2,254 research outputs found
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
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."
This is a technical report on the proceedings of the workshop held July 21 to
July 25, 2008 at the American Institute of Mathematics, Palo Alto, California,
organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We
include a list of open problems coming from applications in 4 different areas:
signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and
holographic algorithms, and entanglement and quantum information theory. We
emphasize the interactions between geometry and representation theory and these
applied areas
Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture
We give improved separations for the query complexity analogue of the
log-approximate-rank conjecture i.e. we show that there are a plethora of total
Boolean functions on input bits, each of which has approximate Fourier
sparsity at most and randomized parity decision tree complexity
. This improves upon the recent work of Chattopadhyay, Mande and
Sherif (JACM '20) both qualitatively (in terms of designing a large number of
examples) and quantitatively (improving the gap from quartic to cubic). We
leave open the problem of proving a randomized communication complexity lower
bound for XOR compositions of our examples. A linear lower bound would lead to
new and improved refutations of the log-approximate-rank conjecture. Moreover,
if any of these compositions had even a sub-linear cost randomized
communication protocol, it would demonstrate that randomized parity decision
tree complexity does not lift to randomized communication complexity in general
(with the XOR gadget)
An algorithm to explore entanglement in small systems
A quantum state's entanglement across a bipartite cut can be quantified with
entanglement entropy or, more generally, Schmidt norms. Using only Schmidt
decompositions, we present a simple iterative algorithm to maximize Schmidt
norms. Depending on the choice of norm, the optimizing states maximize or
minimize entanglement, possibly across several bipartite cuts at the same time
and possibly only among states in a specified subspace.
Recognizing that convergence but not success is certain, we use the algorithm
to explore topics ranging from fermionic reduced density matrices and varieties
of pure quantum states to absolutely maximally entangled states and minimal
output entropy of channels.Comment: Published version, 20 page
Numerical Study of Quantum Resonances in Chaotic Scattering
This paper presents numerical evidence that for quantum systems with chaotic
classical dynamics, the number of scattering resonances near an energy
scales like as . Here, denotes
the subset of the classical energy surface which stays bounded for
all time under the flow generated by the Hamiltonian and denotes
its fractal dimension. Since the number of bound states in a quantum system
with degrees of freedom scales like , this suggests that the
quantity represents the effective number of degrees of
freedom in scattering problems.Comment: 24 pages, including 44 figure
Limitations of semidefinite programs for separable states and entangled games
Semidefinite programs (SDPs) are a framework for exact or approximate
optimization that have widespread application in quantum information theory. We
introduce a new method for using reductions to construct integrality gaps for
SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy
in approximating two particularly important sets in quantum information theory,
where previously no -round integrality gaps were known: the set of
separable (i.e. unentangled) states, or equivalently, the
norm of a matrix, and the set of quantum correlations; i.e. conditional
probability distributions achievable with local measurements on a shared
entangled state. In both cases no-go theorems were previously known based on
computational assumptions such as the Exponential Time Hypothesis (ETH) which
asserts that 3-SAT requires exponential time to solve. Our unconditional
results achieve the same parameters as all of these previous results (for
separable states) or as some of the previous results (for quantum
correlations). In some cases we can make use of the framework of
Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not
only the SoS hierarchy. Our hardness result on separable states also yields a
dimension lower bound of approximate disentanglers, answering a question of
Watrous and Aaronson et al. These results can be viewed as limitations on the
monogamy principle, the PPT test, the ability of Tsirelson-type bounds to
restrict quantum correlations, as well as the SDP hierarchies of
Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published
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Cryptography from tensor problems
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler
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