Shor’s algorithm has been turning heads since Peter Shor introduced it back in 1995. Today, with quantum processors available for public testing, you can experiment with an approach that could one day outclass classical methods in factoring integers. Though current hardware still struggles with large numbers because of the need for thousands of qubits and ultra-low noise levels, working with smaller numbers offers plenty of insight.
This article walks you through the essentials: the order-finding quantum circuit and modular arithmetic. Through a full implementation in IBM’s Qiskit, complete with simulations and tests on quantum hardware, you get a clear look at how the technique works. If you’ve ever battled with complex algorithms, you’ll appreciate this clear, step-by-step guide.
When factoring an integer, classical methods often hit a wall because of their exponential time requirements. Shor’s algorithm, by contrast, achieves the task in polynomial time—a notable demonstration of quantum efficiency that has real implications for public key cryptography, particularly RSA, which underpins much of our secure online communication.
The process begins by selecting a composite number (one that is odd and not a power of a prime) and choosing a random integer. The algorithm then seeks common factors and finds the order of the chosen number in ZN via quantum operations. Although the quantum part, especially the order-finding step, can sound intimidating, our explanation keeps it direct and approachable.
Our implementation, which typically requires around 2n+2 qubits, hints at the quantum advantage for integer factoring. Companies like IBM are already offering quantum platforms—with up to 128 qubits available for free exploratory runs—demonstrating that even if noise issues persist, early experiments (like those in Z15 or Z21) pave the way forward.
While initial implementations can be gate-heavy and sensitive to quantum noise, steady improvements in hardware technology should soon allow for more reliable and complex calculations. As the technology matures, the prospect of using Shor’s algorithm to tackle real-world problems becomes ever more tangible.
