昨天 19:23 武汉大学算法工程师发布于江西

关注

Shor算法工作原理详解

数学原理与问题转化

Shor算法的核心思想是将大整数分解问题转化为寻找函数周期的问题‌1。对于给定的合数N，算法通过以下步骤实现分解：

‌随机选择整数a‌：选取一个与N互质的整数a(1 < a < N)。如果gcd(a,N)≠1，则已经找到N的一个因子‌3。
‌寻找周期r‌：找到最小的正整数r，使得a^r ≡ 1 mod N。这个r称为a模N的阶(order)‌5。
‌因数分解‌：如果r为偶数且a^(r/2) mod N ≠ -1，则计算：p = gcd(a^(r/2) - 1, N)q = gcd(a^(r/2) + 1, N)这两个最大公约数中必有一个是N的非平凡质因数‌

关键洞察在于Shor算法通过将质因数分解问题转化为周期寻找问题，利用了量子计算机在周期寻找方面的指数级加速能力，这是经典计算机无法实现的‌

量子实现与关键组件

Shor算法的量子部分需要两个量子寄存器，并巧妙地利用量子力学特性实现指数级加速‌

‌量子叠加态‌：n个量子比特可同时表示2^n个状态组合，突破了经典计算机的线性存储限制‌
‌量子傅里叶变换(QFT)‌：这是Shor算法的关键量子组件，它能将周期信息编码为相位，并通过测量转化为经典信息。与经典FFT的O(N log N)复杂度相比，量子QFT仅需O((log N)²)门操作，实现指数级加速‌
‌模幂运算‌：这是Shor算法中最计算密集的部分，需要通过量子电路实现高效的模乘操作‌

量子并行计算的硬件实现依赖于量子门操作序列，如Hadamard门、CNOT门等逻辑门的组合构成量子线路的基础操作单元‌ 具体实现步骤

Shor算法的完整实现流程可分为以下几个关键步骤‌22：

‌经典部分‌：随机选择一个与N互质的整数a计算gcd(a, N)，如果不等于1，则已经找到N的一个因子
‌量子部分‌：初始化两个量子寄存器：|0⟩⊗ⁿ|1⟩应用Hadamard门创建叠加态：1/√2ⁿ ∑|x⟩|1⟩执行模指数操作：1/√2ⁿ ∑|x⟩|aˣ mod N⟩应用量子傅里叶变换(QFT)到第一寄存器测量获取r的近似值
‌后处理‌：如果r为奇数，重新选择a如果√(aʳ) ± 1 ≡ 0 (mod N)，则重新选择a否则，计算gcd(√(aʳ) + 1, N)和gcd(√(aʳ) - 1, N)得到N的因数

一个具体的例子是分解N=15：

选择a=2

计算gcd(2,15)=1

寻找阶r：2⁰ mod 15=1, 2¹ mod 15=2, 2² mod 15=4, 2³ mod 15=8, 2⁴ mod 15=1，故r=4

验证条件：r=4为偶数，且2^(4/2) mod 15=1

计算公因数：gcd(4-1,15)=3, gcd(4+1,15)=5

成功分解15=3×5‌2

量子电路设计

Shor算法的量子电路包含以下几个关键模块‌32：

‌初始化模块‌：将量子寄存器初始化为|0⟩⊗ⁿ|1⟩状态。
‌模幂运算模块‌：通过受控门操作实现模乘运算，将相位信息编码到量子态中。
‌量子傅里叶变换模块‌：提取周期信息，这是Shor算法的核心量子组件。
‌测量模块‌：将量子态坍缩为经典信息，获取周期r的近似值。

实验上，IBM的一个小组在2001年展示了Shor算法的实例，使用NMR实验的量子计算机和7个量子位元，将15分解成3×5‌5。此后，其他团队以光学量子位元实验Shor算法，并强调其缠结现象可被观察到‌5。

Shor算法的量子优势在于：

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经典困难性：寻找r的经典算法复杂度为指数级(如试除法)，而Shor算法通过量子相位估计将其降至多项式时间O((log N)³)‌2
量子并行性：叠加态和量子干涉特性使得单次操作可覆盖所有可能输入组合的计算结

总结

Shor算法代表了量子计算对经典计算的革命性突破，它将大整数分解这一经典计算难题转化为量子计算机上可高效解决的周期寻找问题。通过巧妙地结合量子傅里叶变换和模幂运算的量子并行性，Shor算法实现了对最佳经典算法的指数级加速。虽然目前量子计算机的规模和技术限制使得Shor算法还无法实际破解大规模RSA密钥，但其理论突破已经深刻影响了密码学的发展，推动了后量子密码学的研究。随着量子计算技术的进步，Shor算法有望在未来成为破解现有公钥密码体系的有力工具。