Quantum Cellular Automata Dynamics: Integrability, Many-Body Decoherence, and Complex Entanglement

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Complexity science is one of the deep outstanding problems of the 21st century, tapping into such profound questions as the biological origin of consciousness. What are the origins of complexity generally? Does it appear already in quantum mechanics? Do we require a new physical theory, or is complexity an outgrowth of what we know? Noisy intermediate scale quantum (NISQ) computers involve an increasing number of interacting quantum subsystems giving rise to complexity. One way to study such emergent complexity in NISQ computers is with Goldilocks quantum cellular automata (QCA). Goldilocks QCA are dynamic computational rules written into a quantum circuit which involve a trade-off or balance between not too many and not too few in a local neighborhood -- thus the term "Goldilocks". Enhancing the progress of science, this project will connect formerly disparate foundational mathematical and scientific concepts, such as integrability and complexity; point the way to possible beyond-classical computing demonstrations with 2D QCA; and develop new quantum computing paradigms for open quantum systems via irreversible elementary QCA making use of the environment instead of avoiding it, just as living systems do. Additionally, the team proposes a multi-faceted approach to meet broader impact goals. First, they will mentor 2-3 undergraduates per year in research from one of the largest undergraduate physics programs in the US. Second, they will engage internationally by serving on the executive editorial board of the Journal of Physics: Complexity; and by supporting science diplomacy via the U.S. Department of State as a Jefferson Science Fellow alumni. Third, they will engage in educational development in the quantitative study of student collaboration networks in hybrid and virtual class environments; and by creation of a science diplomacy course in the Mines honors program accessible to all STEM students and publicly disseminated. Fourth, they will increase diversity, equity, and inclusion in physics via a supportive, diverse group environment with a track record of success in this area. Finally, at present the quantum workforce is insufficient to meet our national quantum initiative requirements. This program will train a sizeable cadre of students at BS, MS, and PhD levels with practical hands-on experience modeling near-term analog and digital quantum computers, crossing over into quantum networks, to help fill this pressing need. In this project on new insight into entanglement dynamics enabled by NISQ devices, the team will pursue the connections between (i) complex entanglement, (ii) many-body decoherence, and (iii) integrability. QCA provide a practical quantum test-bed, proven in recent work to be realizable on digital quantum computers. (i) They will systematically study the full set of quasi-1D (extended 5-site) and 2D QCA, their complexity properties, and their prospective efficient implementation in digital quantum circuits on the Sycamore chip. Is complex entanglement a result of integrability? Or, are integrable systems, which happen to provably encompass 1D (3-site) Goldilocks QCA, a red herring for complexity? This remains to be resolved. (ii) They will systematically treat the decoherence properties of quasi-1D and 2D Goldilocks and non-Goldilocks QCA under quantum trajectories evolution with realistic conditions for both digital quantum circuits and analog quantum simulators. They will then build on prior studies of the 16 reversible elementary 1D QCA to a representative selection of the remaining 240 irreversible elementary 1D QCA in small quantum systems to see if many-body decoherence properties hold up. This study will include Rule 110, which is classically Turing complete; and point the way to the long-term goal of realizing a quantum version of Conway's game of life. (iii) They will provide a complete proof that 1D 3-site QCA under quite general conditions are integrable, and strong evidence for a conjecture that the proof extends to arbitrary sized neighborhoods. To this end they will study thermalization via a truncated generalized Gibbs ensemble based on newly discovered conserved charges; scrambling via the dynamics of the out-of-time-order correlator; and Page curves and spectral characteristics including many-body scars.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.