Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes

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 ⟦2𝐷,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 strong evidence that sampling from these hypercube IQP circuits is classically hard to simulate even at relatively low depths. We analyze the linear cross-entropy benchmark (XEB) in comparison to the average fidelity and, depending on the local noise rate, find two different asymptotic regimes. 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⁡(𝑑𝐷),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.