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Resource Analysis of Low-Overhead Transversal Architectures for Reconfigurable Atom Arrays

Hengyun Zhou, Casey Duckering, Chen Zhao, Dolev Bluvstein, Madelyn Cain, Aleksander Kubica, Sheng-Tao Wang, Mikhail D. Lukin

arXiv:2505.15907v1quant-ph

TL;DR

Large-scale fault-tolerant architectures must address substantial space-time overheads while managing transversal-gate layout, logical errors, routing, timing, and correlated-decoding complexity. The paper decomposes algorithms into optimized transversal functional units and layouts, and introduces heuristic models for logical performance in transversal circuits. 2048-bit RSA factoring is estimated at 19 million qubits in 5.6 days, representing a 50× time speed-up with no increase in space footprint.

  • Problem

    Large-scale fault-tolerant architectures must address substantial space-time overheads while managing transversal-gate layout, logical errors, routing, timing, and correlated-decoding complexity.

  • Method

    The paper decomposes algorithms into optimized transversal functional units and layouts, and introduces heuristic models for logical performance in transversal circuits.

  • Results

    2048-bit RSA factoring is estimated at 19 million qubits in 5.6 days, representing a 50× time speed-up with no increase in space footprint.

  • Takeaways & Limitations

    Transversal architectures can substantially reduce space-time costs for large-scale fault-tolerant quantum algorithms and may extend to other reconfigurable platforms.

Abstract

from arXiv · show

Neutral atom arrays have recently emerged as a promising platform for fault-tolerant quantum computing. Based on these advances, including dynamically-reconfigurable connectivity and fast transversal operations, 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, 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 50$\times$ speed-up of the run-time compared to existing estimates with similar assumptions, with no increase in space footprint.

I. INTRODUCTION

The paper presents a low-overhead transversal architecture for large-scale fault-tolerant quantum algorithms, addressing layout, logical-error, decoding, and compilation challenges. It develops efficient functional units and models, then demonstrates substantial resource savings for 2048-bit RSA factoring.

  • Architecture and compilation: The architecture decomposes target algorithms into modular subroutines, explicitly lays them out, and re-optimizes algorithm parameters to minimize space-time cost while meeting target logical error rates.The framework compiles subroutines into transversal operations and balances their space-time footprint.
  • Functional units: Efficient transversal constructions cover magic state factories, quantum arithmetic units, and quantum lookup tables while minimizing space-time overhead, atom movement, and correlated-decoding volume.These functional units also support reaction-limited computation for streamlined execution.
  • Logical performance: Heuristic logical-error models account for elevated errors from additional transversal gates and the complexity of correlated decoding, with verification in key limiting cases.The framework treats logical performance as a design constraint alongside layout and execution cost.
  • Resource savings: 19 million qubits and 5.6 days enable 2048-bit RSA factoring, a 50× runtime speed-up with no increase in space footprint versus similar estimates.The comparison is presented as a substantial reduction in space-time cost relative to existing estimates.
  • Scope and outlook: The framework is expected to apply beyond neutral atoms to reconfigurable platforms such as trapped ions and silicon spin qubits, and to support future algorithm, compilation, QEC, and layout improvements.The authors position it as a common framework for guiding large-scale dynamically reconfigurable quantum computers.
  • Transversal operations: Transversal gates provide an O(d) logical clock speed-up, while correlated decoding requires jointly decoding logical qubits within distance d and therefore motivates layouts that limit decoding volume.The architecture uses space-time-efficient layouts to manage the resulting routing, timing, and decoding considerations.

II. BACKGROUND

Dynamically reconfigurable neutral-atom arrays combine long-range atom shuttling and parallel control with QEC methods needed to suppress physical errors. Their dominant architectural costs are atom movement, measurement, space overhead, syndrome-extraction time, and decoding latency.

