Dark path holonomic qudit computation

Nonadiabatic holonomic quantum computation is a method used to implement high-speed quantum gates with non-Abelian geometric phases associated with paths in state space. Due to their noise tolerance, these phases can be used to construct error resilient quantum gates. We extend the holonomic dark path qubit scheme in [M.-Z. Ai et al., Fundam. Res. 2, 661 (2022)] to qudits. Specifically, we demonstrate one- and two-qudit universality by using the dark path technique. Explicit qutrit ($d=3$) gates are demonstrated and the scaling of the number of loops with the dimension $d$ is addressed. This scaling is linear and we show how any diagonal qudit gate can be implemented efficiently in any dimension.

Figure 1

Level setting for realizing a dark path holonomic qutrit gate (left panel). The specific coupling structure may be realized by inducing transitions between hyperfine levels in trapped ions. It allows for a Morris-Shore transformation [30] applied to the qutrit levels $|1\rangle ,|2\rangle ,|3\rangle$, resulting in a single dark energy eigenstate $|\mathcal{D}\rangle$ and two bright states $|b_1\rangle ,|b_2\rangle$, while leaving the auxiliary state $|a\rangle$ untouched (right panel).

Figure 2

Rabi frequencies in the qutrit scheme for zero and nonzero value of $\eta$. Note that only ${\mathrm{\Omega }}_2\left(t\right)$ and ${\mathrm{\Omega }}_a\left(t\right)$ change shape with $\eta$. This extends to the higher-dimensional case in that only the pulse with the largest index and ${\mathrm{\Omega }}_a\left(t\right)$ depend on $\eta$ for arbitrary $d$ [see Eq. (20) below].

Figure 3

The effect of the ${\mathrm{H}}_3$ gate on the initial states $\frac{1}{\sqrt{3}}\left(\right|1\rangle +|2\rangle +|3\rangle )$ (upper panel) and $|1\rangle$ (lower panel) plotted as a function of dimensionless time $t/\tau$. The coupling to the auxiliary state is $\eta =4.0$. Note that since the plots show the populations of the computational, excited, and auxiliary states, phases cannot be seen.

Figure 4

The effect of the ${\mathrm{X}}_3$ gate on the initial states $\frac{1}{\sqrt{2}}\left(\right|2\rangle +|3\rangle )$ (upper panel) and $\frac{1}{\sqrt{38}}\left(5\right|1\rangle +3|2\rangle +2|3\rangle )$ (lower panel) plotted as a function of dimensionless time $t/\tau$. The coupling to the auxiliary state is $\eta =4.0$. Note that since the plots show the populations of the computational, excited, and auxiliary states, phases cannot be seen.

Figure 5

Robustness test, average fidelity of the ${\mathrm{T}}_3,{\mathrm{X}}_3,{\mathrm{H}}_3$, and ${\mathrm{Z}}_3$ gates. The averages are calculated by sampling over 500 randomized initial states with a perturbation $\mathrm{\Omega }\mapsto (1+\delta )\mathrm{\Omega }$ of the Rabi frequency.

Figure 6

Qudit setting in the Morris-Shore basis [30] applied to the qudit levels $|1\rangle ,...,|d\rangle$. As in the qutrit case, the auxiliary state $|a\rangle$ is untouched and a single dark state $|\mathcal{D}\rangle$ emerges in the computational qudit subspace $\mathrm{Span}\left\{\right|1\rangle ,...,|d\rangle \}$.