[Paper Review] A pseudo-simulation of Shor's quantum factoring algorithm

Justin F. Schneiderman, Morgan Stanley+1 more|ArXiv.org|Jun 16, 2002

Quantum Computing Algorithms and Architecture8 references3 citations

TL;DR

This paper presents a Maple-based pseudo-simulation of Shor's quantum factoring algorithm that emulates the measurement outcome of the quantum order-finding step without simulating unitary operations. It enables users to explore factoring integers up to 10 digits by inputting the number to factor and qubit count, producing a detailed history of successful and failed attempts, thereby offering hands-on insight into quantum factoring beyond current quantum hardware capabilities.

ABSTRACT

This paper presents a computer program, written in Maple, that allows a user to simulate certain aspects of Shor's quantum factoring algorithm on a desktop or laptop computer. The program does not simulate the unitary operations carried out by a quantum computer but does faithfully mimic its output at the crucial "readout" step of the order-finding process. The program reqires only two inputs from the user - the number to be factored (which can be up to 10 digits long) and the number of qubits to be used in the factoring (for which a helpful hint is given). The program then returns a detailed history of all its attempts at factoring the number, beginning with its various unsuccessful attempts and ending with the final successful attempt that leads to the correct factors. The structure of the simulation is described, a typical output produced by it is shown, and the factors limiting its performance are discussed. The purpose of this simulation is to provide the user with some "hands-on" experience of how quantum factoring works on integers somewhat larger than can be handled by today's quantum computers.

Motivation & Objective

• To provide a practical, accessible tool for users to experience the behavior of Shor's quantum factoring algorithm on classical computers.
• To simulate the critical measurement (readout) step of the quantum order-finding process, bypassing the need to simulate unitary evolution.
• To enable users to factor integers larger than those feasible on current quantum hardware, up to 10-digit numbers.
• To offer a detailed trace of all factoring attempts—both unsuccessful and successful—thereby illustrating the probabilistic nature of the algorithm.
• To serve as an educational and exploratory tool for understanding quantum factoring dynamics through a non-quantum simulation framework.

Proposed method

• The simulation uses classical computation to model the output distribution of the quantum Fourier transform at the measurement stage of Shor’s algorithm.
• It accepts two inputs: the integer to be factored (up to 10 digits) and the number of qubits used in the quantum register.
• The program performs repeated attempts at finding the period of a random base modulo the input number, mimicking the probabilistic nature of quantum measurement.
• It tracks and logs all attempts, including failures and the final successful factorization, providing a full history of the process.
• The simulation avoids simulating unitary operations, focusing instead on the classical emulation of the measurement outcome distribution.
• The algorithm uses number-theoretic techniques to compute the period and verify whether it leads to a valid factorization of the input number.

Experimental results

Research questions

• RQ1 How can the measurement outcome of Shor’s quantum factoring algorithm be effectively emulated on classical hardware without simulating unitary evolution?
• RQ2 What is the performance and accuracy of a classical simulation that mimics only the readout step of the quantum order-finding process?
• RQ3 To what extent can such a simulation enable users to factor integers larger than those currently feasible on real quantum computers?
• RQ4 How do the number of qubits and the choice of base affect the success rate and convergence of the simulation?
• RQ5 What insights into the probabilistic behavior of Shor’s algorithm can be gained through a detailed trace of successful and failed attempts?

Key findings

• The simulation successfully emulates the measurement outcome of Shor’s algorithm, allowing users to observe the probabilistic nature of period finding without quantum hardware.
• The program provides a complete history of all factoring attempts, including both failures and the final successful factorization, enhancing understanding of the algorithm’s behavior.
• The simulation is limited to factoring integers up to 10 digits due to computational constraints in classical number-theoretic operations.
• The number of qubits required for a given number is guided by a helpful hint in the program, improving usability for non-experts.
• The method avoids simulating unitary operations, significantly reducing computational overhead while preserving the essential output characteristics of the quantum algorithm.
• The simulation demonstrates that classical systems can effectively model key aspects of quantum factoring, offering educational value and insight into quantum algorithmic behavior.