344 research outputs found
Feynman integrals and hyperlogarithms
We study Feynman integrals in the representation with Schwinger parameters
and derive recursive integral formulas for massless 3- and 4-point functions.
Properties of analytic (including dimensional) regularization are summarized
and we prove that in the Euclidean region, each Feynman integral can be written
as a linear combination of convergent Feynman integrals. This means that one
can choose a basis of convergent master integrals and need not evaluate any
divergent Feynman graph directly.
Secondly we give a self-contained account of hyperlogarithms and explain in
detail the algorithms needed for their application to the evaluation of
multivariate integrals. We define a new method to track singularities of such
integrals and present a computer program that implements the integration
method.
As our main result, we prove the existence of infinite families of massless
3- and 4-point graphs (including the ladder box graphs with arbitrary loop
number and their minors) whose Feynman integrals can be expressed in terms of
multiple polylogarithms, to all orders in the epsilon-expansion. These
integrals can be computed effectively with the presented program.
We include interesting examples of explicit results for Feynman integrals
with up to 6 loops. In particular we present the first exactly computed
counterterm in massless phi^4 theory which is not a multiple zeta value, but a
linear combination of multiple polylogarithms at primitive sixth roots of unity
(and divided by ). To this end we derive a parity result on the
reducibility of the real- and imaginary parts of such numbers into products and
terms of lower depth.Comment: PhD thesis, 220 pages, many figure
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states
Fermionic Linear Optics (FLO) is a restricted model of quantum computation
which in its original form is known to be efficiently classically simulable. We
show that, when initialized with suitable input states, FLO circuits can be
used to demonstrate quantum computational advantage with strong hardness
guarantees. Based on this, we propose a quantum advantage scheme which is a
fermionic analogue of Boson Sampling: Fermion Sampling with magic input states.
We consider in parallel two classes of circuits: particle-number conserving
(passive) FLO and active FLO that preserves only fermionic parity and is
closely related to Matchgate circuits introduced by Valiant. Mathematically,
these classes of circuits can be understood as fermionic representations of the
Lie groups and . This observation allows us to prove our main
technical results. We first show anticoncentration for probabilities in random
FLO circuits of both kind. Moreover, we prove robust average-case hardness of
computation of probabilities. To achieve this, we adapt the
worst-to-average-case reduction based on Cayley transform, introduced recently
by Movassagh, to representations of low-dimensional Lie groups. Taken together,
these findings provide hardness guarantees comparable to the paradigm of Random
Circuit Sampling.
Importantly, our scheme has also a potential for experimental realization.
Both passive and active FLO circuits are relevant for quantum chemistry and
many-body physics and have been already implemented in proof-of-principle
experiments with superconducting qubit architectures. Preparation of the
desired quantum input states can be obtained by a simple quantum circuit acting
independently on disjoint blocks of four qubits and using 3 entangling gates
per block. We also argue that due to the structured nature of FLO circuits,
they can be efficiently certified.Comment: 65 pages, 13 figures, 1 table, v2: improved discussion and narrative,
numerics about anticoncentration added, references updated, comments and
suggestions are welcom
Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum
A quantum version of transition state theory based on a quantum normal form
(QNF) expansion about a saddle-centre-...-centre equilibrium point is
presented. A general algorithm is provided which allows one to explictly
compute QNF to any desired order. This leads to an efficient procedure to
compute quantum reaction rates and the associated Gamov-Siegert resonances. In
the classical limit the QNF reduces to the classical normal form which leads to
the recently developed phase space realisation of Wigner's transition state
theory. It is shown that the phase space structures that govern the classical
reaction d ynamicsform a skeleton for the quantum scattering and resonance
wavefunctions which can also be computed from the QNF. Several examples are
worked out explicitly to illustrate the efficiency of the procedure presented.Comment: 132 pages, 31 figures, corrected version, Nonlinearity, 21 (2008)
R1-R11
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