14 research outputs found
Entanglement subvolume law for 2D frustration-free spin systems
Let be a frustration-free Hamiltonian describing a 2D grid of qudits with
local interactions, a unique ground state, and local spectral gap lower bounded
by a positive constant. For any bipartition defined by a vertical cut of length
running from top to bottom of the grid, we prove that the corresponding
entanglement entropy of the ground state of is upper bounded by
. For the special case of a 1D chain, our result provides a
new area law which improves upon prior work, in terms of the scaling with qudit
dimension and spectral gap. In addition, for any bipartition of the grid into a
rectangular region and its complement, we show that the entanglement
entropy is upper bounded as where
is the boundary of . This represents the first subvolume bound on
entanglement in frustration-free 2D systems. In contrast with previous work,
our bounds depend on the local (rather than global) spectral gap of the
Hamiltonian. We prove our results using a known method which bounds the
entanglement entropy of the ground state in terms of certain properties of an
approximate ground state projector (AGSP). To this end, we construct a new AGSP
which is based on a robust polynomial approximation of the AND function and we
show that it achieves an improved trade-off between approximation error and
entanglement
Sub-exponential algorithm for 2D frustration-free spin systems with gapped subsystems
We show that in the setting of the subvolume law of [Anshu, Arad, Gosset '19]
for 2D locally gapped frustration-free spin systems there exists a randomized
classical algorithm which computes the ground states in sub-exponential time.
The running time cannot be improved to polynomial unless SAT can be solved in
randomized polynomial time, as even the special case of classical constraint
satisfaction problems on the 2D grid is known to be NP-hard
An improved 1D area law for frustration-free systems
We present a new proof for the 1D area law for frustration-free systems with
a constant gap, which exponentially improves the entropy bound in Hastings' 1D
area law, and which is tight to within a polynomial factor. For particles of
dimension , spectral gap and interaction strength of at most
, our entropy bound is S_{1D}\le \orderof{1}X^3\log^8 X where
X\EqDef(J\log d)/\epsilon. Our proof is completely combinatorial, combining
the detectability lemma with basic tools from approximation theory.
Incorporating locality into the proof when applied to the 2D case gives an
entanglement bound that is at the cusp of being non-trivial in the sense that
any further improvement would yield a sub-volume law.Comment: 15 pages, 6 figures. Some small style corrections and updated ref
An area law for 2D frustration-free spin systems
We prove that the entanglement entropy of the ground state of a locally
gapped frustration-free 2D lattice spin system satisfies an area law with
respect to a vertical bipartition of the lattice into left and right regions.
We first establish that the ground state projector of any locally gapped
frustration-free 1D spin system can be approximated to within error
by a degree multivariate polynomial in the
interaction terms of the Hamiltonian. This generalizes the optimal bound on the
approximate degree of the boolean AND function, which corresponds to the
special case of commuting Hamiltonian terms. For 2D spin systems we then
construct an approximate ground state projector (AGSP) that employs the optimal
1D approximation in the vicinity of the boundary of the bipartition of
interest. This AGSP has sufficiently low entanglement and error to establish
the area law using a known technique.Comment: version 2: updated affiliation and added some references. 25 pages, 3
figure
Entanglement area law for 1D gauge theories and bosonic systems
We prove an entanglement area law for a class of 1D quantum systems involving
infinite-dimensional local Hilbert spaces. This class of quantum systems
include bosonic models such as the Hubbard-Holstein model, and both U(1) and
SU(2) lattice gauge theories in one spatial dimension. Our proof relies on new
results concerning the robustness of the ground state and spectral gap to the
truncation of Hilbert space, applied within the approximate ground state
projector (AGSP) framework from previous work. In establishing this area law,
we develop a system-size independent bound on the expectation value of local
observables for Hamiltonians without translation symmetry, which may be of
separate interest. Our result provides theoretical justification for using
tensor network methods to study the ground state properties of quantum systems
with infinite local degrees of freedom
From communication complexity to an entanglement spread area law in the ground state of gapped local Hamiltonians
In this work, we make a connection between two seemingly different problems.
The first problem involves characterizing the properties of entanglement in the
ground state of gapped local Hamiltonians, which is a central topic in quantum
many-body physics. The second problem is on the quantum communication
complexity of testing bipartite states with EPR assistance, a well-known
question in quantum information theory. We construct a communication protocol
for testing (or measuring) the ground state and use its communication
complexity to reveal a new structural property for the ground state
entanglement. This property, known as the entanglement spread, roughly measures
the ratio between the largest and the smallest Schmidt coefficients across a
cut in the ground state. Our main result shows that gapped ground states
possess limited entanglement spread across any cut, exhibiting an "area law"
behavior. Our result quite generally applies to any interaction graph with an
improved bound for the special case of lattices. This entanglement spread area
law includes interaction graphs constructed in [Aharonov et al., FOCS'14] that
violate a generalized area law for the entanglement entropy. Our construction
also provides evidence for a conjecture in physics by Li and Haldane on the
entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08]. On the
technical side, we use recent advances in Hamiltonian simulation algorithms
along with quantum phase estimation to give a new construction for an
approximate ground space projector (AGSP) over arbitrary interaction graphs.Comment: 29 pages, 1 figur
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