33 research outputs found
A quantum information approach to statistical mechanics
We review some connections between quantum information and statistical
mechanics. We focus on three sets of results for classical spin models. First,
we show that the partition function of all classical spin models (including
models in different dimensions, different types of many-body interactions,
different symmetries, etc) can be mapped to the partition function of a single
model. Second, we give efficient quantum algorithms to estimate the partition
function of various classical spin models, such as the Ising or the Potts
model. The proofs of these two results are based on a mapping from partition
functions to quantum states and to quantum circuits, respectively. Finally, we
show how classical spin models can be used to describe certain fluctuating
lattices appearing in models of discrete quantum gravity.Comment: 9 pages, 9 figure
Simple universal models capture all classical spin physics
Spin models are used in many studies of complex systems---be it condensed
matter physics, neural networks, or economics---as they exhibit rich
macroscopic behaviour despite their microscopic simplicity.
Here we prove that all the physics of every classical spin model is
reproduced in the low-energy sector of certain `universal models'.
This means that (i) the low energy spectrum of the universal model reproduces
the entire spectrum of the original model to any desired precision, (ii) the
corresponding spin configurations of the original model are also reproduced in
the universal model, (iii) the partition function is approximated to any
desired precision, and (iv) the overhead in terms of number of spins and
interactions is at most polynomial.
This holds for classical models with discrete or continuous degrees of
freedom.
We prove necessary and sufficient conditions for a spin model to be
universal, and show that one of the simplest and most widely studied spin
models, the 2D Ising model with fields, is universal.Comment: v1: 4 pages with 2 figures (main text) + 4 pages with 3 figures
(supplementary info). v2: 12 pages with 3 figures (main text) + 35 pages with
6 figures (supplementary info) (all single column). v2 contains new results
and major revisions (results for spin models with continuous degrees of
freedom, explicit constructions, examples...). Close to published version.
v3: minor typo correcte
Quantum Information Theory and Free Semialgebraic Geometry: One Wonderland Through Two Looking Glasses
We illustrate how quantum information theory and free (i.e. noncommutative)
semialgebraic geometry often study similar objects from different perspectives.
We give examples in the context of positivity and separability, quantum magic
squares, quantum correlations in non-local games, and positivity in tensor
networks, and we show the benefits of combining the two perspectives. This
paper is an invitation to consider the intersection of the two fields, and
should be accessible for researchers from either field.Comment: This overview article will appear in 'Internationale Mathematische
Nachrichten' (IMN), the Journal of the Austrian Mathematical Societ
Describing classical spin Hamiltonians as automata: a new complexity measure
We describe classical spin Hamiltonians as automata and use the
classification of the latter to obtain a new complexity measure of
Hamiltonians. Specifically, we associate a classical spin Hamiltonian to the
formal language consisting of pairs of spin configurations and the
corresponding energy, and classify this language in the Chomsky hierarchy. We
prove that the language associated to (i) effectively zero-dimensional spin
Hamiltonians is regular, (ii) local one-dimensional (1D) spin Hamiltonians is
deterministic context-free, (iii) local two-dimensional (2D) or
higher-dimensional spin Hamiltonians is context-sensitive, and (iv) totally
unbounded spin Hamiltonians is recursively enumerable. It follows that only
highly non-physical spin Hamiltonians [(iv)] correspond to Turing machines. It
also follows that the Ising model without fields is easy or hard if defined on
a 1D or 2D lattice, in contrast to the computational complexity of its ground
state energy problem, where the threshold is found between planar and
non-planar graphs. Our work puts classical spin Hamiltonians at the same level
as automata, and paves the road toward a rigorous comparison of universal spin
models and universal Turing machines.Comment: v3: more results; 24 pages, 9 figures, 9 tables. v2: More results and
extensively rewritten; 18 pages and 7 figures; code of linear bounded
automaton also attached. v1: 13 pages, 7 figures, code of deterministic
pushdown automaton attache
Irreducible forms of Matrix Product States: Theory and Applications
The canonical form of Matrix Product States (MPS) and the associated
fundamental theorem, which relates different MPS representations of a state,
are the theoretical framework underlying many of the analytical results derived
through MPS, such as the classification of symmetry-protected phases in one
dimension. Yet, the canonical form is only defined for MPS without non-trivial
periods, and thus cannot fully capture paradigmatic states such as the
antiferromagnet. Here, we introduce a new standard form for MPS, the
irreducible form, which is defined for arbitrary MPS, including periodic
states, and show that any tensor can be transformed into a tensor in
irreducible form describing the same MPS. We then prove a fundamental theorem
for MPS in irreducible form: If two tensors in irreducible form give rise to
the same MPS, then they must be related by a similarity transform, together
with a matrix of phases. We provide two applications of this result: an
equivalence between the refinement properties of a state and the divisibility
properties of its transfer matrix, and a more general characterisation of
tensors that give rise to matrix product states with symmetries.Comment: 12 page
Continuum limits of Matrix Product States
We determine which translationally invariant matrix product states have a
continuum limit, that is, which can be considered as discretized versions of
states defined in the continuum. To do this, we analyse a fine-graining
renormalization procedure in real space, characterise the set of limiting
states of its flow, and find that it strictly contains the set of continuous
matrix product states. We also analyse which states have a continuum limit
after a finite number of a coarse-graining renormalization steps. We give
several examples of states with and without the different kinds of continuum
limits.Comment: 7 pages, 2 figures. New version: somewhat expanded, some explanations
added. Close to published versio