Error Correction

Abstract: For fault-tolerant quantum memory defined by periodic Pauli measurements, called Floquet codes, we prove that every correctable, undetectable spacetime error occurring during the steady stage is a product of (i) measurement operators inserted at the time of the measurement and (ii) pairs of identical Pauli operators sandwiching a measurement that commutes with the operator. We call such errors benign; they define a binary vector subspace of spacetime errors which properly generalize stabilizers of static Pauli stabilizer codes. Hence, the code distance of a Floquet code is the minimal weight of an undetectable spacetime Pauli error that is not benign. Our results apply more generally to families of dynamical codes for which every instantaneous stabilizer is inferred from measurements in a time interval of bounded length.

Abstract: Fault-tolerant logical entangling gates are essential for scalable quantum computing, but are limited by the error rates and overheads of physical two-qubit gates and measurements. To address this limitation we introduce phantom codes---quantum error-correcting codes that realize entangling gates between all logical qubits in a codeblock purely through relabelling of physical qubits during compilation, yielding perfect fidelity with no spatial or temporal overhead. We present a systematic study of such codes. First, we identify phantom codes using complementary numerical and analytical approaches. We exhaustively enumerate all 2.71 x 10^{10} inequivalent CSS codes up to n=14 and identify additional instances up to n=21 via SAT-based methods. We then construct higher-distance phantom-code families using quantum Reed--Muller codes and the binarization of qudit codes. Across all identified codes, we characterize other supported fault-tolerant logical Clifford and non-Clifford operations. Second, through end-to-end noisy simulations with state preparation, full QEC cycles, and realistic physical error rates, we demonstrate scalable advantages of phantom codes over the surface code across multiple tasks. We observe one–to–two–order-of-magnitude reduction in logical infidelity at comparable qubit overhead for GHZ-state preparation and Trotterized many-body simulation tasks, given a modest preselection acceptance rate. Our work establishes phantom codes as a viable architectural route to fault-tolerant quantum computation with scalable benefits for workloads with dense local entangling structure, and introduces general tools for systematically exploring the broader landscape of quantum error-correcting codes.

Abstract: We present a low-overhead architecture that supports the layout and resource estimation of large-scale fault-tolerant quantum algorithms. Utilizing recent advances in fault tolerance with transversal gate operations, this architecture achieves a run time speed-up on the order of the code distance d, which we find directly translates to run time improvements of large-scale quantum algorithms. Our architecture consists of functional building blocks of key algorithmic subroutines, including magic state factories, quantum arithmetic units, and quantum look-up tables. These building blocks are implemented using efficient transversal operations, and we design space-time-efficient versions of them that minimize interaction distance, thereby reducing atom move times and minimizing the volume for correlated decoding. We further propose models to estimate their logical error performance. We perform resource estimation for a large-scale implementation of Shor's factoring algorithm, one of the prototypical benchmarks for large-scale quantum algorithms, on dynamically reconfigurable neutral atom arrays, finding that 2048-bit RSA factoring can be executed with 19 million qubits in 5.6 days, for 1 ms QEC cycle times. This represents close to 50x speed-up of the run-time compared to existing estimates with similar assumptions, with no increase in space footprint, achieving a genuine reduction of the space-time volume required for error-corrected quantum computation, and bringing the runtime of large-scale algorithms on emerging platforms into a practical regime.