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Enhanced Scalability in Assessing Quantum Integer Factorization Performance

Junseo Lee

TL;DR

The paper addresses evaluating the scalability of integer factorization with Shor's algorithm using a gate-based quantum circuit simulator based on Matrix Product States (MPS). It demonstrates that pre-selecting the parameter $a$ improves the stability and scalability of performance measurements, enabling factorization benchmarks up to $14$-bit numbers under fixed shot counts, and analyzes the quantum order-finding routine, entanglement structure, and register ordering. Key findings show that pre-selection yields more consistent timing and success, while random $a$ leads to variable period $r$ and frequent failures beyond modest problem sizes. The work provides a practical benchmarking approach for quantum factoring on classical simulators, informs parameter-selection strategies and circuit design for near-term hardware, and outlines future directions including time-structure analysis per step and QRAM-inspired reduction in sampling complexity.

Abstract

With the advancement of quantum technologies, there is a potential threat to traditional encryption systems based on integer factorization. Therefore, developing techniques for accurately measuring the performance of associated quantum algorithms is crucial, as it can provide insights into the practical feasibility from the current perspective. In this chapter, we aim to analyze the time required for integer factorization tasks using Shor's algorithm within a gate-based quantum circuit simulator of the matrix product state type. Additionally, we observe the impact of parameter pre-selection in Shor's algorithm. Specifically, this pre-selection is expected to increase the success rate of integer factorization by reducing the number of iterations and facilitating performance measurement under fixed conditions, thus enabling scalable performance evaluation even on real quantum hardware.

Enhanced Scalability in Assessing Quantum Integer Factorization Performance

TL;DR

The paper addresses evaluating the scalability of integer factorization with Shor's algorithm using a gate-based quantum circuit simulator based on Matrix Product States (MPS). It demonstrates that pre-selecting the parameter improves the stability and scalability of performance measurements, enabling factorization benchmarks up to -bit numbers under fixed shot counts, and analyzes the quantum order-finding routine, entanglement structure, and register ordering. Key findings show that pre-selection yields more consistent timing and success, while random leads to variable period and frequent failures beyond modest problem sizes. The work provides a practical benchmarking approach for quantum factoring on classical simulators, informs parameter-selection strategies and circuit design for near-term hardware, and outlines future directions including time-structure analysis per step and QRAM-inspired reduction in sampling complexity.

Abstract

With the advancement of quantum technologies, there is a potential threat to traditional encryption systems based on integer factorization. Therefore, developing techniques for accurately measuring the performance of associated quantum algorithms is crucial, as it can provide insights into the practical feasibility from the current perspective. In this chapter, we aim to analyze the time required for integer factorization tasks using Shor's algorithm within a gate-based quantum circuit simulator of the matrix product state type. Additionally, we observe the impact of parameter pre-selection in Shor's algorithm. Specifically, this pre-selection is expected to increase the success rate of integer factorization by reducing the number of iterations and facilitating performance measurement under fixed conditions, thus enabling scalable performance evaluation even on real quantum hardware.
Paper Structure (11 sections, 1 theorem, 5 equations, 3 figures, 2 tables)

This paper contains 11 sections, 1 theorem, 5 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theorectical Background
  3. Shor's algorithm
  4. Quantum Fourier transform
  5. Matrix product state quantum simulation
  6. Integer Factorization using Selected Parameter
  7. The Effect of Random Parameters
  8. Quantum order finding routine
  9. Entanglement Analysis for MPS
  10. Performance of Shor's Algorithm at Scale
  11. Conclusion & Future Works

Key Result

Lemma 1

Let $f:S^{n}\rightarrow \mathbb{R}$ be a function defined on the $n$-dimensional hypersphere $S^n$ with Lipschitz constant with respect to the euclidean norm $\| \cdot \|$ and a point $\mathbf{x}\in S^{n}$ be chosen uniformly at random. Then for some constant $C>0$.

Figures (3)

  • Figure 1: Simulation results of the quantum period-finding subroutine in Shor’s algorithm when varying $N,~a,~r$
  • Figure 2: Analysis of the degree of entanglement for each register
  • Figure 3: The result of the quantum period-finding subroutine time

Theorems & Definitions (1)

  • Lemma 1: Lévy's inequality 13