantum information}}
{{Fundamental info units}}
A '''qutrit''' (or '''quantum trit''') is a unit of [[quantum information]] that is realized by a 3-level quantum system, that may be in a [[Quantum superposition|superposition]] of three mutually orthogonal [[quantum states]].<ref>{{Cite journal|last1=Nisbet-Jones|first1=Peter B. R.|last2=Dilley|first2=Jerome|last3=Holleczek|first3=Annemarie|last4=Barter|first4=Oliver|last5=Kuhn|first5=Axel|date=2013|title=Photonic qubits, qutrits and ququads accurately prepared and delivered on demand|url=http://stacks.iop.org/1367-2630/15/i=5/a=053007|journal=New Journal of Physics|language=en|volume=15|issue=5|pages=053007|doi=10.1088/1367-2630/15/5/053007|issn=1367-2630|arxiv=1203.5614|bibcode=2013NJPh...15e3007N|s2cid=110606655 }}</ref>

The qutrit is analogous to the classical [[radix]]-3 [[trit (computing)|trit]], just as the [[qubit]], a quantum system described by a superposition of two orthogonal states, is analogous to the classical radix-2 [[bit]].

There is ongoing work to develop quantum computers using qutrits<ref>{{Cite journal |last1=Yurtalan |first1=M. A. |last2=Shi |first2=J. |last3=Kononenko |first3=M. |last4=Lupascu |first4=A. |last5=Ashhab |first5=S. |date=2020-10-27 |title=Implementation of a Walsh-Hadamard Gate in a Superconducting Qutrit |url=https://link.aps.org/doi/10.1103/PhysRevLett.125.180504 |journal=Physical Review Letters |volume=125 |issue=18 |pages=180504 |doi=10.1103/PhysRevLett.125.180504|pmid=33196217 |arxiv=2003.04879 |bibcode=2020PhRvL.125r0504Y |s2cid=128064435 }}</ref><ref>{{Cite journal |last1=Morvan |first1=A. |last2=Ramasesh |first2=V. V. |last3=Blok |first3=M. S. |last4=Kreikebaum |first4=J. M. |last5=O’Brien |first5=K. |last6=Chen |first6=L. |last7=Mitchell |first7=B. K. |last8=Naik |first8=R. K. |last9=Santiago |first9=D. I. |last10=Siddiqi |first10=I. |date=2021-05-27 |title=Qutrit Randomized Benchmarking |url=https://link.aps.org/doi/10.1103/PhysRevLett.126.210504 |journal=Physical Review Letters |volume=126 |issue=21 |pages=210504 |doi=10.1103/PhysRevLett.126.210504|pmid=34114846 |arxiv=2008.09134 |bibcode=2021PhRvL.126u0504M |hdl=1721.1/143809 |osti=1818119 |s2cid=221246177 }}</ref><ref>{{Cite journal |last1=Goss |first1=Noah |last2=Morvan |first2=Alexis |last3=Marinelli |first3=Brian |last4=Mitchell |first4=Bradley K. |last5=Nguyen |first5=Long B. |last6=Naik |first6=Ravi K. |last7=Chen |first7=Larry |last8=Jünger |first8=Christian |last9=Kreikebaum |first9=John Mark |last10=Santiago |first10=David I. |last11=Wallman |first11=Joel J. |last12=Siddiqi |first12=Irfan |date=2022-12-05 |title=High-fidelity qutrit entangling gates for superconducting circuits |journal=Nature Communications |language=en |volume=13 |issue=1 |pages=7481 |doi=10.1038/s41467-022-34851-z |issn=2041-1723 |pmc=9722686 |pmid=36470858|arxiv=2206.07216 |bibcode=2022NatCo..13.7481G }}</ref> and [[qudit]]s in general.<ref name="urlFull Page Reload">{{cite web |url=https://spectrum.ieee.org/qudits-the-real-future-of-quantum-computing |title=Qudits: The Real Future of Quantum Computing? |date=28 June 2017 |publisher=[[IEEE Spectrum]] |access-date=2021-05-24}}</ref><ref>{{Cite journal |last1=Fischer |first1=Laurin E. |last2=Chiesa |first2=Alessandro |last3=Tacchino |first3=Francesco |last4=Egger |first4=Daniel J. |last5=Carretta |first5=Stefano |last6=Tavernelli |first6=Ivano |date=2023-08-28 |title=Universal Qudit Gate Synthesis for Transmons |url=https://link.aps.org/doi/10.1103/PRXQuantum.4.030327 |journal=PRX Quantum |volume=4 |issue=3 |pages=030327 |doi=10.1103/PRXQuantum.4.030327|arxiv=2212.04496 |bibcode=2023PRXQ....4c0327F |s2cid=254408561 }}</ref><ref name="Nguyen Goss Siva Kim 2023 q896">{{cite journal | last1=Nguyen | first1=Long B. | last2=Goss | first2=Noah | last3=Siva | first3=Karthik | last4=Kim | first4=Yosep | last5=Younis | first5=Ed | last6=Qing | first6=Bingcheng | last7=Hashim | first7=Akel | last8=Santiago | first8=David I. | last9=Siddiqi | first9=Irfan | title=Empowering a qudit-based quantum processor by traversing the dual bosonic ladder | journal=Nature Communications | date=2024 | volume=15 | doi=10.1038/s41467-024-51434-2 | pmid=39160166 | arxiv=2312.17741 }}</ref>

