46 research outputs found
Quantum Hamiltonian Complexity
Constraint satisfaction problems are a central pillar of modern computational
complexity theory. This survey provides an introduction to the rapidly growing
field of Quantum Hamiltonian Complexity, which includes the study of quantum
constraint satisfaction problems. Over the past decade and a half, this field
has witnessed fundamental breakthroughs, ranging from the establishment of a
"Quantum Cook-Levin Theorem" to deep insights into the structure of 1D
low-temperature quantum systems via so-called area laws. Our aim here is to
provide a computer science-oriented introduction to the subject in order to
help bridge the language barrier between computer scientists and physicists in
the field. As such, we include the following in this survey: (1) The
motivations and history of the field, (2) a glossary of condensed matter
physics terms explained in computer-science friendly language, (3) overviews of
central ideas from condensed matter physics, such as indistinguishable
particles, mean field theory, tensor networks, and area laws, and (4) brief
expositions of selected computer science-based results in the area. For
example, as part of the latter, we provide a novel information theoretic
presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.Comment: v4: published version, 127 pages, introduction expanded to include
brief introduction to quantum information, brief list of some recent
developments added, minor changes throughou
Complexity Classification of Local Hamiltonian Problems
The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P, NP-complete, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. The third of these classes contains NP and is contained within StoqMA. The characterisation holds even if S does not contain any 1-local terms, for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. These results are a quantum analogue of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.Some of this work was completed while AM was at the
University of Cambridge. TC is supported by the Royal
Society.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/FOCS.2014.2