Okay, let's outline the quantum algorithm designed to efficiently factor large numbers: **Shor's Algorithm**, developed by Peter Shor in 1994. It's crucial to understand that Shor's algorithm doesn't _directly_ find the factors. Instead, it efficiently solves a related problem: **finding the period** of a specific function. Knowing this period allows us to deduce the factors using classical computation. **The Problem:** Factor a large composite integer N. The Core Idea (Classical Reduction): Factoring N can be reduced to finding the order or period r of a randomly chosen integer a modulo N (where 1<a<N and gcd(a,N)=1). The period r is the smallest positive integer such that ar≡1(modN). Once we find such an r: 1. If r is odd, the method fails for this choice of a. We need to pick a different a and try again. 2. If r is even, we compute x=ar/2(modN). 3. If x≡−1(modN) (i.e., x=N−1), the method fails for this choice of a. Pick a different a. 4. Otherwise (r is even and ar/2≡−1(modN)), we have found non-trivial factors of N by calculating: - factor1​=gcd(x−1,N) - factor2​=gcd(x+1,N) The classical difficulty lies in finding the period r efficiently. This is where the quantum computer comes in. **Shor's Algorithm Steps (Quantum Period Finding):** Let n=⌈log2​N⌉ be the number of bits needed to represent N. We typically need about 2n qubits for the first register and n qubits for the second. 1. **Classical Pre-check:** - Choose a random integer a such that 1<a<N. - Calculate d=gcd(a,N) using the classical Euclidean algorithm. - If d>1, then d is a non-trivial factor of N, and we are done. - If d=1, proceed to the quantum steps. 2. **Quantum State Preparation:** - Initialize two quantum registers. Let the first register (input register) have t qubits, where 2t is large enough (e.g., N2≤2t<2N2, so t≈2n). Let the second register (output register) have n qubits. - Apply the Hadamard gate H to each qubit in the first register to create a uniform superposition of all integers from 0 to 2t−1: ∣ψ1​⟩=2t![](data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702%0Ac-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14%0Ac0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54%0Ac44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10%0As173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429%0Ac69,-144,104.5,-217.7,106.5,-221%0Al0 -0%0Ac5.3,-9.3,12,-14,20,-14%0AH400000v40H845.2724%0As-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7%0Ac-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z%0AM834 80h400000v40h-400000z"></path></svg>)​1​x=0∑2t−1​∣x⟩∣0⟩ 3. **Modular Exponentiation (Quantum Oracle):** - Apply a quantum circuit that computes the modular exponentiation f(x)=ax(modN). This operation maps ∣x⟩∣0⟩ to ∣x⟩∣ax(modN)⟩. Apply this to the superposition state: ∣ψ2​⟩=2t![](data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702%0Ac-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14%0Ac0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54%0Ac44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10%0As173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429%0Ac69,-144,104.5,-217.7,106.5,-221%0Al0 -0%0Ac5.3,-9.3,12,-14,20,-14%0AH400000v40H845.2724%0As-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7%0Ac-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z%0AM834 80h400000v40h-400000z"></path></svg>)​1​x=0∑2t−1​∣x⟩∣ax(modN)⟩ - This step entangles the two registers. The function f(x) is periodic with the period r we seek. 4. **Measure the Second Register:** - Measure the second register. Let the outcome be some value k. - This measurement collapses the state of the first register into a superposition of only those values of x for which ax(modN)=k. These x values are separated by the period r. If x0​ is the smallest such x, the state of the first register becomes (approximately): ∣ψ3​⟩≈M![](data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702%0Ac-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14%0Ac0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54%0Ac44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10%0As173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429%0Ac69,-144,104.5,-217.7,106.5,-221%0Al0 -0%0Ac5.3,-9.3,12,-14,20,-14%0AH400000v40H845.2724%0As-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7%0Ac-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z%0AM834 80h400000v40h-400000z"></path></svg>)​1​j=0∑M−1​∣x0​+jr⟩ where M≈2t/r is the number of inputs x mapping to the measured k. The first register now holds a state whose periodicity is r. 5. **Quantum Fourier Transform (QFT):** - Apply the Quantum Fourier Transform to the first register. The QFT transforms the periodic state into a state where the amplitudes are concentrated near multiples of 2t/r. ∣ψ4​⟩=QFT∣ψ3​⟩ 6. **Measure the First Register:** - Measure the first register. Let the outcome be y. This measured value y is likely to be close to an integer multiple of 2t/r. That is: 2ty​≈rc​ for some integer c. 7. **Classical Post-processing (Finding r):** - Use the measured value y and the known value 2t. Use the classical **Continued Fractions Algorithm** on the fraction y/2t to find the best rational approximation c/r with a denominator r less than N. - The denominator r obtained from the continued fractions algorithm is a strong candidate for the period. - **Verification:** Check if ar≡1(modN). If not, r might be a factor of the true period, or the measurement y was unlucky. Also, check if r is even and if ar/2≡−1(modN). - If the candidate r works and satisfies the conditions, calculate the factors gcd(ar/2−1,N) and gcd(ar/2+1,N). - If the candidate r fails (e.g., it's odd, ar≡1(modN), or it leads to trivial factors 1,N), repeat the entire algorithm (steps 1-7) with a different random base a. The probability of success in each run is reasonably high. **Efficiency:** - The most computationally intensive classical algorithms for factoring (like the General Number Field Sieve) have sub-exponential complexity. - Shor's algorithm, running on an ideal quantum computer, has polynomial complexity, roughly O((logN)3). The quantum parts (Modular Exponentiation and QFT) are the critical steps that provide the speedup. - This exponential speedup is why Shor's algorithm poses a significant threat to current public-key cryptography systems like RSA, which rely on the classical difficulty of factoring large numbers. **Requirements:** Shor's algorithm requires a large-scale, fault-tolerant quantum computer capable of executing deep circuits with high fidelity. Building such a machine is a major ongoing challenge in physics and engineering. Current quantum computers are too small and noisy (NISQ era) to factor cryptographically relevant numbers using Shor's algorithm.