qutrit

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 superposition of three mutually orthogonal quantum states.{{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 }}

The qutrit is analogous to the classical radix-3 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{{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 }}{{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 }}{{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 }} and qudits in general.{{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}}{{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 }}{{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 | issue=1 | page=7117 | doi=10.1038/s41467-024-51434-2 | pmid=39160166 | pmc=11333499 | arxiv=2312.17741 }}

Representation

A qutrit has three orthonormal basis states or vectors, often denoted $|0\rangle$ , $|1\rangle$ , and $|2\rangle$ in Dirac or bra–ket notation.

These are used to describe the qutrit as a superposition state vector in the form of a linear combination of the three orthonormal basis states:

: $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle + \gamma |2\rangle$ ,

where the coefficients are complex probability amplitudes, such that the sum of their squares is unity (normalization):

: $| \alpha |^2 + | \beta |^2 + | \gamma |^2 = 1 \,$

The qubit's orthonormal basis states $\{|0\rangle,|1\rangle\}$ span the two-dimensional complex Hilbert space $H_2$ , 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 $H_3$ spanned by the qutrit's basis $\{|0\rangle,|1\rangle,|2\rangle\}$ ,{{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 }} which can be realized by a three-level quantum system.

An n-qutrit register can represent 3ⁿ different states simultaneously, i.e., a superposition state vector in 3ⁿ-dimensional complex Hilbert space.{{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 }}

Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions.{{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 }} In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit.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])

Qutrit quantum gates

The quantum logic gates operating on single qutrits are $3 \times 3$ unitary matrices and gates that act on registers of $n$ qutrits are $3^n \times 3^n$ unitary matrices (the elements of the unitary groups U(3) and U(3ⁿ) respectively).{{cite book|author=Colin P. Williams |year=2011 |title=Explorations in Quantum Computing |publisher=Springer |isbn=978-1-84628-887-6 |pages=22–23}}

The rotation operator gates{{efn|This can be compared with the three rotation operator gates for qubits. We get eight linearly independent rotation operators by selecting appropriate $\Theta$ . For example, we get the 1st rotation operator for SU(3) by setting $\Theta_1 \ne 0$ and all others to zero.}} for SU(3) are $\operatorname{Rot}(\Theta_1, \Theta_2, \dots, \Theta_8)=\exp \left( -i\sum_{a=1}^8 \Theta_a \frac{\lambda_a}{2} \right)$ , where $\lambda_a$ is the a{{'}}th Gell-Mann matrix, and $\Theta_a$ is a real value. The Lie algebra of the matrix exponential is provided here. The same rotation operators are used for gluon interactions, where the three basis states are the three colors {{nowrap|( $|0\rangle=\text{red}, |1\rangle=\text{green}, |2\rangle=\text{blue}$ )}} of the strong interaction.{{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}}{{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}}{{efn|Note: $U(3) = U(1) \times SU(3).$ Quarks 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.{{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}}}}

The global phase shift gate for the qutrit{{efn|Comparable with the global phase shift gate for qubits.}} is $\operatorname{Ph}(\delta) = \begin{bmatrix} e^{i\delta} & 0 & 0 \\ 0 & e^{i\delta} & 0 \\ 0 & 0 & e^{i\delta} \end{bmatrix} = \exp \left( i\delta I \right) = e^{i\delta}I$ where the phase factor $e^{i\delta}$ is called the global phase.

This phase gate performs the mapping $|\Psi\rangle \mapsto e^{i\delta}|\Psi\rangle$ and together with the 8 rotation operators is capable of expressing any single-qutrit gate in U(3), as a series circuit of at most 9 gates.

Notes

References

External links

{{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 }}

Category:Units of information

Category:Quantum information science

Category:Quantum computing

Category:Ternary computers