==Representation==
A qutrit has three orthonormal [[Basis (linear algebra)|basis]] states or [[vector space|vector]]s, often denoted <math>|0\rangle</math>, <math>|1\rangle</math>, and <math>|2\rangle</math> in Dirac or [[bra–ket notation]].
These are used to describe the qutrit as a [[Superposition principle|superposition]] state vector in the form of a linear combination of the three orthonormal basis states:
:<math>|\psi\rangle = \alpha |0\rangle + \beta |1\rangle + \gamma |2\rangle</math>,
where the coefficients are complex [[probability amplitude]]s, such that the sum of their squares is unity (normalization):
: <math>| \alpha |^2 + | \beta |^2 + | \gamma |^2 = 1 \,</math>

The [[qubit]]'s orthonormal basis states <math>\{|0\rangle,|1\rangle\}</math> span the two-dimensional complex [[Hilbert space]] <math>H_2</math>, corresponding to spin-up and spin-down of a [[spin-1/2]] particle. Qutrits require a Hilbert space of higher dimension, namely the three-dimensional <math>H_3</math> spanned by the qutrit's basis <math>\{|0\rangle,|1\rangle,|2\rangle\}</math>,<ref>{{Cite journal|last=Byrd|first=Mark|date=1998|title=Differential geometry on SU(3) with applications to three state systems|journal=Journal of Mathematical Physics|language=en|volume=39|issue=11|pages=6125–6136|doi=10.1063/1.532618|issn=0022-2488|arxiv=math-ph/9807032|bibcode=1998JMP....39.6125B |s2cid=17645992 }}</ref> which can be realized by a three-level quantum system.

An ''n''-qutrit [[quantum register|register]] can represent 3<sup>''n''</sup> different states simultaneously, i.e., a superposition state vector in 3<sup>''n''</sup>-dimensional complex Hilbert space.<ref>{{Cite journal|last1=Caves|first1=Carlton M.|last2=Milburn|first2=Gerard J.|date=2000|title=Qutrit entanglement|journal=Optics Communications|volume=179|issue=1–6|pages=439–446|doi=10.1016/s0030-4018(99)00693-8|issn=0030-4018|arxiv=quant-ph/9910001|bibcode=2000OptCo.179..439C |s2cid=27185877 }}</ref>

Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to [[decoherence]] under certain environmental interactions.<ref>{{Cite journal |doi = 10.1103/PhysRevB.70.014435|title = Parity effects in spin decoherence|journal = Physical Review B|volume = 70|issue = 1|pages = 014435|year = 2004|last1 = Melikidze|first1 = A.|last2 = Dobrovitski|first2 = V. V.|last3 = De Raedt|first3 = H. A.|last4 = Katsnelson|first4 = M. I.|last5 = Harmon|first5 = B. N.|bibcode = 2004PhRvB..70a4435M|arxiv = quant-ph/0212097| s2cid=56567962 }}</ref> In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an [[quantum entanglement|entanglement]] with a [[qubit]].<ref>B. P. Lanyon,1 T. J. Weinhold, N. K. Langford, J. L. O'Brien, K. J. Resch, A. Gilchrist, and A. G. White,  ''Manipulating Biphotonic Qutrits'', Phys. Rev. Lett. '''100''', 060504 (2008) ([http://link.aps.org/abstract/PRL/v100/e060504 link])</ref>

