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.[1]

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 and qubits with multiple states.[2]

Representation

A qutrit has three orthonormal basis states or vectors, often denoted ${\displaystyle |0\rangle }$, ${\displaystyle |1\rangle }$, and ${\displaystyle |2\rangle }$ 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:

${\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle +\gamma |2\rangle }$,

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

${\displaystyle |\alpha |^{2}+|\beta |^{2}+|\gamma |^{2}=1\,}$

The qubit's orthonormal basis states ${\displaystyle \{|0\rangle ,|1\rangle \}}$ span the two-dimensional complex Hilbert space ${\displaystyle H_{2}}$, 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 ${\displaystyle H_{3}}$ spanned by the qutrit's basis ${\displaystyle \{|0\rangle ,|1\rangle ,|2\rangle \}}$,[3] which can be realized by a three-level quantum system. However, not all three-level quantum systems are qutrits.[4]

An n-qutrit register can represents 3n different states simultaneously, i.e., a superposition state vector in 3n-dimensional complex Hilbert space.[5]

Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions.[6] In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit.[7]

Qutrit quantum gates

The quantum logic gates operating on single qutrits are ${\displaystyle 3\times 3}$ unitary matrices and gates that act on registers of ${\displaystyle n}$ qutrits are ${\displaystyle 3^{n}\times 3^{n}}$ unitary matrices (the elements of the unitary groups U(3) and U(3n) respectively).[8]

The rotation operator gates[a] for SU(3) are ${\displaystyle \operatorname {Rot} (\Theta )=\exp \left(-i\sum _{a=1}^{8}\Theta _{a}{\frac {\lambda _{a}}{2}}\right)}$, where ${\displaystyle \lambda _{a}}$ is the a'th Gell-Mann matrix, ${\displaystyle \Theta _{a}}$ is a real value (with period ${\displaystyle 4\pi }$), and ${\displaystyle \Theta =(\Theta _{1},\Theta _{2},\dots ,\Theta _{8})}$. 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 (${\displaystyle |0\rangle ={\text{red}},|1\rangle ={\text{green}},|2\rangle ={\text{blue}}}$) of the strong interaction.[9][b]

The global phase shift gate for the qutrit[c] is ${\displaystyle \operatorname {Ph} (\delta )={\begin{bmatrix}e^{i\delta }&0&0\\0&e^{i\delta }&0\\0&0&e^{i\delta }\end{bmatrix}}=\exp \left(i\delta I\right)=e^{i\delta }I}$ where the phase factor ${\displaystyle e^{i\delta }}$ is called the global phase.

This phase gate performs the mapping ${\displaystyle |\Psi \rangle \mapsto e^{i\delta }|\Psi \rangle }$ 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

1. ^ This can be compared with the three rotation operator gates for qubits. We get eight linearly independent rotation operators by selecting appropriate ${\displaystyle \Theta }$. For example, we get the 1st rotation operator for SU(3) by setting ${\displaystyle \Theta _{1}\neq 0}$ and all others to zero.
2. ^ Note: ${\displaystyle U(3)=U(1)\times SU(3).}$ Quarks and gluons have color charge interactions in SU(3), not U(3), meaning their color charge can not have global phase. If they could have global phase, it would mean that there would be a 9th gluon, but there is only 8.[10] Qutrits can however have global phase.
3. ^ Comparable with the global phase shift gate for qubits.

References

1. ^ Nisbet-Jones, Peter B. R.; Dilley, Jerome; Holleczek, Annemarie; Barter, Oliver; Kuhn, Axel (2013). "Photonic qubits, qutrits and ququads accurately prepared and delivered on demand". New Journal of Physics. 15 (5): 053007. arXiv:1203.5614. Bibcode:2013NJPh...15e3007N. doi:10.1088/1367-2630/15/5/053007. ISSN 1367-2630. S2CID 110606655.
2. ^ "Qudits: The Real Future of Quantum Computing?". IEEE Spectrum. 28 June 2017. Retrieved 2021-05-24.
3. ^ Byrd, Mark (1998). "Differential geometry on SU(3) with applications to three state systems". Journal of Mathematical Physics. 39 (11): 6125–6136. arXiv:math-ph/9807032. Bibcode:1998JMP....39.6125B. doi:10.1063/1.532618. ISSN 0022-2488. S2CID 17645992.
4. ^ "Quantum systems: three-level vs qutrit". Physics Stack Exchange. Retrieved 2018-07-25.
5. ^ Caves, Carlton M.; Milburn, Gerard J. (2000). "Qutrit entanglement". Optics Communications. 179 (1–6): 439–446. arXiv:quant-ph/9910001. Bibcode:2000OptCo.179..439C. doi:10.1016/s0030-4018(99)00693-8. ISSN 0030-4018. S2CID 27185877.
6. ^ Melikidze, A.; Dobrovitski, V. V.; De Raedt, H. A.; Katsnelson, M. I.; Harmon, B. N. (2004). "Parity effects in spin decoherence". Physical Review B. 70 (1): 014435. arXiv:quant-ph/0212097. Bibcode:2004PhRvB..70a4435M. doi:10.1103/PhysRevB.70.014435. S2CID 56567962.
7. ^ 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) (link)
8. ^ Colin P. Williams (2011). Explorations in Quantum Computing. Springer. pp. 22–23. ISBN 978-1-84628-887-6.
9. ^ David J. Griffiths (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. pp. 283–288, 366–369. ISBN 978-3-527-40601-2.
10. ^ Ethan Siegel (Nov 18, 2020). "Why Are There Only 8 Gluons?". Forbes.