Foundations and Frontiers of Quantum Computational Complexity: A Unified Spectral-Proof Roadmap

John Watrous

shifa liu

This paper establishes a complete and rigorous multi-layer axiomatic classification system for quantum biology, unifying all major empirical phenomena—including photosynthetic coherence, magnetoreception, olfactory vibration, enzyme tunneling, and DNA quantum effects—within a single, extensible hierarchical framework. The system comprises seven primary layers: Kingdom, Phylum, Class, Order, Family, Genus, and Species, augmented by a set of cross-layer consistency conditions. We present a fully formalized foundation, including 53 independent core axioms, 11 compatibility conditions, and 7 cross-layer consistency axioms, totaling 87 logically independent statements, together with a duality groupoid that captures physical and biological equivalences. We prove the well-definedness and completeness of the classification, the uniqueness of the species construction up to physical equivalence, and the logical independence of the axiom system. A systematic gap analysis reveals infinitely many unexplored branches; we develop an algorithmic procedure for generating their axiom systems and a formal parameter validation protocol. We detail 28 concrete predictions, each with a complete seven-layer axiom system, a main theorem, and a rigorous proof (each ≥ 14 steps; total > 500 proof steps). We further prove strong rigidity for a class of supersymmetric quantum biological systems, establish the NP-completeness and QMA-completeness of the classification decision problem, and construct non-standard quantum biological vacua via forcing with a fully justified spectral decomposition. All previously identified open problems and conjectures are resolved and promoted to theorems. The system provides a unified language, a rigorous mathematical foundation, and a roadmap for future theoretical and experimental research in quantum biology.

Christoph Becher, Weibo Gao, Swastik Kar +15 more

Quantum technologies are poised to move the foundational principles of quantum physics to the forefront of applications. This roadmap identifies some of the key challenges and provides insights on materials innovations underlying a range of exciting quantum technology frontiers. Over the past decades, hardware platforms enabling different quantum technologies have reached varying levels of maturity. This has allowed for first proof-of-principle demonstrations of quantum supremacy, for example quantum computers surpassing their classical counterparts, quantum communication with reliable security guaranteed by laws of quantum mechanics, and quantum sensors uniting the advantages of high sensitivity, high spatial resolution, and small footprints. In all cases, however, advancing these technologies to the next level of applications in relevant environments requires further development and innovations in the underlying materials. From a wealth of hardware platforms, we select representative and promising material systems in currently investigated quantum technologies. These include both the inherent quantum bit systems as well as materials playing supportive or enabling roles, and cover trapped ions, neutral atom arrays, rare earth ion systems, donors in silicon, color centers and defects in wide-band gap materials, two-dimensional materials and superconducting materials for single-photon detectors. Advancing these materials frontiers will require innovations from a diverse community of scientific expertise, and hence this roadmap will be of interest to a broad spectrum of disciplines.

Daowen Qiu

Christoph Becher, Weibo Gao, Swastik Kar +17 more

Abstract Quantum technologies are poised to move the foundational principles of quantum physics to the forefront of applications. This roadmap identifies some of the key challenges and provides insights on material innovations underlying a range of exciting quantum technology frontiers. Over the past decades, hardware platforms enabling different quantum technologies have reached varying levels of maturity. This has allowed for first proof-of-principle demonstrations of quantum supremacy, for example quantum computers surpassing their classical counterparts, quantum communication with reliable security guaranteed by laws of quantum mechanics, and quantum sensors uniting the advantages of high sensitivity, high spatial resolution, and small footprints. In all cases, however, advancing these technologies to the next level of applications in relevant environments requires further development and innovations in the underlying materials. From a wealth of hardware platforms, we select representative and promising material systems in currently investigated quantum technologies. These include both the inherent quantum bit systems and materials playing supportive or enabling roles, and cover trapped ions, neutral atom arrays, rare earth ion systems, donors in silicon, color centers and defects in wide-band gap materials, two-dimensional materials and superconducting materials for single-photon detectors. Advancing these materials frontiers will require innovations from a diverse community of scientific expertise, and hence this roadmap will be of interest to a broad spectrum of disciplines.

Andrew Chi-Chih Yao