Mathematical models of quantum computation

Гузик Вячеслав Филиппович, Гушанский Сергей Михайлович, Кубраков Евгений Сергеевич

A hardware approach for the quantum computing modeling was considered. A common mathematical model of the quantum computer was described; a methodic of mathematical modeling of the quantum computing was shown. During the mathematical methodic description the most resource-intensive parts of the model are allocated. Considered questions connected with data parallelizing with a hardware accelerator, which is used for quantum computer modeling. A common scheme of the accelerator’s work algorithm was shown. Suggested more compact data format for accelerator implementation, that will reduce time costs spent by elementary arithmetic operations. Proposed the use of the optimization algorithm for data minimization during the calculations in the phase of transforming model of quantum register with quantum gates in quantum computer simulators. Shown results of optimization algorithm execution and compared results with classic mathematical approach.

Ryan Bennink, Olena Burkovska, Konstantin Pieper +2 more

This review is designed to introduce mathematicians and computational scientists to quantum computing (QC) through the lens of uncertainty quantification (UQ) by presenting a mathematically rigorous and accessible narrative for understanding how noise and intrinsic randomness shape quantum computational outcomes in the language of mathematics. By grounding quantum computation in statistical inference, we highlight how mathematical tools such as probabilistic modeling, stochastic analysis, Bayesian inference, and sensitivity analysis, can directly address error propagation and reliability challenges in today's quantum devices. We also connect these methods to key scientific priorities in the field, including scalable uncertainty-aware algorithms and characterization of correlated errors. The purpose is to narrow the conceptual divide between applied mathematics, scientific computing and quantum information sciences, demonstrating how mathematically rooted UQ methodologies can guide validation, error mitigation, and principled algorithm design for emerging quantum technologies, in order to address challenges and opportunities present in modern-day quantum high performance and fault-tolerant quantum computing paradigms.

Mingsheng Ying

Peter P. Rohde

Colin P. Williams

Пуховский Валерий Николаевич, Sergey M. Gushanskiy, Viktor Potapov

In the past few decades, there has been an acute problem of creating a quantum computer that uses quantum mechanical effects such as quantum parallelism and quantum entanglement for computations. Using these mechanisms, the quantum computer is able to solve some of the NP-class problems in polynomial time. A hardware approach to modeling quantum computing is considered, and a general mathematical model of the operation of a quantum computer is described, and a technique for mathematical modeling of quantum computing is presented. At the stage of consideration of the method of mathematical modeling, the most resource-intensive parts of the model are indicated. Also, issues related to data parallelization on a hardware accelerator, which are planned to be used to simulate quantum computing, were considered, and algorithms for the operation of this type of accelerator were given. A more compact data format is proposed, which is recommended to be used when implementing the accelerator. Using the proposed format, it is possible to reduce the resources spent on elementary arithmetic operations. The possibility of introducing an optimization algorithm to minimize the time spent in computations, as well as to speed up the execution of quantum algorithms in simulators of a quantum computer at the stage of the effect of quantum gates on the quantum register model, is proposed. The results of the optimization algorithm and its comparison with the classical mathematical approach using a software model are presented.