  Realizing computationally complex quantum circuits in the presence of noise
and imperfections is a challenging task. While fault-tolerant quantum computing
provides a route to reducing noise, it requires a large overhead for generic
algorithms. Here, we develop and analyze a hardware-efficient, fault-tolerant
approach to realizing complex sampling circuits. We co-design the circuits with
the appropriate quantum error correcting codes for efficient implementation in
a reconfigurable neutral atom array architecture, constituting what we call a
fault-tolerant compilation of the sampling algorithm. Specifically, we consider
a family of $[[2^D , D, 2]]$ quantum error detecting codes whose transversal
and permutation gate set can realize arbitrary degree-$D$ instantaneous quantum
polynomial (IQP) circuits. Using native operations of the code and the atom
array hardware, we compile a fault-tolerant and fast-scrambling family of such
IQP circuits in a hypercube geometry, realized recently in the experiments by
Bluvstein et al. [Nature 626, 7997 (2024)]. We develop a theory of
second-moment properties of degree-$D$ IQP circuits for analyzing hardness and
verification of random sampling by mapping to a statistical mechanics model. We
provide evidence that sampling from hypercube IQP circuits is classically hard
to simulate and analyze the linear cross-entropy benchmark (XEB) in comparison
to the average fidelity. To realize a fully scalable approach, we first show
that Bell sampling from degree-$4$ IQP circuits is classically intractable and
can be efficiently validated. We further devise new families of
$[[O(d^D),D,d]]$ color codes of increasing distance $d$, permitting exponential
error suppression for transversal IQP sampling. Our results highlight
fault-tolerant compiling as a powerful tool in co-designing algorithms with
specific error-correcting codes and realistic hardware.
