6,792 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
Classification of the phases of 1D spin chains with commuting Hamiltonians
We consider the class of spin Hamiltonians on a 1D chain with periodic
boundary conditions that are (i) translational invariant, (ii) commuting and
(iii) scale invariant, where by the latter we mean that the ground state
degeneracy is independent of the system size. We correspond a directed graph to
a Hamiltonian of this form and show that the structure of its ground space can
be read from the cycles of the graph. We show that the ground state degeneracy
is the only parameter that distinguishes the phases of these Hamiltonians. Our
main tool in this paper is the idea of Bravyi and Vyalyi (2005) in using the
representation theory of finite dimensional C^*-algebras to study commuting
Hamiltonians.Comment: 8 pages, improved readability, added exampl
Graph states as ground states of many-body spin-1/2 Hamiltonians
We consider the problem whether graph states can be ground states of local
interaction Hamiltonians. For Hamiltonians acting on n qubits that involve at
most two-body interactions, we show that no n-qubit graph state can be the
exact, non-degenerate ground state. We determine for any graph state the
minimal d such that it is the non-degenerate ground state of a d-body
interaction Hamiltonian, while we show for d'-body Hamiltonians H with d'<d
that the resulting ground state can only be close to the graph state at the
cost of H having a small energy gap relative to the total energy. When allowing
for ancilla particles, we show how to utilize a gadget construction introduced
in the context of the k-local Hamiltonian problem, to obtain n-qubit graph
states as non-degenerate (quasi-)ground states of a two-body Hamiltonian acting
on n'>n spins.Comment: 10 pages, 1 figur
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