Weak Schur sampling with logarithmic quantum memory

The quantum Schur transform maps the computational basis of a system of $n$ qudits onto a \textit{Schur basis}, which spans the minimal invariant subspaces of the representations of the unitary and the symmetric groups acting on the state space of $n$ $d$-level systems. We introduce a new algorithm for the task of weak Schur sampling. Our algorithm efficiently determines both the Young label which indexes the irreducible representations and the multiplicity label of the symmetric group. There are two major advantages of our algorithm for weak Schur sampling when compared to existing approaches which proceed via quantum Schur transform algorithm or Generalized Phase Estimation algorithm. First, our algorihtm is suitable for streaming applications and second it is exponentially more efficient in its memory usage. We show that an instance of our weak Schur sampling algorithm on $n$ qubits to accuracy $\epsilon$ requires only $O(\log_2n)$ qubits of memory and $O(n^3\log_2(\frac{n}{\epsilon}))$ gates from the Clifford+T set. Further, we show that our weak Schur sampling algorithm on $n$ qudits decomposes into $O\big(dn^{2d}\log_2^p\big(\frac{n^{2d}}{\epsilon}\big)\big)$ gates from an arbitrary fault-tolerant qudit universal set, for $p\approx 4$, and requires a memory of $O(\log_dn)$ qudits to implement.