51,936 research outputs found
Modeling Quantum Behavior in the Framework of Permutation Groups
Quantum-mechanical concepts can be formulated in constructive finite terms
without loss of their empirical content if we replace a general unitary group
by a unitary representation of a finite group. Any linear representation of a
finite group can be realized as a subrepresentation of a permutation
representation. Thus, quantum-mechanical problems can be expressed in terms of
permutation groups. This approach allows us to clarify the meaning of a number
of physical concepts. Combining methods of computational group theory with
Monte Carlo simulation we study a model based on representations of permutation
groups.Comment: 8 pages, based on plenary lecture at Mathematical Modeling and
Computational Physics 2017, Dubna, July 3--7, 201
Combinatorial Approach to Modeling Quantum Systems
Using the fact that any linear representation of a group can be embedded into
permutations, we propose a constructive description of quantum behavior that
provides, in particular, a natural explanation of the appearance of complex
numbers and unitarity in the formalism of quantum mechanics. In our approach,
the quantum behavior can be explained by the fundamental impossibility to trace
the identity of indistinguishable objects in their evolution. Any observation
only provides information about the invariant relations between such objects.
The trajectory of a quantum system is a sequence of unitary evolutions
interspersed with observations -- non-unitary projections. We suggest a scheme
to construct combinatorial models of quantum evolution. The principle of
selection of the most likely trajectories in such models via the large numbers
approximation leads in the continuum limit to the principle of least action
with the appropriate Lagrangians and deterministic evolution equations.Comment: 12 pages (12+ for version 2), based on plenary lecture at
Mathematical Modeling and Computational Physics 2015, Stara Lesna, High Tatra
Mountains, Slovakia, Jully 13--17, 201
Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems
We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise
sense, to probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs).
We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD, we can approximate its termination probabilities (i.e., its matrix) to within bits of precision (i.e., within additive error ), in time polynomial in \underline{both} the encoding size of the QBD and in , in the unit-cost rational arithmetic RAM model of computation. Specifically,
we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved.
In fact, we observe (based on recent results) that for the more general TL-QBDs such a polynomial upper bound on Newton's method fails badly. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for strongly connected monotone systems of polynomial equations.
We show that the quantitative termination decision problem for QBDs (namely, ``is ?'') is at least as hard as long standing open problems in the complexity of exact numerical computation, specifically the square-root sum problem. On the other hand, it follows from our earlier results for RMCs that any non-trivial approximation of termination probabilities for TL-QBDs is sqrt-root-sum-hard
- …