  • Neutral atom arrays: Neutral-atom arrays encode qubits in optically trapped atoms, enabling long coherence, high-fidelity gates, dynamic reconfiguration, long-range connectivity, and parallel transversal operations.Optical tweezers support flexible rearrangement, while atom shuttling enables connectivity beyond 2D-local codes.
  • Neutral atom arrays: Atom movement and qubit measurement are the dominant timescales, with movement time scaling as the square root of travel distance.The movement model uses effective acceleration and deceleration during the trajectory.
  • Quantum error correction: A QEC code [[n, k, d]] uses n physical qubits to encode k logical qubits, while d characterizes detectable and correctable errors.Below threshold, logical error rates decrease exponentially with code distance for many codes.
  • Quantum error correction: Space-time volume captures both qubit-encoding overhead and the time cost of repeated syndrome extraction, while decoder reaction time limits conditional operations.Standard constructions often incur a time overhead of order d, and decoding closes the measurement-to-correction loop.

II.3. Surface Codes

The paper uses surface-code transversal operations and modular algorithmic layouts to reduce fault-tolerant execution time and manage routing, movement, and decoding costs. Shor’s factoring implementation is decomposed into windowed arithmetic, quantum lookup, and addition subroutines that can be optimized as reusable building blocks.

  • II.3. Surface Codes: Transversal gates support one syndrome-extraction round per logical operation, whereas surface-code deformation and lattice surgery typically require d rounds.They apply independent physical gates across matching qubits, preventing errors from spreading within a code block.
  • II.3. Surface Codes: Correlated decoding enables O(1) syndrome-extraction rounds per transversal operation but requires jointly decoding logical qubits within circuit distance d.The architecture limits decoding volume through layouts designed around interaction distance.
  • Compilation framework: The compilation framework decomposes algorithms into building blocks, compiles them into transversal surface-code operations, and optimizes explicit layouts for space-time efficiency.The approach includes magic-state preparation, quantum adders, and quantum lookup tables, while direct atom movement improves routing flexibility but makes long moves slower.
  • Example: Factoring: Shor’s algorithm is dominated by modular exponentiation, which the paper implements through controlled multipliers and windowed arithmetic using quantum lookup tables and additions.Classically computing grouped coefficients and loading them through QROM reduces non-Clifford gates on the quantum processor.
  • Example: Quantum Chemistry: The same functional building blocks are relevant to quantum chemistry, although detailed layout and resource estimates for those applications are left for future work.The paper presents factoring as the main evaluated benchmark and discusses transferability rather than completing a chemistry estimate.

Logical Error Model with Transversal Gates

The paper develops a heuristic logical-error model for transversal circuits by combining syndrome-extraction and transversal-gate noise, then calibrates it against numerical data. The model supports optimization of syndrome-extraction frequency and space-time cost.

  • The heuristic model combines error contributions from syndrome extraction and transversal gates, with transversal gates assigned decoder-dependent weights.The model assumes additive logical error rates across qubits and syndrome-extraction rounds, while accounting for the more complex decoding caused by transversal gates.
  • The decoding factor α captures how much transversal CNOTs increase the effective noise rate handled by error correction.Larger α corresponds to a more challenging decoding problem and a higher effective error rate.
  • The model reproduces the usual memory limit when the number of syndrome-extraction rounds per CNOT becomes large.This behavior is required for consistency with the conventional memory regime.
  • The optimized number of syndrome-extraction rounds per CNOT is typically at most one and depends on physical error rate and decoder factor.Increasing the decoder factor causes only a small space-time-volume increase at physical error rates of 0.1%.

Trade-offs

The architecture trades execution time against active space-time volume by parallelizing sequential circuit segments with Bell pairs. Clifford segments can be parallelized broadly, while non-Clifford dependencies remain limited by reaction time.

  • Bell-pair teleportation bends circuit segments across space-time, allowing sequential blocks to execute in parallel and reducing active execution time.Bell-basis outcomes determine the corrections applied to the later block.
  • Clifford circuits can be implemented in constant depth using Bell-state preparation and measurement, whereas non-Clifford operations retain sequential measurement dependencies.Those dependencies require time equal to the reaction time tr to resolve.

