Experimenting with D-Wave Quantum Annealers on Prime Factorization problems

Abstract

This paper builds on top of a paper we have published very recently, in which we have proposed a novel approach to prime factorization (PF) by quantum annealing, where 8,219,999=32,749x251 was the highest prime product we were able to factorize -- which, to the best of our knowledge is the largest number which was ever factorized by means of a quantum device. The series of annealing experiments which led us to these results, however, did not follow a straight-line path; rather, they involved a convoluted trial-and-error process, full of failed or partially-failed attempts and backtracks, which only in the end drove us to find the successful annealing strategies. In this paper, we delve into the reasoning behind our experimental decisions and provide an account of some of the attempts we have taken before conceiving the final strategies that allowed us to achieve the results. This involves also a bunch of ideas, techniques, and strategies we investigated which, although turned out to be inferior wrt. those we adopted in the end, may instead provide insights to a more-specialized audience of D-Wave users and practitioners. In particular, we show the following insights: ($i$) different initialization techniques affect performances, among which flux biases are effective when targeting locally-structured embeddings; ($ii$) chain strengths have a lower impact in locally-structured embeddings compared to problem relying on global embeddings; ($iii$) there is a trade-off between broken chain and excited CFAs, suggesting an incremental annealing offset remedy approach based on the modules instead of single qubits. Thus, by sharing the details of our experiences, we aim to provide insights into the evolving landscape of quantum annealing, and help people access and effectively use D-Wave quantum annealers.

Figures (2)

• Figure 1: (Left) Comparison of different chain strengths $c, c \in \{1, 1.5, 2\}$, for QA to factor integers of $3+3$ bits up to $11+8$ bits, with the annealing time $T_a=10\mu s$ and 1,000 samples set for each problem instance. (Right) Excitations distribution of chains (first column) and CFAs (second column) for factoring 10 integers of $8+8$ bits tested in the previous experiments, with chain strength equal to $c\in\{1, 1.5, 2\}$ (respectively top, middle, and bottom row).
• Figure 2: Fixing results for factoring 16-bit 113,507 in Table \ref{['tab: local_fixing']}. Each coordinate corresponds to the CFA in the multiplier circuit. The numbers in the figures in the leftmost and rightmost columns represent the number of excitations out of 1000 samples, whereas the number in the figure in the middle denotes the anneal offset used in the whole fixing process, for advancing the annealing schedule of the specific CFA. Notice the less homogeneous distribution of excitations of CFAs in the leftmost figure compared to the rightmost figure. (Left). Before the fixing. (Middle). The fixing history. (Right). After the fixing.