scholarly journals A Schematic Definition of Quantum Polynomial Time Computability

2020 ◽  
pp. 1-29
Author(s):  
Tomoyuki Yamakami
Keyword(s):  
Polynomial Time ◽  
Quantum Polynomial ◽  
Definition Of

scholarly journals Mildly Short Vectors in Cyclotomic Ideal Lattices in Quantum Polynomial Time

2021 ◽  
Vol 68 (2) ◽  
pp. 1-26
Author(s):  
Ronald Cramer ◽  
Léo Ducas ◽  
Benjamin Wesolowski
Keyword(s):  
Polynomial Time ◽  
Ideal Lattices ◽  
Quantum Polynomial

scholarly journals EQUIVALENCE OF LABELED MARKOV CHAINS

2008 ◽  
Vol 19 (03) ◽  
pp. 549-563 ◽  
Author(s):  
LAURENT DOYEN ◽  
THOMAS A. HENZINGER ◽  
JEAN-FRANÇOIS RASKIN
Keyword(s):  
Markov Chains ◽  
Markov Decision Processes ◽  
Polynomial Time ◽  
Finite Sequence ◽  
Equivalence Problem ◽  
Decision Processes ◽  
Probabilistic Automata ◽  
Alternative Algorithm ◽  
Markov Decision ◽  
Definition Of

We consider the equivalence problem for labeled Markov chains (LMCs), where each state is labeled with an observation. Two LMCs are equivalent if every finite sequence of observations has the same probability of occurrence in the two LMCs. We show that equivalence can be decided in polynomial time, using a reduction to the equivalence problem for probabilistic automata, which is known to be solvable in polynomial time. We provide an alternative algorithm to solve the equivalence problem, which is based on a new definition of bisimulation for probabilistic automata. We also extend the technique to decide the equivalence of weighted probabilistic automata. Then, we consider the equivalence problem for labeled Markov decision processes (LMDPs), which asks given two LMDPs whether for every scheduler (i.e. way of resolving the nondeterministic decisions) for each of the processes, there exists a scheduler for the other process such that the resulting LMCs are equivalent. The decidability of this problem remains open. We show that the schedulers can be restricted to be observation-based, but may require infinite memory.

scholarly journals Alternating Turing machines and the analytical hierarchy

10.29007/t77g ◽  
2018 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Recursion Theory ◽  
Turing Machines ◽  
Arithmetical Hierarchy ◽  
Definition Of ◽  
Alternating Turing Machines ◽  
Insight Into ◽  
Computably Enumerable

We use notions originating in Computational Complexity to provide insight into the analogies between computational complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is Pi-1-1. Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyper-arithmetical (Delta-1-1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy..The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the correspondence between the polynomial-time and the arithmetical hierarchies, as well as that between the computably-enumerable, the inductive (Pi-1-1), and the PSpace languages.

scholarly journals Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 106 ◽  
Author(s):  
Tomoyuki Morimae ◽  
Yuki Takeuchi ◽  
Harumichi Nishimura
Keyword(s):  
Quantum Computing ◽  
Polynomial Time ◽  
Probability Distributions ◽  
Quantum Circuits ◽  
Universal Models ◽  
Reversible Circuit ◽  
Universal Quantum ◽  
Quantum Polynomial ◽  
Output Probability ◽  
Probabilistic Polynomial Time

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.

Quantum polynomial-time fixed-point attack for RSA

2018 ◽  
Vol 15 (2) ◽  
pp. 25-32 ◽  
Author(s):  
Yahui Wang ◽  
Huanguo Zhang ◽  
Houzhen Wang
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Quantum Polynomial

scholarly journals LATTICED SIMULATION RELATIONS AND GAMES

2010 ◽  
Vol 21 (02) ◽  
pp. 167-189 ◽  
Author(s):  
ORNA KUPFERMAN ◽  
YOAD LUSTIG
Keyword(s):  
Formal Methods ◽  
Polynomial Time ◽  
Partially Ordered ◽  
Multiple View ◽  
A Value ◽  
Lattice Element ◽  
Kripke Structures ◽  
Simulation Relations ◽  
Truth Value ◽  
Definition Of

Multi-valued Kripke structures are Kripke structures in which the atomic propositions and the transitions are not Boolean and can take values from some set. In particular, latticed Kripke structures, in which the elements in the set are partially ordered, are useful in abstraction, query checking, and reasoning about multiple view-points. The challenges that formal methods involve in the Boolean setting are carried over, and in fact increase, in the presence of multi-valued systems and logics. We lift to the latticed setting two basic notions that have been proven useful in the Boolean setting. We first define latticed simulation between latticed Kripke structures. The relation maps two structures M1 and M2 to a lattice element that essentially denotes the truth value of the statement "every behavior of M1 is also a behavior of M2". We show that latticed-simulation is logically characterized by the universal fragment of latticed µ-calculus, and can be calculated in polynomial time. We then proceed to defining latticed two-player games. Such games are played along graphs in which each transition have a value in the lattice. The value of the game essentially denotes the truth value of the statement "the ∨-player can force the game to computations that satisfy the winning condition". An earlier definition of such games involved a zig-zagged traversal of paths generated during the game. Our definition involves a forward traversal of the paths, and it leads to better understanding of multi-valued games. In particular, we prove a min-max property for such games, and relate latticed simulation with latticed games.

POINT AND LINE SEGMENT RECONSTRUCTION FROM VISIBILITY INFORMATION

2000 ◽  
Vol 10 (02) ◽  
pp. 189-200 ◽  
Author(s):  
S. K. WISMATH
Keyword(s):  
Polynomial Time ◽  
Line Segment ◽  
Related Problem ◽  
Parallel Line ◽  
Visibility Graph ◽  
Reconstruction Problem ◽  
System Of Linear Inequalities ◽  
Line Segments ◽  
Definition Of ◽  
Set Of Points

In general, visibility reconstruction problems involve determining a set of objects in the plane that exhibit a specified set of visibility constraints. In this paper, an algorithm is presented for reconstructing a set of parallel line segments from specified visibility information contained in an extended endpoint visibility graph. The algorithm runs in polynomial time and relies on simple vector arithmetic to generate a system of linear inequalities. A related problem, solvable with the same technique, is the point reconstruction problem, in which the cyclic ordering and the x-coordinates of a set of points is specified. A second contribution is the definition of an extension of the visibility graph called the Stab Graph, which contains extra visibility information.

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