23,930 research outputs found
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
Exponential renormalization
Moving beyond the classical additive and multiplicative approaches, we
present an "exponential" method for perturbative renormalization. Using Dyson's
identity for Green's functions as well as the link between the Faa di Bruno
Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the
composition of formal power series is analyzed. Eventually, we argue that the
new method has several attractive features and encompasses the BPHZ method. The
latter can be seen as a special case of the new procedure for renormalization
scheme maps with the Rota-Baxter property. To our best knowledge, although very
natural from group-theoretical and physical points of view, several ideas
introduced in the present paper seem to be new (besides the exponential method,
let us mention the notions of counterfactors and of order n bare coupling
constants).Comment: revised version; accepted for publication in Annales Henri Poincar
Distance matrices of a tree: two more invariants, and in a unified framework
Graham-Pollak showed that for the distance matrix of a tree ,
det depends only on its number of edges. Several other variants of ,
including directed/multiplicative/- versions were studied, and always,
det depends only on the edge-data.
We introduce a general framework for bi-directed weighted trees, with
threefold significance. First, we improve on state-of-the-art for all known
variants, even in the classical Graham-Pollak case: we delete arbitrary pendant
nodes (and more general subsets) from the rows/columns of , and show these
minors do not depend on the tree-structure.
Second, our setting unifies all known variants (with entries in a commutative
ring). We further compute in closed form the inverse of , extending a result
of Graham-Lovasz [Adv. Math. 1978] and answering a question of Bapat-Lal-Pati
[Lin. Alg. Appl. 2006].
Third, we compute a second function of the matrix : the sum of all its
cofactors, cof. This was worked out in the simplest setting by
Graham-Hoffman-Hosoya (1978), but is relatively unexplored for other variants.
We prove a stronger result, in our general setting, by computing cof for
minors as above, and showing these too depend only on the edge-data.
Finally, we show our setting is the 'most general possible', in that with
more freedom in the edgeweights, det and cof depend on the tree
structure. In a sense, this completes the study of the invariant det
(and cof) for trees with edge-data in a commutative ring.
Moreover: for a bi-directed graph we prove multiplicative
Graham-Hoffman-Hosoya type formulas for det, cof, . We
then show how this subsumes their 1978 result. The final section introduces and
computes a third, novel invariant for trees and a Graham-Hoffman-Hosoya type
result for our "most general" distance matrix .Comment: 42 pages, 2 figures; minor edits in the proof of Theorems A and 1.1
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