Search Results (3)

We develop detailed quaternionic and octonionic frameworks for quantum computation grounded on normed division algebras. Our central result is to prove the polynomial computational equivalence of quaternionic and complex quantum models: Computation over $\mathbb{H}$ is polynomially equivalent to the standard complex quantum circuit model and hence captures the same complexity class BQP up to polynomial reductions. Over $\mathbb{H}$, we construct a complete model—quaternionic qubits on right $\mathbb{H}$-modules with quaternion-valued inner products, unitary dynamics, associative tensor products, and universal gate sets—and establish polynomial equivalence with the standard complex model; routes for implementation at fidelities exceeding $99\%$ via pulse-level synthesis on current hardware are discussed. Over $\mathbb{O}$, non-associativity yields path-dependent evolution, ambiguous adjoints/inner products, non-associative tensor products, and possible failure of energy conservation outside associative sectors. We formalize these obstructions and systematize four mitigation strategies: Confinement to associative subalgebras, $G_2$-invariant codes, dynamical decoupling of associator terms, and a seven-factor algebraic decomposition for gate synthesis. The results delineate the feasible quaternionic regime from the constrained octonionic landscape and point to applications in symmetry-protected architectures, algebra-aware simulation, and hypercomplex learning. Full article

The vast majority of existing neural networks operate by rules set within the algebra of real numbers. However, as theoretical understanding of the fundamentals of neural networks and their practical applications grow stronger, new problems arise, which require going beyond such algebra. Various tasks come to light when the original data naturally have complex-valued formats. This situation is encouraging researchers to explore whether neural networks based on complex numbers can provide benefits over the ones limited to real numbers. Multiple recent works have been dedicated to developing the architecture and building blocks of complex-valued neural networks. In this paper, we generalize models by considering other types of hypercomplex numbers of the second order: dual and double numbers. We developed basic operators for these algebras, such as convolution, activation functions, and batch normalization, and rebuilt several real-valued networks to use them with these new algebras. We developed a general methodology for dual and double-valued gradient calculations based on Wirtinger derivatives for complex-valued functions. For classical computer vision (CIFAR-10, CIFAR-100, SVHN) and signal processing (G2Net, MusicNet) classification problems, our benchmarks show that the transition to the hypercomplex domain can be helpful in reaching higher values of metrics, compared to the original real-valued models. Full article

Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers; (3) a method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N; and (4) the possibility of constructing a normed commutative algebra of other N-dimensional hypercomplex numbers with division over the field of real numbers. The article also shows that when using specific forms of representation of unit vectors, the product of vectors has the property of commutativity. Normed commutative algebras of N-dimensional hypercomplex numbers can be widely used to solve many topical scientific problems in the field of theoretical physics for modeling force fields with various types of symmetry, in cryptography for developing a number of new cryptographic programs using hypercomplex number algebras with different values of dimension, and in many other areas of fundamental and applied sciences. Full article