143 research outputs found
No distributed quantum advantage for approximate graph coloring
We give an almost complete characterization of the hardness of -coloring
-chromatic graphs with distributed algorithms, for a wide range of models
of distributed computing. In particular, we show that these problems do not
admit any distributed quantum advantage. To do that: 1) We give a new
distributed algorithm that finds a -coloring in -chromatic graphs in
rounds, with . 2) We prove that any distributed
algorithm for this problem requires rounds.
Our upper bound holds in the classical, deterministic LOCAL model, while the
near-matching lower bound holds in the non-signaling model. This model,
introduced by Arfaoui and Fraigniaud in 2014, captures all models of
distributed graph algorithms that obey physical causality; this includes not
only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL,
even with a pre-shared quantum state.
We also show that similar arguments can be used to prove that, e.g.,
3-coloring 2-dimensional grids or -coloring trees remain hard problems even
for the non-signaling model, and in particular do not admit any quantum
advantage. Our lower-bound arguments are purely graph-theoretic at heart; no
background on quantum information theory is needed to establish the proofs
Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
The quantum chromatic number of a graph is sandwiched between its
chromatic number and its clique number, which are well known NP-hard
quantities. We restrict our attention to the rank-1 quantum chromatic number
, which upper bounds the quantum chromatic number, but is
defined under stronger constraints. We study its relation with the chromatic
number and the minimum dimension of orthogonal representations
. It is known that . We
answer three open questions about these relations: we give a necessary and
sufficient condition to have , we exhibit a class of
graphs such that , and we give a necessary and
sufficient condition to have . Our main tools are
Kochen-Specker sets, collections of vectors with a traditionally important role
in the study of noncontextuality of physical theories, and more recently in the
quantification of quantum zero-error capacities. Finally, as a corollary of our
results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum
chromatic number, we give a family of Kochen-Specker sets of growing dimension.Comment: 12 page
Classical simulation of short-time quantum dynamics
Recent progress in the development of quantum technologies has enabled the
direct investigation of dynamics of increasingly complex quantum many-body
systems. This motivates the study of the complexity of classical algorithms for
this problem in order to benchmark quantum simulators and to delineate the
regime of quantum advantage. Here we present classical algorithms for
approximating the dynamics of local observables and nonlocal quantities such as
the Loschmidt echo, where the evolution is governed by a local Hamiltonian. For
short times, their computational cost scales polynomially with the system size
and the inverse of the approximation error. In the case of local observables,
the proposed algorithm has a better dependence on the approximation error than
algorithms based on the Lieb-Robinson bound. Our results use cluster expansion
techniques adapted to the dynamical setting, for which we give a novel proof of
their convergence. This has important physical consequences besides our
efficient algorithms. In particular, we establish a novel quantum speed limit,
a bound on dynamical phase transitions, and a concentration bound for product
states evolved for short times.Comment: 23 pages, 5 figures, comments welcom
Renormalization: an advanced overview
We present several approaches to renormalization in QFT: the multi-scale
analysis in perturbative renormalization, the functional methods \`a la
Wetterich equation, and the loop-vertex expansion in non-perturbative
renormalization. While each of these is quite well-established, they go beyond
standard QFT textbook material, and may be little-known to specialists of each
other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added,
minor improvements; v3: some changes to sect. 5, refs. adde
What Makes a Distributed Problem Truly Local?
International audienceIn this talk we attempt to identify the characteristics of a task of distributed network computing, which make it easy (or hard) to solve by meansof fast local algorithms. We look at specific combinatorial tasks within the LOCAL model of distributed computation, and rephrase some recent algorithmic results in a framework of constraint satisfaction. Finally, we discuss the issue of efficient computability for relaxed variants of the LOCAL model, involving the so-called non-signaling property
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