*Proceeding Paper* **Evaluation of Computer Technologies for Calculation of Exact Approximations of Statistics Probability Distributions †**

**Andrey Melnikov 1, Ilya Levin 2, Aleksey Dordopulo 2,\* and Lyubov Slasten <sup>2</sup>**

	- levin@superevm.ru (I.L.); lmslasten@yandex.ru (L.S.)

**\*** Correspondence: dordopulo@superevm.ru; Tel.: +7-8634-612111

† Presented at the 15th International Conference "Intelligent Systems" (INTELS'22), Moscow, Russia, 14–16 December 2022.

**Abstract:** This paper is devoted to the evaluation of the hardware resources of computer systems for solving a computationally expensive problem such as the calculation of the probability distributions of statistics by the second multiplicity method based on Δ-exact approximations of samples with a size of 320–1280 characters and an alphabet power of 128–256 characters. The accuracy is Δ = 10−<sup>5</sup> and the total solution time should not exceed 30 days or 2.592 <sup>×</sup> 106 seconds for 24/7 computing. Owing to the use of the properties of the second multiplicity method, the computational complexity of the calculations can be brought within the range of 9.68 <sup>×</sup> <sup>10</sup>22–1.60 <sup>×</sup> <sup>10</sup><sup>52</sup> operations with the number of tested vectors at 6.50 <sup>×</sup> <sup>10</sup>23–1.39 <sup>×</sup> <sup>10</sup>50. The solution of this problem for the specified parameters of samples during the given time requires hardware resources which cannot be provided by modern computer technology such as processors, graphics accelerators and programmable logic integrated circuits. Therefore, in this paper, we analyze the possibilities of promising quantum and photon technologies for solving the problem with the given parameters. The main advantage of quantum computer systems is the high speed of calculations for all possible parameter values. However, quantum acceleration will not be achieved to calculate the probability distributions of statistics due to the need to check all the obtained solutions. Here, the number of obtained solutions corresponds to the dimension of the problem. In addition, due to the current development level of quantum hardware components, it is impossible to create and use 120 qubit quantum computers for the solution of the considered problem. Photon computers can provide high computation speeds at low power consumptions and require the smallest number of nodes to solve the considered problem. However, unsolved problems with the physical implementation of efficient memory elements and the lack of available hardware components make the use of photon computer technologies impossible for calculating the probability distributions of statistics in the near future (5–7 years). Therefore, it is most reasonable to use hybrid computer systems containing nodes of different architectures. To solve the problem on various hardware platforms (general purpose processors, GPUs and FPGAs) and configurations of hybrid computer systems, we suggest to use the architecture-independent high-level programming language SET@L. The language combines the representation of calculations as sets and collections (based on the alternative set theory of P. Vopenka), the absolutely parallel form of the problem represented as an information graph and the paradigm of aspect-oriented programming.

**Keywords:** probability; statistic; exact distribution; exact approximation; algorithmic complexity; quantum calculations; photonic technologies; architecture-independent programming; Set@l language
