Adapting permanent‑based techniques to stabilizer rank

Investigate whether techniques that prove super‑polynomial lower bounds on the approximate coherent state rank via the border complexity of multi‑linear formulas for the permanent can be adapted to derive lower bounds on the approximate stabilizer rank of qubit states; in particular, determine whether Boson Sampling can be compiled into qubit circuits in a way that leverages the permanent’s algebraic complexity to lower bound stabilizer rank.

Background

The paper draws parallels between coherent state rank in bosonic systems and stabilizer rank in qubit systems. While it obtains super‑polynomial lower bounds for κ(|1⟩{⊗n}), super‑polynomial lower bounds for stabilizer rank remain unknown. The authors suggest a route via compiling Boson Sampling into qubit models to transfer permanent‑based hardness to stabilizer rank but leave it open.

References

Given the similarities between the two problems, it is natural to wonder whether the proof techniques of \cref{thm:tensor_fock_lowerbound_csr} for bounding the approximate coherent state rank can be adapted to bound the approximate stabilizer rank. In particular, can we compile Boson Sampling using qubits to leverage the algebraic complexity of the permanent to lower bound the stabilizer rank? We also leave this question as an interesting open problem.

— Lower Bounds on Coherent State Rank  (2604.00766 - Cottier et al., 1 Apr 2026) in Discussion and open questions