The computation of Littlewood-Richardson coefficients (to which the Kostka number computation reduces parsimoniously) is sharp-$\mathsf{P}$-hard for binary encoding by a reduction to Knapsack [16]. Binary encoding encodes the length of each row in the input integer partition in binary; this means that for a fixed input size $n$, the order of the underlying symmetric group ranges from $S_n$ to $S_{2^n}$. Fixing the order of $S_n$, as we do here, is equivalent to using unary encoding of the input. It is open if Kostka and LR coefficients are sharp-$\mathsf{P}$-hard with unary encoding. See [14].

The maximally isotypic subspace of a given type is the subspace of $\mathcal{H}$ containing all irreps of such type.

The irreps $\alpha$, $\beta$ can be labeled with $O(\log |G|)$-many bits each, so that the input size $n$ relates nontrivially to the order of the group $G$. Since there are at most $|G|$ distinct irreducible representations, such labeling always exists.

We made a distinction between a hypothesis and a conjecture. By a hypothesis, we mean claim that we strongly believe in but cannot prove. A conjecture labels a claim that we cannot prove and have no strong beliefs about.

There is no polynomial time randomized algorithm that (with high probability) computes $g_{\lambda \mu \nu }$ for $\mu ,\nu ,\lambda ⊢n$ and $d_{\nu }/(d_{\lambda }{d}_{\mu })\ge 1/\mathrm{poly}(n)$. The key challenge then becomes to find a nicely parametrized set of inputs such that $d_{\nu }$, $d_{\lambda }$, $d_{\mu }$ are all superpolynomial, their ratio is bounded by a polynomial, and the partitions do not fall into one of the special cases that can be simulated easily (such as hooks). As long as $\lambda$, $\mu$ have number of parts that scale with $n$, the quantum algorithm can do better than the bound in Eq. (15) as well as the state-of-the-art classical algorithms, but the new result in [39] supersedes this obstruction.