Recent progress on concatenated codes, especially many-hypercube codes, achieves unprecedented space efficiency. Yet two critical challenges persist in practice. First, these codes lack efficient implementations of addressable logical gates. Second, the required high degree of parallelism and long-range interactions pose significant challenges for current hardware platforms. In this paper, we propose an efficient compilation approach for concatenated codes, specifically many-hypercube codes, targeted at neutral atom arrays, which provide the necessary parallelism and long-range interactions. Our approach builds on two key innovations. First, we introduce Automorphism-assisted Hierarchical Addressing (AHA) logical CNOT gates that significantly reduce spacetime overhead compared to conventional distillation-based methods. Second, we develop Virtual Atom Intermediate Representation (VAIR) that enables level-wise optimization and legalization. We implement these innovations in ConiQ, a hardware-aware quantum compiler designed to compile fault-tolerant quantum circuits for neutral atom arrays using many-hypercube codes. Our evaluation demonstrates that ConiQ achieves up to$2000 \times$reduction in spacetime overhead and up to$10^{6} \times$reduction in compilation time compared to state-of-the-art compilers, with our AHA gates providing an additional overhead reduction of up to$20 \times$. These results establish concatenated codes as a promising approach for fault-tolerant quantum computing in the near future.
Recent progress on concatenated codes, especially many-hypercube codes, achieves unprecedented space efficiency. Yet two critical challenges persist in practice. First, these codes lack efficient implementations of addressable logical gates. Second, the required high degree of parallelism and long-range interactions pose significant challenges for current hardware platforms. In this paper, we propose an efficient compilation approach for concatenated codes, specifically many-hypercube codes, targeted at neutral atom arrays, which provide the necessary parallelism and long-range interactions. Our approach builds on two key innovations. First, we introduce Automorphism-assisted Hierarchical Addressing (AHA) logical CNOT gates that significantly reduce spacetime overhead compared to conventional distillation-based methods. Second, we develop Virtual Atom Intermediate Representation (VAIR) that enables level-wise optimization and legalization. We implement these innovations in ConiQ, a hardware-aware quantum compiler designed to compile fault-tolerant quantum circuits for neutral atom arrays using many-hypercube codes. Our evaluation demonstrates that ConiQ achieves up to2000×reduction in spacetime overhead and up to106×reduction in compilation time compared to state-of-the-art compilers, with our AHA gates providing an additional overhead reduction of up to20×. These results establish concatenated codes as a promising approach for fault-tolerant quantum computing in the near future.
Many-hypercube codes, concatenated [[n,n−2,2]] quantum error-detecting codes (n is even), have recently been proposed as high-rate quantum codes suitable for fault-tolerant quantum computing. While the original many-hypercube codes with n=6 can achieve remarkably high encoding rates (about 30% and 20% at concatenation levels 3 and 4, respectively), they have large code block sizes at high levels (216 and 1296 physical qubits per block at levels 3 and 4, respectively), making not only experimental realization difficult but also logical error rates per code block high. Toward earlier experimental realization and lower logical error rates, here we comprehensively investigate smaller many-hypercube codes with [[6,4,2]] and/or [[4,2,2]] codes, where, e.g., D6,4,4 denotes the many-hypercube code using [[6,4,2]] at level 1 and [[4,2,2]] at levels 2 and 3. As a result, we found a counterintuitive fact that D6,4,4 (D6,6,4,4) can achieve lower logical error rates per code block than D4,4,4 (D4,4,4,4), despite its higher encoding rate and larger code block size. Focusing on level 3, we also developed efficient fault-tolerant encoders realizing about 60% overhead reduction while maintaining or even improving the performance, compared to the original design. Using them, we numerically confirmed that D6,4,4 also achieves the best performance for logical controlled-NOT gates in a circuit-level noise model. These results are important for targeting a high-rate code toward early experimental realization of efficient fault-tolerant quantum computing.
To achieve scalable universal quantum computing, we need to implement a universal set of logical gates fault-tolerantly, for which the main difficulty lies with non-Clifford gates. We demonstrate that several characteristic features of the reconfigurable atom array platform are inherently well-suited for addressing this key challenge, potentially leading to significant advantages in fidelity and efficiency. Specifically, we consider a series of different strategies, including magic state distillation, concatenated code array, and fault-tolerant logical multi-controlled- Z gates, leveraging key platform features such as nonlocal connectivity, parallel gate action, collective mobility, and native multi-controlled- Z gates. Our analysis provides valuable insights into the efficient experimental realization of logical gates, serving as a guide for the full-cycle demonstration of fault-tolerant quantum computation with reconfigurable atom arrays.
