Modular Quantum Shor Compilation

---
name: modular-quantum-shor-compilation
description: >
  Distributed compilation of Shor's algorithm on modular atomic quantum processors.
  Methodology for large-scale integer factorization across multiple quantum modules with
  optimized inter-module communication and intra-module clock rates.
  Use when: compiling Shor's algorithm for distributed quantum hardware, designing modular quantum architectures,
  optimizing quantum communication between modules, analyzing resource requirements for large-scale factoring,
  or planning fault-tolerant quantum cryptography attacks.
  Trigger words: Shor's algorithm, quantum factoring, modular quantum processor, distributed quantum compilation,
  RSA factoring, quantum cryptography, inter-module communication, Bell pairs, atomic processor.
---

# Modular Quantum Shor Compilation

## Overview

Methodology for compiling and optimizing Shor's algorithm across modular atomic quantum processors.
Addresses the challenge of distributing ~10^6 physical qubits across multiple interconnected modules
while minimizing the overhead from inter-module communication.

Based on: "Factoring 2048-bit RSA integers with a half-million-qubit modular atomic processor" (arXiv: 2605.03951, 2026-05-08)

## Architecture

### CPU-Inspired Modular Design

The processor architecture organizes quantum modules analogous to CPU cores:
- **Modules**: Each contains a subset of physical qubits with local operations
- **Inter-module links**: Bell pair distribution channels for remote operations
- **Measurement units**: Local measurement with specified latency (e.g., 1 ms)

### Key Parameters

| Parameter | Value (2048-bit RSA) | Impact |
|---|---|---|
| Total qubits | ~500,000 | Hardware scale |
| Bell pair rate | 10^5 /sec | Communication bandwidth |
| Measurement time | 1 ms | Gate latency |
| Time overhead vs single-module | 16% | Communication efficiency |

## Compilation Strategy

### Step 1: Problem Decomposition

Decompose the factoring problem into module-local and cross-module operations:
- **Modular exponentiation**: Core of Shor's algorithm, requires most gates
- **Quantum Fourier Transform (QFT)**: Requires cross-module entanglement
- **Measurement and classical post-processing**: Determines factors from output

### Step 2: Qubit Mapping

Map logical qubits to physical locations across modules:
- **Data qubits**: Distributed to minimize cross-module operations
- **Ancilla qubits**: Placed near frequently accessed data qubits
- **Communication qubits**: Dedicated qubits for Bell pair distribution

### Step 3: Communication Optimization

Optimize the interplay between inter-module communication and intra-module clock rate:
- **Pipelining**: Overlap communication with local computation
- **Batching**: Group remote operations to amortize Bell pair setup cost
- **Scheduling**: Order operations to minimize idle time waiting for remote results

### Step 4: Gate Compilation

Compile logical gates into module-local and cross-module primitives:
- **Local gates**: Direct execution within a module
- **Remote CNOT**: Teleportation-based using pre-distributed Bell pairs
- **Measurement-based**: Use measurement outcomes to control subsequent operations

## Performance Analysis

### Resource Scaling

For N-bit RSA integer factorization:
- **Physical qubits**: O(N^2) with surface code error correction
- **Logical gates**: O(N^3) for modular exponentiation
- **Communication cost**: Scales with the fraction of cross-module operations

### Time Complexity

The distributed compilation achieves:
- **16% time overhead** vs ideal single-module for 2048-bit RSA
- **Linear scaling** of overhead with communication latency
- **Sub-linear scaling** with number of modules (due to pipelining)

## Practical Considerations

### Error Correction

- Surface code or similar QEC required for fault tolerance
- Logical error rate must be below algorithm threshold
- Error correction overhead dominates physical qubit count

### Communication Bottlenecks

- Bell pair distribution rate limits remote gate throughput
- Measurement latency affects feedback-dependent operations
- Network topology affects worst-case communication distance

### Verification

- Classical verification of factoring result is O(N^2)
- Quantum volume benchmarks validate module performance
- Cross-module entanglement fidelity must exceed threshold

## Pitfalls

- Underestimating communication overhead can negate parallelism benefits
- Module size must balance local computation vs communication frequency
- Error correction resource estimates vary significantly by code choice
- Classical preprocessing (selecting smoothness bounds) affects quantum resource needs