There are many different forms of multidimensional mathematics: complex numbers, quaternions, octonions, and others. All of them share a common property: the absence of zero divisors. In 2019, Kazakh mathematician M. Abenov published a monograph, “Four-Dimensional Mathematics: Methods and Applications,” on four-dimensional mathematics with zero divisors. The four-dimensional mathematics he developed was a natural generalization of one-dimensional and two-dimensional (complex numbers) mathematics. An important advantage of four-dimensional mathematics is the commutativity and associativity of multiplication, which allowed him to develop not only four-dimensional algebra but also four-dimensional mathematical analysis, topology, and differential and integral calculus.

The authors of this paper, building on M. Abenov’s idea, developed a four-dimensional algebra with complex components, as well as eight-dimensional and sixteen-dimensional algebras with real and complex components.

Four-dimensional algebras with complex components ideally describe the pure states of two-qubit quantum systems. Accordingly, eight-dimensional and sixteen-dimensional algebras are adequate mathematical models of three-qubit and four-qubit quantum systems. Entire spaces of two-, three-, and four-qubit gates are defined. The concept of a unitary state of quantum systems is introduced, and an algorithm is derived that allows transitions from one unitary state to any other unitary state in a single step for n-qubit quantum systems.

This fact opens up new possibilities for the application of the generated gates in quantum machine learning.