Quantum algorithms and approximating polynomials for composed functions with shared inputs

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m-bit Boolean function and consider an n-bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f), we give an algorithm for evaluating F using O~(Q(f)⋅n) quantum queries. This improves on the bound of O(Q(f)⋅n) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f.By recursively applying our composition theorems, we obtain a nearly optimal O~(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0 circuits.As an additional consequence, we show that AC0∘⊕ circuits of depth d+1 require size Ω~(n1/(1−2−d))≥ω(n1+2−d) to compute the Inner Product function even on average. The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).

Deployment the Spoke of a Large-Sized Transformable Refl ector Using a Sequential Optimization Algorithm

The use of large-sized opening reflectors close-packed in spacecraft is associated with spreading the spokes at a given angle, extending the fragments of the spokes and adjusting the shape of the radio-reflecting mesh. The problem of optimizing these processes with automatic output of the reflector to the deployed working state is urgent. The optimal control problem of spreading spokes of a large-sized space-based reflector with respect to bending vibrations is investigated in the article. The optimization process is complicated by ensuring convergence of iterative procedure for control finding. The bending vibrations of the spokes complicate the task of spreading. That makes it difficult to fix spokes when reaching the stops. In this paper the mathematical model of spoke dynamics is improved with respect to spoke’s bend change in length and in time, the model takes into account the presence of stop and retainer devices and an actuator. It is proposed to consider a hierarchy of two target composed functions and develop an algorithm for sequential optimization for a smooth exit to the stops. It is suggested to include the terminal condition for the angular spreading rate in the first criterion. A study was carried out using mathematical simulation for the process of turning the spoke by a given angle at small values of the angular velocity at the final moment of time taking into account bending vibrations. The exact values of the weight coefficients included in the target composed functions are found. Weight coefficients influence on transient processes is investigated. The performance of the algorithm was checked when the value of the optimization interval was changed. The comparison of the results of simulation modeling with control options using the PID controller, application of an algorithm with a predictive model and an algorithm with optimal correction of the control structure, revealed by means of the maximum principle, was carried out. The results of simulation modeling foe spokes spreading process using the sequential optimization algorithm demonstrate the achievement of the required accuracy with permissible tolerance residual vibrations. The developed algorithm of sequential optimization forms control in a real time and it is recommended to use it in more complex solutions under random disturbances using measurement process and optimization of observation intervals.