On Efficiently Solvable Cases of Quantum k-SAT

Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem k-SAT, Quantum k-SAT is $$\hbox {QMA}_1$$ QMA 1 -complete (for $$k\ge 3$$ k ≥ 3 , and where $$\hbox {QMA}_1$$ QMA 1 is a quantum generalization of NP with one-sided error), and thus likely intractable. But whereas k-SAT has been well-studied for special tractable cases, as well as from a “parameterized complexity” perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a “dimer covering” or “matching”; such systems are known to be frustration-free, but it remains open whether one can efficiently compute a ground state. Our results fall into three directions, all of which relate to the “dimer covering” setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two interaction terms or clauses. (2) We give a “parameterized algorithm” for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. An explicit family of hypergraphs, denoted Crash, for which such a speedup is obtained is introduced. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a “dimer covering”. We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.

Multiharmonic Hamiltonian models with applications to first-order resonances

ABSTRACT In this work, two multiharmonic Hamiltonian models for mean motion resonances are formulated and their applications to first-order resonances are discussed. For the kp:k resonance, the usual critical argument φ = kλ − kpλp + (kp − k)ϖ is taken as the resonant angle in the first model, while the second model is characterized by a new critical argument σ = φ/kp. Based on canonical transformations, the resonant Hamiltonians associated with these two models are formulated. It is found that the second Hamiltonian model holds two advantages in comparison with the first model: (i) providing a direct correspondence between phase portraits and Poincaré sections, and (ii) presenting new phase-space structures where the zero-eccentricity point is a visible saddle point. Then, the second Hamiltonian model is applied to the first-order inner and outer resonances, including the 2:1, 3:2, 4:3, 2:3, and 3:4 resonances. The phase-space structures of these first-order resonances are discussed in detail and then the libration centres and associated resonant widths are identified analytically. Simulation results show that there are pericentric and apocentric libration zones where the libration centres diverge away from the nominal resonance location as the eccentricity approaches zero and, in particular, the resonance separatrices do not vanish at arbitrary eccentricities for both the inner and outer (first-order) resonances.

Network Hamiltonian models reveal pathways to amyloid fibril formation

Amyloid fibril formation is central to the etiology of a wide range of serious human diseases, such as Alzheimer’s disease and prion diseases. Despite an ever growing collection of amyloid fibril structures found in the Protein Data Bank (PDB) and numerous clinical trials, therapeutic strategies remain elusive. One contributing factor to the lack of progress on this challenging problem is incomplete understanding of the mechanisms by which these locally ordered protein aggregates self-assemble in solution. Many current models of amyloid deposition diseases posit that the most toxic species are oligomers that form either along the pathway to forming fibrils or in competition with their formation, making it even more critical to understand the kinetics of fibrillization. A recently introduced topological model for aggregation based on network Hamiltonians is capable of recapitulating the entire process of amyloid fibril formation, beginning with thousands of free monomers and ending with kinetically accessible and thermodynamically stable amyloid fibril structures. The model can be parameterized to match the five topological classes encompassing all amyloid fibril structures so far discovered in the PDB. This paper introduces a set of network statistical and topological metrics for quantitative analysis and characterization of the fibrillization mechanisms predicted by the network Hamiltonian model. The results not only provide insight into different mechanisms leading to similar fibril structures, but also offer targets for future experimental exploration into the mechanisms by which fibrils form.

Empirical Line Lists in the ExoMol Database

The ExoMol database aims to provide comprehensive molecular line lists for exoplanetary and other hot atmospheres. The data are expanded by inclusion of empirically derived line lists taken from the literature for a series of diatomic molecules, namely CH, NH, OH, AlCl, AlF, OH + , CaF, MgF, KF, NaF, LiCl, LiF, MgH, TiH, CrH, FeH, C 2 , CP, CN, CaH, and triplet N 2 . Generally, these line lists are constructed from measured spectra using a combination of effective rotational Hamiltonian models for the line positions and ab initio (transition) dipole moments to provide intensities. This work results in the inclusion of 22 new molecules (36 new isotopologues) in the ExoMol database.

Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

We present an information geometric analysis of off-resonance effects on classes of exactly solvable generalized semi-classical Rabi systems. Specifically, we consider population transfer performed by four distinct off-resonant driving schemes specified by su 2 ; ℂ time-dependent Hamiltonian models. For each scheme, we study the consequences of a departure from the on-resonance condition in terms of both geodesic paths and geodesic speeds on the corresponding manifold of transition probability vectors. In particular, we analyze the robustness of each driving scheme against off-resonance effects. Moreover, we report on a possible tradeoff between speed and robustness in the driving schemes being investigated. Finally, we discuss the emergence of a different relative ranking in terms of performance among the various driving schemes when transitioning from on-resonant to off-resonant scenarios.