Tag Archives: Computational Complexity

Quantum encoding is suitable for matched filtering [CL]

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http://arxiv.org/abs/2204.04159


Matched filtering is a powerful signal searching technique used in several employments from radar and communications applications to gravitational-wave detection. Here we devise a method for matched filtering with the use of quantum bits. Our method’s asymptotic time complexity does not depend on template length and, including encoding, is $\mathcal{O}(L(\log_2L)^2)$ for a data with length $L$ and a template with length $N$, which is classically $\mathcal{O}(NL)$. Hence our method has superior time complexity over the classical computation for long templates. We demonstrate our method with real quantum hardware on 4 qubits and also with simulations.

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D. Veske, C. Tüysüz, M. Amico, et. al.
Mon, 11 Apr 22
43/61

Comments: 4 pages + 3 figures. Comments are welcome

Numerical verification of the microscopic time reversibility of Newton's equations of motion: Fighting exponential divergence [IMA]

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http://arxiv.org/abs/1802.00970


Numerical solutions to Newton’s equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible. This is due to the exponential growth of errors introduced by the integration scheme and the numerical round-off in the least significant figure. This secular growth of error is sometimes attributed to the increase in entropy of the system even though Newton’s equations of motion are strictly time reversible. We demonstrate that when numerical errors are reduced to below the physical perturbation and its exponential growth during integration the microscopic reversibility is retrieved. Time reversibility itself is not a guarantee for a definitive solution to the chaotic N-body problem. However, time reversible algorithms may be used to find initial conditions for which perturbed trajectories converge rather than diverge. The ability to calculate such a converging pair of solutions is a striking illustration which shows that it is possible to compute a definitive solution to a highly unstable problem. This works as follows: If you (i) use a code which is capable of producing a definitive solution (and which will therefore handle converging pairs of solutions correctly), (ii) use it to study the statistical result of some other problem, and then (iii) find that some other code produces a solution S with statistical properties which are indistinguishable from those of the definitive solution, then solution S may be deemed veracious.

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S. Zwart and T. Boekholt
Tue, 6 Feb 18
28/62

Comments: Accepted for publication in Communications in Nonlinear Science and Numerical Simulation. Calculations are performed with Brutus as part of the AMUSE framework. Simultion data will become available online

Cross-identification of stellar catalogs with multiple stars: Complexity and Resolution [CL]

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http://arxiv.org/abs/1710.09417


In this work, I present an optimization problem which consists of assigning entries of a stellar catalog to multiple entries of another stellar catalog such that the probability of such assignment is maximum. I prove that the problem is NP-Hard and show a way of modeling this problem as a maximum weighted stable set problem. A real application is solved in this way through integer programming.

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D. Severin
Fri, 27 Oct 17
27/60

Comments: N/A

On the minimal accuracy required for simulating self-gravitating systems by means of direct N-body methods [IMA]

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http://arxiv.org/abs/1402.6713


The conservation of energy, linear momentum and angular momentum are important drivers for our physical understanding of the evolution of the Universe. These quantities are also conserved in Newton’s laws of motion under gravity \citep{Newton:1687}. Numerical integration of the associated equations of motion is extremely challenging, in particular due to the steady growth of numerical errors (by round-off and discrete time-stepping, \cite{1981PAZh….7..752B,1993ApJ…415..715G,1993ApJ…402L..85H,1994LNP…430..131M}) and the exponential divergence \citep{1964ApJ…140..250M,2009MNRAS.392.1051U} between two nearby solution. As a result, numerical solutions to the general N-body problem are intrinsically questionable \citep{2003gmbp.book…..H,1994JAM….61..226L}. Using brute force integrations to arbitrary numerical precision we demonstrate empirically that ensembles of different realizations of resonant 3-body interactions produce statistically indistinguishable results. Although individual solutions using common integration methods are notoriously unreliable, we conjecture that an ensemble of approximate 3-body solutions accurately represents an ensemble of true solutions, so long as the energy during integration is conserved to better than 1/10. We therefore provide an independent confirmation that previous work on self-gravitating systems can actually be trusted, irrespective of the intrinsic chaotic nature of the N-body problem.

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S. Zwart and T. Boekholt
Fri, 28 Feb 14
12/54