Search

In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on $n$-bits in time $\text{poly}(n) 2^{n/2}$. In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of $\Omega(2^{n/2})$ has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is \#P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach., Comment: 29+18 pages, 3 figures

Nostradamus, inspired by the French astrologer and reputed seer, is a detailed study exploring relations between environmental factors and changes in the stock market. In this paper, we analyze associative correlation and causation between environmental elements (including natural disasters, climate and weather conditions) and stock prices, using historical stock market data, historical climate data, and various climate indicators such as carbon dioxide emissions. We have conducted our study based on the US financial market, global climate trends, and daily weather records to demonstrate a significant relationship between climate and stock price fluctuation. Our analysis covers both short-term and long-term rises and dips in company stock performances. Lastly, we take four natural disasters as a case study to observe the effect they have on people's emotional state and their influence on the stock market., Comment: 13 pages, 13 figures; updated abstract; updated format to Springer LNCS

We analyze the computational complexity of the video game "CELESTE" and prove that solving a generalized level in it is NP-Complete. Further, we also show how, upon introducing a small change in the game mechanics (adding a new game entity), we can make it PSPACE-complete., Comment: Keywords: complexity analysis, NP completeness, algorithmic analysis, game analysis