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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 of the sampling algorithm. Specifically, we consider a family of $\text{⟦}{2}^D,D,2\text{⟧}$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 [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 $\text{⟦}O\left(d^D\right),D,d\text{⟧}$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. 

Quantum error correction (QEC) is believed to be essential for the realization of large-scale quantum computers. However, due to the complexity of operating on the encoded `logical' qubits, understanding the physical principles for building fault-tolerant quantum devices and combining them into efficient architectures is an outstanding scientific challenge. Here we utilize reconfigurable arrays of up to 448 neutral atoms to implement all key elements of a universal, fault-tolerant quantum processing architecture and experimentally explore their underlying working mechanisms. We first employ surface codes to study how repeated QEC suppresses errors, demonstrating 2.14(13)x below-threshold performance in a four-round characterization circuit by leveraging atom loss detection and machine learning decoding. We then investigate logical entanglement using transversal gates and lattice surgery, and extend it to universal logic through transversal teleportation with 3D [[15,1,3]] codes, enabling arbitrary-angle synthesis with logarithmic overhead. Finally, we develop mid-circuit qubit re-use, increasing experimental cycle rates by two orders of magnitude and enabling deep-circuit protocols with dozens of logical qubits and hundreds of logical teleportations with [[7,1,3]] and high-rate [[16,6,4]] codes while maintaining constant internal entropy. Our experiments reveal key principles for efficient architecture design, involving the interplay between quantum logic and entropy removal, judiciously using physical entanglement in logic gates and magic state generation, and leveraging teleportations for universality and physical qubit reset. These results establish foundations for scalable, universal error-corrected processing and its practical implementation with neutral atom systems. 

Abstract The exploration of topologically-ordered states of matter is a long-standing goal at the interface of several subfields of the physical sciences. Such states feature intriguing physical properties such as long-range entanglement, emergent gauge fields and non-local correlations, and can aid in realization of scalable fault-tolerant quantum computation. However, these same features also make creation, detection, and characterization of topologically-ordered states particularly challenging. Motivated by recent experimental demonstrations, we introduce a paradigm for quantifying topological states—locally error-corrected decoration (LED)—by combining methods of error correction with ideas of renormalization-group flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly suitable for realistic experiments. We demonstrate the power of LED using numerical simulations of the toric code under a variety of perturbations. We subsequently apply it to an experimental realization, providing new insights into a quantum spin liquid created on a Rydberg-atom simulator. Finally, we extend LED to generic topological phases, including those with non-abelian order.