Qutrit

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).

The rotation operator gates for SU(3) are ${\displaystyle \mathrm{Rot}({\mathrm{\Theta }}_1,{\mathrm{\Theta }}_2,\dots ,{\mathrm{\Theta }}_8)=\exp \left(-i\sum _{a=1}^8{\mathrm{\Theta }}_a\frac{{\lambda }_a}{2}\right)}$, where ${\displaystyle {\lambda }_a}$ is the a'th Gell-Mann matrix, and ${\displaystyle {\mathrm{\Theta }}_a}$ 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 (${\displaystyle |0⟩=\text{red},|1⟩=\text{green},|2⟩=\text{blue}}$) of the strong interaction.

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

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

Error corrected ternary quantum computing

Fault-tolerant implementation of a specific qutrit gate can be more or less challenging, depending on its place in the so-called Clifford hierarchy.

Let ${\displaystyle \omega =e^{2\pi i/3}}$ be the third root of unity. Consider the subgroup ${\displaystyle P_3}$ of ternary Pauli gates with two generators ${\displaystyle X=|1⟩⟨0|+|2⟩⟨1|+|0⟩⟨2|}$ and ${\displaystyle Z=|0⟩⟨0|+\omega |1⟩⟨1|+{\omega }^2|2⟩⟨2|}$. Operators in the Pauli subgroup are in general the least expensive to implement. The multi-qutrit Pauli subgroup consists of arbitrary tensor products of single-qutrit Pauli operators.

A significantly larger group of Clifford gates consists of all the gates that stabilize the Pauli subgroup. As shown in , the following gates can be chosen as generators of the Clifford group: ${\displaystyle \mathrm{H}=\frac{1}{{\omega }^2-\omega }\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right]}$

${\displaystyle S=\omega |0⟩⟨0|+{\omega }^2|1⟩⟨1|+|2⟩⟨2|}$ and ${\displaystyle CZ=\text{diag}(1,1,1,1,\omega ,{\omega }^2,1,{\omega }^2,\omega )}$

For any fixed set of qutrits the Clifford group is finite and therefore is not sufficient for universal quantum computation. Adding one single non-Clifford gate to a set of its generators, however, ensures quantum universality, but requires resource-consuming techniques such as magic state distillation.

Arguably, the simplest non-Clifford gate to consider is ${\displaystyle T=|0⟩⟨0|+|1⟩⟨1|+\sqrt{\omega }|2⟩⟨2|}$. Injecting ${\displaystyle T}$ as a new generator is equivalent to being able to prepare the quantum ("magic") state ${\displaystyle \hat{T}=|0⟩+|1⟩+\sqrt{\omega }|2⟩}$ from a standard qutrit basis state. This, however, cannot be done perfectly using qutrit Clifford group alone (because e.g.${\displaystyle \sqrt{\omega }}$ does not belong to the ${\displaystyle \mathbb{Q}(\omega )}$ numeric field).