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\begin{document}

\title{Separator Barriers for Quantum LDPC Syndrome Extraction}

\author{TODO}
\affiliation{TODO}
\date{\today}

\begin{abstract}
Good quantum LDPC codes promise asymptotically efficient fault tolerance, but their syndrome extraction circuits must still be implemented on hardware with constrained connectivity. We prove a static two-dimensional depth barrier for this task and place it in a broader separator and cut-rank framework. For theorem-level Quantum Tanner code families with $k=\Theta(n)$ and $d=\Theta(n)$, the low-dimensional syndrome-depth tradeoff gives depth $\Omega(n^{3/2}/m)$ for any two-dimensional local implementation on $m$ physical qubits, and hence $\Omega(\sqrt n)$ depth in linear space. We then isolate the separator mechanism behind this obstruction: local Clifford stabilizer-measurement circuits for bounded-weight local-expander QLDPC codes on bounded-degree hardware with weighted separator profile $s_{\hw}(N)$ require depth $\Omega(n/(\Delta_{\hw}s_{\hw}(N)))$, yielding $\Omega(n/\sqrt N)$ for fixed-minor-free hardware families. To connect these geometric statements to compiler-native lower bounds, we define an intrinsic stabilizer cut-rank functional $\chi_L(\cS)$, give its rank formula, and package it as a canonical symmetric submodular demand object $T_{\mathrm{sub}}(\cS)$. In the measurement-free SWAP-only model, depth is lower-bounded by the associated submodular cut congestion. These results prove the intended static two-dimensional barrier while leaving the full classical routing and hidden-variable realization questions open.
\end{abstract}

\maketitle

\input{sections/introduction}
\input{sections/models}
\input{sections/static_2d_barrier}
\input{sections/separator_barriers}
\input{sections/stabilizer_cut_rank}
\input{sections/submodular_cd}
\input{sections/discussion}

\appendix
\input{appendices/proof_details}
\input{appendices/route_d_frontier}

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