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<p>Due to the reversibility constraints of quantum circuits, the development of modular operations in quantum modular multiplication has been significantly limited. In this work, we propose an optimized modular operation scheme based on Barrett reduction and implement quantum circuits for three versions of Barrett reduction, including our optimized approach. The proposed quantum modular operation circuit can be integrated with quantum-quantum and quantum-classical multiplication circuits. Additionally, we analyze the quantum resource requirements of the proposed circuit and compare the results across the three versions. Our optimized folding Barrett reduction scheme demonstrates superior performance. Furthermore, general quantum modular reduction relies on recursive subtractors and controlled adders to avoid division operation, resulting in a T-depth of approximately <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_4987_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(2^n)\)</EquationSource> </InlineEquation>, where <i>n</i> is the bit length of the modulus. In contrast, our approach achieves a T-depth of only <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_4987_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(n^2)\)</EquationSource> </InlineEquation> by utilizing the Barrett reduction.</p>

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Optimized quantum folding Barrett reduction for quantum modular multipliers

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月2日

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  • Jian Zhang,
  • Seong-Min Cho,
  • Changyeol Lee,
  • Seung-Hyun Seo

摘要

Due to the reversibility constraints of quantum circuits, the development of modular operations in quantum modular multiplication has been significantly limited. In this work, we propose an optimized modular operation scheme based on Barrett reduction and implement quantum circuits for three versions of Barrett reduction, including our optimized approach. The proposed quantum modular operation circuit can be integrated with quantum-quantum and quantum-classical multiplication circuits. Additionally, we analyze the quantum resource requirements of the proposed circuit and compare the results across the three versions. Our optimized folding Barrett reduction scheme demonstrates superior performance. Furthermore, general quantum modular reduction relies on recursive subtractors and controlled adders to avoid division operation, resulting in a T-depth of approximately \(O(2^n)\) , where n is the bit length of the modulus. In contrast, our approach achieves a T-depth of only \(O(n^2)\) by utilizing the Barrett reduction.