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Major update (Jan. 2023): new record, and new hope!

A new approach to factoring using quantum computers has recently been proposed: Bao Yan, Ziqi Tan, Shijie Wei, Haocong Jiang, Weilong Wang, Hong Wang, Lan Luo, Qianheng Duan, Yiting Liu, Wenhao Shi, Yangyang Fei, Xiangdong Meng, Yu Han, Zheng Shan, Jiachen Chen, Xuhao Zhu, Chuanyu Zhang, Feitong Jin, Hekang Li, Chao Song, Zhen Wang, Zhi Ma, H. Wang, Gui-Lu Long, Factoring integers with sublinear resources on a superconducting quantum processor (arXiv.2212.12372). They report factoring any 48-bit integer, including experimentally factoring 261980999226229=15538213×16860433 with help of a 10-bit quantum computer, by an algorithm which required number of qbits is nearly linear (even slightly sub-linear) with the bit size of the integer factored. They perform on a quantum computer the hard part (a Short Vector Problem) of some known factorization method. The approach seems interesting. It's discussed here.

Update (Jan. 2023): new record; new hope?

A new approach to factoring using quantum computers has recently been proposed: Bao Yan, Ziqi Tan, Shijie Wei, Haocong Jiang, Weilong Wang, Hong Wang, Lan Luo, Qianheng Duan, Yiting Liu, Wenhao Shi, Yangyang Fei, Xiangdong Meng, Yu Han, Zheng Shan, Jiachen Chen, Xuhao Zhu, Chuanyu Zhang, Feitong Jin, Hekang Li, Chao Song, Zhen Wang, Zhi Ma, H. Wang, Gui-Lu Long, Factoring integers with sublinear resources on a superconducting quantum processor (arXiv.2212.12372). They report factoring any 48-bit integer, including experimentally factoring 261980999226229=15538213×16860433 with help of a 10-bit quantum computer, by an algorithm which required number of qbits is nearly linear (even slightly sub-linear) with the bit size of the integer factored. They perform on a quantum computer the hard part (a Short Vector Problem) of some known factorization method. The approach is discussed here.

This answer starts with two classes of factorization efforts: stunts and adiabatic quantum computing records. Neither use Shor's algorithm, part of the question's title.

This answer starts with stunt records. See final paragraph for the the largest meaningful record. None of these use Shor's algorithm, part of the question's title.

Major update (Jan. 2023): new hope!

Major update (Jan. 2023): new record, and new hope!