Standard approaches to quantum error correction for fault-tolerant quantum computing are based on encoding a single logical qubit into many physical ones, resulting in asymptotically zero encoding rates and therefore huge resource overheads. To overcome this issue, high-rate quantum codes, such as quantum low-density parity-check codes, have been studied over the past decade. In this case, however, it is difficult to perform logical gates in parallel while maintaining low overheads. Here, we propose concatenated high-rate small-size quantum error-detecting codes as a family of high-rate quantum codes. Their simple structure allows for a geometrical interpretation using hypercubes corresponding to logical qubits. We thus call them many-hypercube codes. They can realize both high rates, e.g., 30% (64 logical qubits are encoded into 216 physical ones), and parallelizability of logical gates. Developing dedicated decoder and encoders, we achieve high error thresholds even in a circuit-level noise model. Thus, the many-hypercube codes will pave the way to high-performance fault-tolerant quantum computing.
Achieving fault-tolerant quantum computing at a practical scale demands quantum error correction (QEC) codes with high encoding rates. Quantum low-density parity-check (qLDPC) codes emerge as a promising candidate, especially given the rise of neutral atom arrays that provide dynamic long-range connectivity via atom movements. In general, synthesizing valid and efficient physical execution plans for QEC is a provably hard combinatorial problem, forming a critical compilation bottleneck that worsens as code sizes grow. To overcome this complexity, we focus on an important product family of qLDPC codes with dimension-reduction properties, and propose ONEX. This framework decomposes complex 2D physical execution planning into independent 1D subproblems, each solved to optimal execution depth within practical compilation time. First, we formulate the 1D execution plan with an explicit satisfiability modulo theories (SMT) encoding. This protocol produces provably depth-optimal solutions with substantial duration reduction. Second, we develop a multi-stage compilation pipeline featuring anytime optimization, movement compaction, and iterative feedback. This pipeline maintains practical wall-clock times while providing progressive refinement and on-demand retrieval of quality solutions. Third, we evaluate ONEX in the application of hypergraph product (HGP) code memory mapped onto neutral atom arrays, achieving 3.7x to 6.1x and 29.8x to 42.1x higher clock rates than the constructive 1D algorithm and the general 2D compiler, respectively, while scaling efficiently to codes with 2,500 data qubits. Finally, we extend ONEX to zoned layouts, revealing architectural insights into the associated trade-offs, and demonstrate its applicability to the broader lifted-product (LP) code family through a representative example.
Quantum computers have the potential to solve certain problems exponentially faster than classical computers. However, realizing this potential in practice is a major challenge, as it requires precise control over large-scale quantum systems operating at extremely low error rates. Quantum error correction (QEC) provides a path to achieving such error rates, but its substantial resource overhead poses a significant practical barrier. In addition, identifying which computational problems offer a provable quantum advantage—and which remain fundamentally intractable—remains an open question. This thesis presents progress towards addressing both of these challenges. The first part of this thesis describes advances that substantially reduce the resource overhead of QEC. We begin by presenting realizations of logical circuits in dynamically reconfigurable arrays of neutral atoms. By jointly decoding the logical qubits, we reduce the cost of implementing such transversal Clifford circuits by a factor proportional to the code distance. We then develop new theories of fault tolerance to extend these savings to universal quantum computation with magic state inputs. Finally, we introduce techniques for fast correlated decoding, enabling practical implementations of these improvements in experimental hardware. Collectively, these advances accelerate progress toward large-scale computation and reduce the cost of QEC by over an order of magnitude. The second part of the thesis explores the relative power of quantum and classical computation in combinatorial optimization, a class of problems that is ubiquitous in science and engineering and foundational to the theory of computational complexity. We experimentally implement the optimized quantum adiabatic algorithm (QAA) in neutral atom arrays and observe evidence of a superlinear speedup over classical simulated annealing on certain hard problem instances. To interpret these results, we develop a theoretical framework to compare the performance of QAA with a broad class of classical Markov chain Monte Carlo algorithms. We identify conditions under which a quantum quadratic speedup is achievable and propose modifications to the QAA to reliably realize this advantage. Together, these contributions advance both the practical implementation and theoretical understanding of quantum algorithms.
Quantum error correction is widely believed to be essential for large-scale quantum computation, but the required qubit overhead remains a central challenge. Quantum low-density parity-check codes can substantially reduce this overhead through high-rate encodings, yet finite-size instances with practical logical error rates often achieve encoding rates only around or below 1/10. Here, building on a recent ultra-high-rate construction by Kasai, we identify new structural conditions on the underlying affine permutation matrices that make encoding rates exceeding 1/2 compatible with efficient implementation on reconfigurable neutral atom arrays. These conditions define a co-designed family of ultra-high-rate quantum codes that supports efficient syndrome extraction and atom rearrangement under realistic parallel control constraints. Using a hierarchical decoder with high accuracy and good throughput, we study the performance under a circuit-level noise model with p=0.1%, achieving per-logical-per-round error rates of 1.3−0.9+3.0×10−13 with a [[2304,1156,≤14]] code and 2.9−1.5+3.1×10−11 with a [[1152,580,≤12]] code. We compare these codes against a heuristic Pareto frontier for finite-blocklength codes relating block length, encoding rate, and logical error rates, and find that our codes lie near the frontier. These results approach the teraquop regime, highlighting the promise of this code family for practical ultra-high-rate quantum error correction.