[Paper Review] Quantum Annealing for Prime Factorization

Shuxian Jiang, Keith A. Britt+3 more|arXiv (Cornell University)|Apr 8, 2018

Quantum Computing Algorithms and Architecture22 references3 citations

TL;DR

This paper presents a resource-efficient quantum annealing approach to prime factorization by mapping integer factorization to an Ising Hamiltonian using ancillary variables to reduce higher-order couplings. The method achieves factorization of 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits, respectively, and validates the approach experimentally on the D-Wave 2000Q processor with 10,000 samples per run, demonstrating feasibility for larger integers as hardware scales.

ABSTRACT

We have developed a framework to convert an arbitrary integer factorization problem to an executable Ising model by first writing it as an optimization function and then transforming the k-bit coupling ($k\geq 3$) terms to quadratic terms using ancillary variables. The method is efficient and uses $\mathcal{O}( ext{log}^2(N))$ binary variables (qubits) for finding the factors of integer $N$. The method was tested using the D-Wave 2000Q for finding an embedding and determining the prime factors for a given composite number. As examples, we present quantum annealing results for factoring 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits respectively. The method is general and could be used to factor larger numbers

Motivation & Objective

• To develop a general, resource-efficient method for mapping integer factorization to a quantum annealing-compatible Ising Hamiltonian.
• To reduce higher-order (k≥3) couplings in the factorization problem to quadratic terms using ancillary variables, enabling implementation on current quantum annealers.
• To demonstrate experimental feasibility of the method on the D-Wave 2000Q processor for small integers.
• To enable scalable factorization of larger integers as quantum hardware with more qubits and higher precision becomes available.
• To provide a pre-processing framework for noisy factorization algorithms, improving efficiency in practical applications like RSA.

Proposed method

• The factorization problem is first expressed as a minimization function of binary variables representing the bits of the factors.
• Higher-order coupling terms (k≥3) arising from multiplication are transformed into quadratic terms using ancillary variables, reducing the problem to a quadratic Ising Hamiltonian.
• A modified multiplication method is introduced to limit coefficient ranges without increasing qubit count, improving hardware compatibility.
• The resulting Ising Hamiltonian is embedded into the D-Wave 2000Q’s Chimera graph using the SAPI findembedding algorithm based on Cai, Macready, and Roy’s randomized method.
• The logical Ising parameters are translated into physical biases and couplings, and the problem is executed on the D-Wave 2000Q with 10,000 samples and a standard annealing schedule.
• Measured output strings are decoded into candidate factors, and solutions are validated by energy ranking and factor verification.

Experimental results

Research questions

• RQ1 Can integer factorization be efficiently mapped to a quadratic Ising Hamiltonian suitable for quantum annealing with minimal qubit overhead?
• RQ2 How can higher-order coupling terms (k≥3) in the factorization Hamiltonian be reduced to quadratic form without increasing the number of required qubits?
• RQ3 What is the experimental viability of this method on current-generation quantum annealing hardware like the D-Wave 2000Q?
• RQ4 How does the method scale in terms of qubit count for larger integers, and what is the estimated resource requirement for factoring RSA-768?
• RQ5 Can this approach serve as a pre-processing tool for improving the efficiency of classical or noisy quantum factorization algorithms?

Key findings

• The method uses O(log²N) logical qubits to factor an integer N, demonstrating scalability with logarithmic resource growth.
• The factorization of 15 was successfully executed using only 4 logical qubits on the D-Wave 2000Q, with high-probability convergence to the correct factors.
• The method achieved correct factorization of 143 (12 qubits), 59989 (59 qubits), and 376289 (94 qubits), confirming correctness across increasing problem sizes.
• For RSA-768, the method estimates a requirement of approximately 147,456 qubits, indicating a feasible path for future scaling.
• The algorithm was implemented using the XACC framework with SAPI integration, and 10,000 annealing runs per instance were used to ensure statistical confidence in the results.
• The approach is general and hardware-agnostic, applicable to any quantum annealer with comparable qubit count, connectivity, and parameter precision.