Factory

The architecture prepares non-Clifford resource states with an 8T-to-CCZ factory and cultivated |T⟩ inputs, while arranging the factory to reduce layout and movement costs. Its parallelization strategy addresses sequential dependencies in factory execution.

  • Factory: Bell pairs parallelize repeated circuit segments in the factory and related gadgets, but non-Clifford measurement dependencies impose reaction-time offsets.The parallel execution resolves dependent measurements only after the relevant prior information is available.
  • Factory: The selected 8T-to-CCZ factory converts eight |T⟩ states into one |CCZ⟩ state using transversal operations and error detection.The factory uses the transversal non-Clifford gate of the [[8,3,2]] code after preparing entanglement through factory CNOT layers.
  • Factory: The factory CNOT layout is synthesized as a one-dimensional arrangement that requires no qubit re-ordering.This simplifies control requirements while syndrome-extraction rounds grow the |T⟩ states to the factory’s full code distance.
  • Factory: Because the 8T-to-CCZ factory provides only quadratic suppression, high-quality |T⟩ inputs are prepared with the cultivation scheme from Ref. [97].The cultivated output is converted from a grafted color-code/surface-code structure into a regular d-by-d surface-code patch by extending and measuring out the color-code region.

Arithmetic

The paper compiles arithmetic and quantum look-up-table subroutines into space-time-conscious layouts using low-cost reversible circuits, bridge qubits, and measurement-based resources. The designs reduce interaction-distance and fan-out burdens while preserving compatibility with transversal operations.

  • Arithmetic: The Cuccaro ripple-carry adder is selected for its low T count, low space requirement, and steady magic-state consumption.It uses repeated MAJ and UMA blocks to propagate carries, restore input bits, and compute sum bits.
  • Arithmetic: MAJ and UMA blocks use one Toffoli each, implemented with distilled |CCZ⟩ states and conditional Clifford corrections based on measurement outcomes.Bridge qubits parallelize different blocks, while extra CZ ancillas implement the conditional corrections.
  • Arithmetic: The oblivious carry runway splits the linear ripple-carry computation into segments, with runway separation and padding controlling space, factory count, and error suppression.The linear structure otherwise creates larger T-depth and ties up idle quantum resources.
  • Look-up tables: Quantum look-up tables load classically calculated data into target registers from address states and are used with O(log n)-sized windows for modular exponentiation.Each table entry requires one Toffoli and one CNOT on average, excluding the larger fan-out cost for relevant factoring register sizes.
  • Look-up tables: Measurement-based GHZ resource states assist CNOT fan-out, avoiding the long atom moves that a naive log-depth fan-out implementation would require.The construction prepares GHZ qubits with helper ancillas, then uses transversal CNOTs and X-basis measurements to perform the fan-out.

IV. EVALUATION

The evaluation optimizes a transversal architecture for 2048-bit factoring, balancing syndrome-extraction schedules, algorithm parameters, layout, and resource costs. It compares the resulting implementation with optimized surface-code and lattice-surgery estimates while identifying dominant cost and error contributors.

  • Overall resource evaluation: Over an order of magnitude reduction in space-time cost is achieved relative to existing estimates with similar assumptions.The evaluation compares against highly optimized surface-code and lattice-surgery compilations for the same 2048-bit factoring problem.
  • Syndrome-extraction optimization: Around 1 SE round per transversal gate balances factory space-time volume and logical error, while storage uses a QEC round every 8 ms under a 10 s coherence-time assumption.The optimal storage frequency is largely independent of code distance and occurs when idle errors become comparable to gate-error contributions.
  • Algorithm optimization: The evaluation searches algorithm parameters to minimize total space-time volume, because faster Clifford operations make reaction time more limiting in the transversal architecture.The selected parameters differ from prior compilations because of changed timescales, transversal operations, and improved magic-state preparation.
  • Overall resource evaluation: Close to 50× runtime reduction is obtained against existing estimates, using a 900 µs baseline QEC cycle for comparison.The architecture exploits pipelining between ancilla measurement and transversal-gate movement, whereas the lattice-surgery baseline cannot pipeline without increasing qubit usage.
  • Resource-cost breakdown: Approximately 1.07 × 10^6 lookup-additions dominate the workload, with each lookup taking 0.17 seconds and each addition 0.28 seconds.CNOT fan-out dominates lookup space and logical-error cost, while magic-state factories dominate addition cost.