==Qutrit quantum gates==
The [[quantum logic gates]] operating on single qutrits are <math>3 \times 3</math> [[unitary matrix|unitary matrices]] and gates that act on [[Quantum register|register]]s of <math>n</math> qutrits are <math>3^n \times 3^n</math> unitary matrices (the elements of the [[Unitary group|unitary groups]] U(3) and U(3<sup>n</sup>) respectively).<ref>{{cite book|author=Colin P. Williams |year=2011 |title=Explorations in Quantum Computing |publisher=[[Springer Science+Business Media|Springer]] |isbn=978-1-84628-887-6 |pages=22–23}}</ref>

The rotation operator gates{{efn|This can be compared with the three [[List of quantum logic gates#Rotation operator gates|rotation operator gates for qubits]]. We get eight [[Linear independence|linearly independent]] rotation operators by selecting appropriate <math>\Theta</math>. For example, we get the 1st rotation operator for SU(3) by setting <math>\Theta_1 \ne 0</math> and all others to zero.}} for [[SU(3)]] are <math>\operatorname{Rot}(\Theta_1, \Theta_2, \dots, \Theta_8)=\exp \left( -i\sum_{a=1}^8 \Theta_a \frac{\lambda_a}{2} \right)</math>, where <math>\lambda_a</math> is the ''a''{{'}}th [[Gell-Mann matrices|Gell-Mann matrix]], and <math>\Theta_a</math> is a [[Real number|real]] value. The [[Lie algebra]] of the [[matrix exponential]] is provided [[Special_unitary_group#Lie_algebra_3|here]].  The same rotation operators are used for [[gluon]] interactions, where the three basis states are the [[color charge|three colors]] {{nowrap|(<math>|0\rangle=\text{red}, |1\rangle=\text{green}, |2\rangle=\text{blue}</math>)}} of the [[strong interaction]].<ref>{{cite book|author=[[David J. Griffiths]] |title=Introduction to Elementary Particles (2nd ed.) |publisher=[[John Wiley & Sons]] |date=2008 |isbn=978-3-527-40601-2 |pages=283–288,366–369}}</ref><ref name="Scherer-Schindler">{{cite arXiv|author1=Stefan Scherer |author2=Matthias R. Schindler|title=A Chiral Perturbation Theory Primer|eprint=hep-ph/0505265|date=31 May 2005|page=1–2}}</ref>{{efn|Note: <math>U(3) = U(1) \times SU(3).</math> [[Quark]]s and gluons have [[color charge]] interactions in SU(3), not U(3), meaning there are no pure phase shift rotations allowed for gluons. If such rotations were allowed, it would mean that there would be a 9th gluon.<ref>{{cite web |url=https://www.forbes.com/sites/startswithabang/2020/11/18/why-are-there-only-8-gluons/ |title=Why Are There Only 8 Gluons?|author=[[Ethan Siegel]]|date=Nov 18, 2020|website=[[Forbes]]}}</ref>}}

The global phase shift gate for the qutrit{{efn|Comparable with the [[List of quantum logic gates#Identity gate and global phase|global phase shift gate for qubits]].}} is <math>\operatorname{Ph}(\delta) = \begin{bmatrix} e^{i\delta} & 0 & 0 \\ 0 & e^{i\delta} & 0 \\ 0 & 0 & e^{i\delta} \end{bmatrix} = \exp \left( i\delta I \right) = e^{i\delta}I</math> where the [[phase factor]] <math>e^{i\delta}</math> is called the ''global phase''.

This phase gate performs the mapping <math>|\Psi\rangle \mapsto e^{i\delta}|\Psi\rangle</math> and together with the 8 rotation operators is capable of expressing any single-qutrit gate in [[Unitary group|U(3)]], as a [[Quantum logic gate#Serially wired gates|series circuit]] of at most 9 gates.

==See also==
* [[Gell-Mann matrices]]
* [[Generalizations of Pauli matrices]]
* [[Mutually unbiased bases]]
* [[Quantum computing]]
* [[Radix economy]]
* [[Ternary computing]]

==Notes==
{{notelist}}

==References==
{{reflist}}

==External links==
{{Wiktionary}}
* {{cite web |url= http://www.physorg.com/news123244300.html |title= Physicists Demonstrate Qubit-Qutrit Entanglement |first= Lisa |last= Zyga |publisher= [[Physorg.com]] |date= Feb 26, 2008 |access-date= Mar 3, 2008 |url-status= |archive-date= Feb 29, 2008 |archive-url= https://web.archive.org/web/20080229001836/http://www.physorg.com/news123244300.html }}


{{quantum computing}}

[[Category:Units of information]]
[[Category:Quantum information science]]
[[Category:Quantum computing]]
[[Category:Ternary computers]]