A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustrationfree 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 to, at least, partially answer this question. Here we 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 hardcore lattice gas at negative fugacity on the Hamiltonian's interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.quantum satisfiability | local Hamiltonian | hardcore lattice gas | critical exponents | universality A n overwhelming majority of systems of physical interest can be described via local Hamiltonians:[1]Here, the "k-local" operator Π i acts on a k-tuple of the microscopic degrees of freedom, best thought of as qudits for the computer scientists among our readers or spins for the physicists. The M operators define an interaction (hyper)graph G, displayed in Fig. 1. A surprisingly diverse and important class of such model Hamiltonians is defined by the additional property of being "frustrationfree": The ground state jψi of H is a simultaneous ground state of each and every Π i . This class comprises both commuting Hamiltonians-for which ½Π i , Π j = 0 ∀i, j-such as the toric code, general quantum error correcting codes, and Levin-Wen models (1-3), and noncommuting ones, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) and Rokhsar-Kivelson models (4-6). Their particular usefulness is also related to the fact that many of these examples can be viewed as "local parent Hamiltonians" for generalized matrix product states (7). In general, frustration-free conditions provide analytic control of ground state properties in otherwise largely inaccessible quantum problems.Determining whether a given Hamiltonian H is frustration-free is well known in quantum complexity theory as the quantum satisfiability...
Distinguishing different topologically ordered phases and characterizing phase transitions between them is a difficult task due to the absence of local order parameters. In this paper, we use a combination of analytical and numerical approaches to distinguish two such phases and characterize a phase transition between them. The "toric code" and "double semion" models are simple lattice models exhibiting Z2 topological order. Although both models express the same topological ground state degeneracies and entanglement entropies, they are distinct phases of matter because their emergent quasi-particles obey different statistics. For a 1D model, we tune a phase transition between these two phases and obtain an exact solution to the entire phase diagram, finding a second-order Ising×Ising transition. We then use exact diagonalization to study the 2D case and find indications of a first-order transition. We show that the quasi-particle statistics provides a robust indicator of the distinct topological orders throughout the whole phase diagram.
We show that neutral anyonic excitations have a signature in spectroscopic measurements of materials: The low-energy onset of spectral functions near the threshold follows universal power laws with an exponent that depends only on the statistics of the anyons. This provides a route, using experimental techniques such as neutron scattering and tunneling spectroscopy, for detecting anyonic statistics in topologically ordered states such as gapped quantum spin liquids and hypothesized fractional Chern insulators. Our calculations also explain some recent theoretical results in spin systems
Quantum dimer models typically arise in various low energy theories like those of frustrated antiferromagnets. We introduce a quantum dimer model on the kagome lattice which stabilizes an alternative Z2 topological order, namely the so-called "double semion" order. For a particular set of parameters, the model is exactly solvable, allowing us to access the ground state as well as the excited states. We show that the double semion phase is stable over a wide range of parameters using numerical exact diagonalization. Furthermore, we propose a simple microscopic spin Hamiltonian for which the low-energy physics is described by the derived quantum dimer model.
Condensed matter systems provide alternative 'vacua' exhibiting emergent low-energy properties drastically different from those of the standard model. A case in point is the emergent quantum electrodynamics (QED) in the fractionalized topological magnet known as quantum spin ice, whose magnetic monopoles set it apart from the familiar QED of the world we live in. Here, we show that the two greatly differ in their fine-structure constant α, which parametrizes how strongly matter couples to light: αQSI is more than an order of magnitude greater than αQED ≈ 1/137. Furthermore, αQSI, the emergent speed of light, and all other parameters of the emergent QED, are tunable by engineering the microscopic Hamiltonian. We find that αQSI can be tuned all the way from zero up to what is believed to be the strongest possible coupling beyond which QED confines. In view of the small size of its constrained Hilbert space, this marks out quantum spin ice as an ideal platform for studying exotic quantum field theories and a target for quantum simulation. The large αQSI implies that experiments probing candidate condensed-matter realizations of quantum spin ice should expect to observe phenomena arising due to strong interactions.
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