Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A broad range of quantum optimisation problems can be phrased as the question whether a specific system has a ground state at zero energy, i.e. whether its Hamiltonian is frustration free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms in order to, at least, partially ansr this question. Here  prove a general criterion -- a sufficient condition -- under which a local Hamiltonian is guaranteed to be frustration free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hard-core lattice gas at negative fugacity on the Hamiltonian's interaction graph which as a statistical mechanics problem is of interest in their own right.  concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass theory which permit us to obtain new bounds on the SAT/UNSAT transition in random quantum satisfiability. These also lead us to natural conjectures for when such bounds will be exact (tight), as ll as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this underscores the por of marrying classical statistical mechanics with quantum computation and complexity theory.
['Or Sattath', 'Siddhardh C. Morampudi', 'C. R. Laumann', 'Roderich Moessner']
I am writing a research paper on a idea When must a local Hamiltonian be frustration free
[{"from": "human", "value": "I am writing a research paper on a idea When must a local Hamiltonian be frustration free"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A broad range of quantum optimisation problems can be phrased as the question whether a specific system has a ground state at zero energy, i.e. whether its Hamiltonian is frustration free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms in order to, at least, partially ansr this question. Here  prove a general criterion -- a sufficient condition -- under which a local Hamiltonian is guaranteed to be frustration free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hard-core lattice gas at negative fugacity on the Hamiltonian's interaction graph which as a statistical mechanics problem is of interest in their own right.  concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass theory which permit us to obtain new bounds on the SAT/UNSAT transition in random quantum satisfiability. These also lead us to natural conjectures for when such bounds will be exact (tight), as ll as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this underscores the por of marrying classical statistical mechanics with quantum computation and complexity theory."}]