Henry Lamm

Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension $q\gg 100$ or code length $\mathcal{N}\gg 100$. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the $\mathbb{F}_{2^{2m}}$ alphabet with $m \geq 3$ that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as $q=64$. Further, we identify a promising code with parameters $[[42,14,6]]_{64}$. Finally, we show that a distilled $\ket{CCZ}_{2^{2m}}$ can be reduced to a $\ket{CCZ}_{2^n}$ state for arbitrary $n$ with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.

We present two deterministic compilation algorithms for single-qutrit unitaries with $\mathcal{O}(\log \frac{1}{\varepsilon})$ gate depth. Each algorithm selects a nearby approximation to the target unitary and then exactly synthesizes the approximation over the Clifford + $\mathbf{R}$ basis. The first algorithm exhaustively searches over the group; while the second algorithm searches only for Householder reflections. The exhaustive search algorithm yields an average $\mathbf{R}$ count of $\yintm + \slopem \log_{10}(1 / \varepsilon)$, albeit with a time complexity of $\mathcal{O}(\varepsilon^{\pgfmathprintnumber[fixed,precision=2]{\fullcomplexity}})$. The Householder search algorithm results in a larger average $\mathbf{R}$ count of $\yint + \slope \log_{10}(1 / \varepsilon)$ at a reduced time complexity of $\mathcal{O}(\varepsilon^{\pgfmathprintnumber[fixed,precision=2]{\householdercomplexity}})$, greatly extending the reach in $\varepsilon$. These costs correspond asymptotically to 35\% and 69\% more non-Clifford gates compared to synthesizing the same unitary with two qubits. Such initial results are encouraging for using the $\mathbf{R}$ gate as the non-transversal gate for qutrit-based computation.