Distributed Shor's Algorithm Compilation

• The paper demonstrates how distributed compilation enables partitioning of Shor's algorithm into modular tasks for efficient prime factorization across quantum processing units.
• It outlines a methodology that employs task decomposition—using windowed modular exponentiation and DAG-based scheduling—to optimize resource allocation and minimize idle times.
• The analysis quantifies trade-offs between inter-module communication and intra-module computation, illustrating scalable execution for large RSA instances.

Distributed compilation of Shor’s algorithm encompasses the algorithmic, architectural, and workflow transformations required to execute prime factorization across multiple quantum processing units (QPUs) or modules, rather than a monolithic quantum computer. This paradigm is driven by the substantial resource costs associated with large-scale instances (e.g., factoring 2048-bit RSA integers), which necessitate modular hardware and advanced compilation strategies that efficiently orchestrate inter-module communication, resource allocation, and workflow pipelining. Distributed approaches enable quantum factorization with reduced per-device qubit and coherence requirements, improved flexibility in accommodating hardware constraints, and more scalable implementations aligned with foreseeable neutral-atom and superconducting quantum architectures.

1. Modular Architectures and Communication Models

Modern distributed implementations of Shor’s algorithm partition computation across a set of QPUs, each responsible for a distinct workload and interconnected via photonic Bell-pair links or operationally equivalent mechanisms. Canonical examples, such as the CPU-inspired modular atomic processor architecture (Xue et al., 5 May 2026), utilize:

• A central quantum-memory module (storing most logical qubits, error-protected with qLDPC codes).
• N active QPU modules (surface code encoding), where $N$ can range from a few (e.g., 6 for RSA-2048) to dozens.
• High-fidelity photonic links generate $\sim10^5$ purified Bell pairs/s between adjacent modules.
• Specialized subregions within each QPU—Bell-pair factories and reservoirs, Clifford-factory (magic-state distillation), and dynamic cache regions for modular arithmetic.
• Logical qubit patches of size $d\times d$ ($d=25$ at $p_{\text{phys}}=10^{-3}$), CNOTs via Rydberg drive-by gates, with single logical CNOT times $t_{\text{CNOT}}\simeq10(d+1)\,\mu$s.

Distributed protocols also leverage multi-channel ebit generation for parallel remote-gate execution, with careful modeling of inter-module operation latency ($t_{\text{inter}}$) relative to intra-module circuit timing ($t_{\text{intra}}$) (Schmidt et al., 28 Mar 2025).

2. Compilation and Task Decomposition Strategies

Distributed Shor compilation is predicated on decomposing the global period-finding circuit into mid-level computational tasks and further mapping those tasks to distinct hardware modules. Key aspects include:

• At the highest level: prepare a uniform superposition $|e⟩$ in the exponent register, realize a sequence of controlled modular exponentiations $g^e \bmod N$, apply the inverse quantum Fourier transform (QFT), and measure.
• Modular exponentiation is restructured into $\sim10^5$0 blocks using residue arithmetic and windowed modular exponentiation in small prime moduli, reducing the Toffoli count to $\sim10^5$1 for $\sim10^5$2-bit moduli (Xue et al., 5 May 2026).
• Each modular addition is broken into atomic sub-steps: Load$\sim10^5$3, Add, Wrap (via lookup), and Unload. These decompose to ladders of CNOTs and two lookup operations per addition.
• Task distribution leverages inter-module GHZ-state creation for data loading (costing 1 Bell pair per load) and nearest-neighbor qubit teleportation for addition overflows.

Mid-level abstractions expose idle times between high-level tasks. Compiler workflows construct directed acyclic graphs (DAGs) over these tasks, enabling static timing analysis and explicit optimization of execution sequence and resource utilization (Schmidt et al., 28 Mar 2025).

3. Communication–Computation Trade-offs and Scheduling

Optimizing distributed execution requires balancing communication bottlenecks and module clock rates. The critical dimensionless parameter $\sim10^5$4 quantifies inter- vs. intra-module latency. Analysis yields:

• A "free modularization" regime in which $\sim10^5$5—i.e., negligible overhead—obtains if $\sim10^5$6.
• For neutral-atom systems with measurement time $\sim10^5$7 ms, $\sim10^5$8 (i.e., $\sim10^5$9 Bell pairs/s suffices); for $d\times d$0s, $d\times d$1.
• Protocols minimize idle time by overlapping magic state distillation, buffer loading, modular addition, teleportation, and cleanup. Dynamic patch reallocation further smooths peak demand (Xue et al., 5 May 2026).
• For task-pipelined architectures, scheduling ensures that up to $d\times d$2 ebit channels can maintain near-optimal utilization, subject to $d\times d$3 (where $d\times d$4 is the average controlled-unitary execution time) (Schmidt et al., 28 Mar 2025).

