A recent paper by Jordan et al. introduced Decoded Quantum Interferometry (DQI), a novel quantum algorithm that uses the quantum Fourier transform to reduce linear optimization problems -- max-XORSAT and max-LINSAT -- to decoding problems. In this paper, we extend DQI to optimization problems involving quadratic constraints, which we call max-QUADSAT. Leveraging a connection to quadratic Gauss sums, we give an efficient algorithm to prepare the DQI state for max-QUADSAT (DISCLAIMER: A mistake was found in one of the algorithm's steps, which invalidates this result unless a fix is found). To demonstrate that our algorithm achieves a quantum advantage, we introduce the Quadratic Optimal Polynomial Intersection (quadratic-OPI) problem, a restricted variant of OPI for which, to our knowledge, the standard DQI framework offers no algorithmic speedup. We show that quadratic-OPI is an instance of max-QUADSAT and use our algorithm to optimize it. Lastly, we present a new generalized proof of the "semicircle law" for the fraction of satisfied constraints, generalizing it to any DQI state of problems where the distribution of the number of satisfied constraints for a random assignment is sufficiently close to a binomial distribution. This condition holds exactly for the DQI state of max-LINSAT, and approximately holds in the max-QUADSAT case, with the approximation becoming exponentially better as the problem size increases. This establishes performance guarantees for our algorithm.
Quantum Circuit Design for Decoded Quantum Interferometry
Graduate School of Media and Governance, Keio University Shonan Fujisawa Campus（庆应义塾大学湘南藤泽校区媒体治理研究生院）；Faculty of Environment and Information Studies, Keio University Shonan Fujisawa Campus（庆应义塾大学湘南藤泽校区环境信息学部）
MED-X Institute, the First Affiliated Hospital of Xi’an Jiaotong University（西安交通大学第一附属医院 MED-X 研究所）；Shaanxi Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, College of Physics, Xi’an Jiaotong University（西安交通大学物理学院 陕西省量子信息与量子光电器件重点实验室）
Towards solving industrial integer linear programs with Decoded Quantum Interferometry
BMW Group（宝马集团）；Technical University of Munich, TUM School of Natural Sciences, Physics Department（慕尼黑工业大学自然科学学院物理系）；Boston Consulting Group and BCG X AI Science Institute（波士顿咨询集团及BCG X人工智能科学研究所）
Global Technology Applied Research, JPMorganChase（摩根大通全球技术应用研究）；Harvard University（哈佛大学）；Google Quantum AI（谷歌量子人工智能）；Sandia National Laboratories（桑迪亚国家实验室）
Tight inapproximability of max-LINSAT and implications for decoded quantum interferometry
Freie Universität Berlin（柏林自由大学）；Technische Universität Berlin（柏林工业大学）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料与能源中心）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希赫兹研究所）
Weighted performance bounds for decoded quantum interferometry with imperfect decoding
Chengdu University of Information Technology（成都信息工程大学）；Advanced Cryptography and System Security Key Laboratory of Sichuan Province（四川省密码学与系统安全重点实验室）；SUGON Industrial Control and Security Center（曙光工业控制与安全中心）；National Key Laboratory of Security Communication, Institute of Southwestern Communication（西南通信研究所安全通信国家重点实验室）
Approximate sampling from decoded quantum interferometry via Markov chain Monte Carlo methods
Freie Universität Berlin（柏林自由大学）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希赫兹研究所）；IBM Research（IBM研究院）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料能源研究中心）
The Ohio State University（俄亥俄州立大学）；Harvard University（哈佛大学）；Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局）；Singapore University of Technology and Design（新加坡科技设计大学）
Institute for Quantum Information and Matter, Caltech（加州理工学院量子信息物质研究所）；Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）；Sloan School of Management, MIT（麻省理工学院斯隆管理学院）；Department of Mathematics, MIT（麻省理工学院数学系）
A provable quantum advantage for approximate optimization via decoded quantum interferometry
Freie Universität Berlin（柏林自由大学）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希赫兹研究所）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料与能源中心）
Approximability limits for bounded-degree max-LINSAT and implications for decoded quantum interferometry
Freie Universität Berlin（柏林自由大学）；Technische Universität Berlin（柏林工业大学）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料能源中心）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希赫兹研究所）
Quantum Decoding Algorithms: Quantum Speedups in Optimization
University of Copenhagen（哥本哈根大学）；New York University Shanghai（纽约大学上海分校；纽约大学上海NYU-ECNU物理研究所）；NYU-ECNU Institute of Physics at NYU Shanghai（阿布扎比纽约大学研究中心（CQTS）量子与拓扑系统中心）；Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, New York University Abu Dhabi（纽约大学物理系）；Department of Physics, New York University
