3 research outputs found
Difficult instances of the counting problem for 2-quantum-SAT are very atypical
The problem 2-quantum-satisfiability (2-QSAT) is the generalisation of the
2-CNF-SAT problem to quantum bits, and is equivalent to determining whether or
not a spin-1/2 Hamiltonian with two-body terms is frustration-free. Similarly
to the classical problem 2-SAT, the counting problem #2-QSAT of determining the
size (i.e. the dimension) of the set of satisfying states is #P-complete.
However, if we consider random instances of #2-QSAT in which constraints are
sampled from the Haar measure, intractible instances have measure zero. An
apparent reason for this is that almost all two-qubit constraints are
entangled, which more readily give rise to long-range constraints.
We investigate under which conditions product constraints also give rise to
efficiently solvable families of #2-QSAT instances. We consider #2-QSAT
involving only discrete distributions over tensor product operators, which
interpolates between classical #2-SAT and #2-QSAT involving arbitrary product
constraints. We find that such instances of #2-QSAT, defined on Erdos--Renyi
graphs or bond-percolated lattices, are asymptotically almost surely
efficiently solvable except to the extent that they are biased to resemble
monotone instances of #2-SAT.Comment: 25 pages, 2 figures. Fixed errata concerning frustrated figure eights
(relating to the junction probability), and the threshold for a decoupled
regime on bond-percolated 3D cubic lattice
Recommended from our members
Many-Body Quantum Dynamics and Non-Equilibrium Phases of Matter
Isolated, many-body quantum systems, evolving under their intrinsic dynamics, exhibit a multitude of exotic phenomena and raise foundational questions about statistical mechanics. A flurry of theoretical work has been devoted to understanding how these systems reach thermal equilibrium in the absence of coupling to an external bath and, when thermalization does not occur, investigating the emergent non-equilibrium phases of matter. With the advent of synthetic quantum systems, such as ultra-cold atoms in optical lattices or trapped ions, these questions are no longer academic and can be directly studied in the laboratory. This dissertation explores the non-equilibrium phenomena that stem from the interplay between interactions, disorder, symmetry, topology, and external driving. First, we study how strong disorder, leading to many-body localization, can arrest the heating of a Floquet system and stabilize symmetry-protected topological order that does not have a static analogue. We analyze its dynamical and entanglement properties, highlight its duality to a discrete time crystal, and propose an experimental implementation in a cold-atom setting.Quenched disorder and the many-body localized state are crucial ingredients in protecting macroscopic quantum coherence. We explore the stability of many-body localization in two and higher dimensions and analyze its robustness to rare regions of weak disorder.We then study a second example of non-thermal behavior, namely integrability. We show that a class of random spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram, which includes regions of integrability, classical chaos, and of a novel integrable structure whose conservation laws are reminiscent of the integrals of motion found in a many-body localized phase.The third group of disordered, non-ergodic systems we consider, spin glasses, have fascinating connections to complexity theory and the hardness of constraint satisfaction. We define a statistical ensemble that interpolates between the classical and quantum limits of such a problem and show that there exists a sharp boundary separating satisfiable and unsatisfiable phases