IV.3. Sensitivity Analysis

The sensitivity analysis measures how decoding performance, coherence, movement, and reaction times affect space-time volume. It finds bounded degradation for moderate decoder-threshold changes, stronger penalties at short coherence times, and diminishing gains from faster control once lookup fan-out dominates.

  • Changes in physical error rates and decoder: A CNOT-threshold decrease from 0.86% to 0.6% increases space-time volume by only 50%.The exponential reduction of logical errors with code distance limits the increase caused by a lower effective threshold.
  • Changes in physical error rates and decoder: Space-time volume accelerates upward when coherence time falls below 1 s.Coherence time serves as a rough proxy for characteristic system error rates in this analysis.
  • Changes in physical-operation timescales: Increasing atom acceleration improves space-time volume by enabling more operations per unit time.The same analysis uses movement timescales to estimate effects such as changed grid spacing.
  • Changes in physical-operation timescales: Reducing reaction time lowers cost, but gains are eventually bottlenecked by the large CNOT fan-out volume of the lookup table.This identifies lookup-table fan-out as a limiting cost after control latency is reduced.

IV.3.3. Changes in qubit number constraints

The qubit-count analysis exposes a continuous runtime–space trade-off at comparable total space-time volume. Dense qLDPC storage may reduce footprint, but gate operations and longer-distance moves constrain the estimate.

  • Changes in qubit number constraints: A continuous trade-off permits different qubit counts and runtimes with comparable total space-time volume.The total space-time volume begins increasing more sharply when the qubit count drops below 15 million.
  • Changes in qubit number constraints: Below 15 million qubits, further footprint reduction increases total space-time volume more strongly.The analysis considers both qubit-constrained execution and accelerated execution when qubit number is less important.
  • Further Optimization with Dense qLDPC Code: Dense qLDPC storage is estimated to reduce the space footprint by about 20% if execution time remains unchanged.Only 4–6 million qubits are idling, so the estimate assumes a tenfold storage-footprint reduction from dense qLDPC codes.
  • Further Optimization with Dense qLDPC Code: The qLDPC estimate remains incomplete because selective logical gates are challenging and longer-distance moves may increase QEC cycle time.The paper leaves detailed analysis of improved gate constructions and movement overhead to future work.

V. PRIOR WORK

The paper positions its contribution as a detailed resource analysis of transversal architectures for large-scale neutral-atom algorithms, addressing a gap in prior estimates. It reports substantially larger savings than complementary qLDPC analyses while identifying hardware and modeling directions for future work.

  • Comparison with qLDPC approaches: The architecture achieves 50× space-time reduction compared with 2× reductions reported for prior hybrid store-compute qLDPC analyses.The comparison is made against complementary qLDPC approaches rather than as a replacement for them.
  • Large-scale fault-tolerant resource estimation: Detailed resource estimation for large-scale algorithms in high-threshold transversal architectures was previously lacking, and this work addresses that gap.The paper argues that transversal architectures can substantially reduce space-time cost on compatible platforms.
  • Contribution: The reported runtimes improve by over an order of magnitude relative to prior state-of-the-art estimates and may improve further.The analysis covers layout, logical performance, and space-time cost for a low-overhead neutral-atom architecture.
  • Discussions and outlook: The framework is intended to support future improvements in control, readout, decoding, layout, algorithms, and distributed quantum-computing hardware.The paper also suggests direct subroutine simulations and scaled-down experimental implementations as next steps.
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