4. Phase Estimation and Order-Finding in Distributed Settings

Several distributed strategies specifically target phase estimation (central to order-finding in Shor’s algorithm):

• Blockwise/Windowed QPE: Decomposes standard QPE into $d\times d$5 shallow, independent circuit blocks of $d\times d$6 counting qubits and $d\times d$7-qubit work registers. Each block is dispatched to an available QPU, with overlaps $d\times d$8 for robust phase bit stitching (Shukla et al., 5 Sep 2025). Classical postprocessing stitches outcomes via overlap consistency, and the approach supports adaptive rebalancing of window sizes or overlaps for error mitigation.
• Distributed Phase Estimation (DPE): Splits the phase estimation register among $d\times d$9 nodes, each estimating a window of $d=25$0 bits, with overlaps (typically 3 bits) for classical combination. DPE eliminates the need for cross-node quantum gates during phase estimation. Sequential teleportation of the data register ensures all nodes act on the same eigenstate, and classical postprocessing (CorrectAndCombine) reconstructs the full global phase estimate (Xiao et al., 2023).

Distributed PE strategies reduce per-node qubit requirements to $d=25$1, achieving substantial memory savings and allowing for flexible resource allocation and device-limited scaling (Xiao et al., 2023, Shukla et al., 5 Sep 2025).

5. Resource Analysis and Performance Results

Resource requirements for state-of-the-art distributed Shor compilers are quantitatively analyzed for large cryptographic instances:

RSA-bits	Code Distance $d=25$2	Qubits per QPU $d=25$3	Total Qubits $d=25$4	Monolithic Time (days) $d=25$5	6-QPU Distributed Time (days) $d=25$6
1024	23	$d=25$7	$d=25$8	17.1	27.4
2048	25	$d=25$9	$p_{\text{phys}}=10^{-3}$0	164.5	190.4
4096	27	$p_{\text{phys}}=10^{-3}$1	$p_{\text{phys}}=10^{-3}$2	1062.1	1085.3

For RSA-2048, a 6-QPU distributed system at $p_{\text{phys}}=10^{-3}$3 ($p_{\text{phys}}=10^{-3}$4 Bell pairs/s) achieves factoring in $p_{\text{phys}}=10^{-3}$5 days (16% slower than monolithic), using $p_{\text{phys}}=10^{-3}$6 physical qubits (Xue et al., 5 May 2026). The distributed overhead is dominated by communication and state distillation costs; computation accounts for $p_{\text{phys}}=10^{-3}$7 days, communication for $p_{\text{phys}}=10^{-3}$8.

Empirical studies for smaller $p_{\text{phys}}=10^{-3}$9 (up to 64 bits) confirm that alternating two-data-qubit schemes achieve near-optimal delay with minimal qubit overhead for architectures with non-negligible measurement/reset latency (e.g., neutral atoms), while iterative single-data-qubit designs are optimal for platforms with fast measurement (e.g., superconducting) (Schmidt et al., 28 Mar 2025).

6. Classical Postprocessing, Synchronization, and Error Handling

Distributed implementations introduce novel patterns of postprocessing and synchronization:

• Overlap-based stitching of QPE phase blocks (windowed or DPE) ensures global phase integrity, exploits redundancy, and supports pruning of error-prone combinations (Shukla et al., 5 Sep 2025, Xiao et al., 2023).
• Classical communication is confined to block result aggregation and data-register teleportation (where needed), for a complexity $t_{\text{CNOT}}\simeq10(d+1)\,\mu$0 in both quantum and classical channels. No all-to-all quantum connectivity is required—linear chains and classical stars suffice (Xiao et al., 2023).
• Synchronization between modules must accommodate both quantum- and classical-latency, with device clocks or peer-to-peer heartbeats to manage inter-module handoff especially during data-register teleportation (Xiao et al., 2023).
• Trade-offs include the need to pre-share EPR pairs, store registers coherently during teleport wait, and mitigate serialization overhead as module count $t_{\text{CNOT}}\simeq10(d+1)\,\mu$1 grows.

7. Design Guidelines and Future Directions

Best practices for distributed Shor compilation and implementation include:

• Begin compilation from a task-graph (DAG) abstraction to expose and minimize idle intervals and hardware bottlenecks (Schmidt et al., 28 Mar 2025).
• On monolithic architectures with $t_{\text{CNOT}}\simeq10(d+1)\,\mu$2, iterative single-data-qubit designs minimize resource requirements; for architectures with longer measurement/reset, alternating two-qubit schemes restore parallelism effectively.
• In distributed compilation, choose the number of QPUs ($t_{\text{CNOT}}\simeq10(d+1)\,\mu$3 or $t_{\text{CNOT}}\simeq10(d+1)\,\mu$4) to balance per-module load and communication volume, targeting the regime where $t_{\text{CNOT}}\simeq10(d+1)\,\mu$5.
• Use windowed modular exponentiation and residue arithmetic for Toffoli gate and depth minimization at scale (Xue et al., 5 May 2026).
• Dynamic resource allocation within modules (e.g., reallocating patch regions between CCX factories and caches) further smooths workload peaks (Xue et al., 5 May 2026).
• For NISQ devices, use blockwise phase estimation and aggressive qubit reuse, combined with classical stitching and adaptive error correction (Shukla et al., 5 Sep 2025).

Looking forward, distributed paradigms will be increasingly critical as the quantum hardware scaling frontier is approached, with emerging blueprints (Xue et al., 5 May 2026) providing end-to-end recipes encompassing error correction, compilation, communication, and execution for distributed prime factorization at and beyond cryptographic sizes.