Standard quantum information is formulated over complex Hilbert spaces, where the pure state space of a single qubit is geometrically encoded by the first Hopf fibration S3→S2. Beyond this familiar setting, the normed division algebras C, H and O provide a hierarchy of increasingly rich algebraic and topological structures that has motivated several extensions of quantum theory and quantum computation. This review synthesizes the literature connecting quaternionic and octonionic frameworks, Clifford algebras, spinors, projective spaces and Hopf fibrations. We emphasize a central conceptual point that is often blurred in the literature: the second and third Hopf fibrations play two distinct roles, namely, (i) as kinematical descriptions of hypercomplex single-particle state spaces such as the quaternionic projective line HP1≃S4, and (ii) as entanglement-sensitive descriptions of multi-qubit complex systems, especially two- and three-qubit Hilbert spaces. On the algebraic side, Clifford and geometric algebras provide a natural language for rotations, spinors, and gate synthesis, while quaternionic Hilbert modules furnish a mathematically consistent extension of standard qubit kinematics and dynamics. By contrast, octonionic models face major obstructions due to non-associativity, which affects inner products, tensor products, spectral theory and circuit composition. We therefore distinguish carefully between robust results, partial constructions and speculative directions. The outcome is a unified geometric review of hypercomplex quantum information, together with a map of open problems at the interface of topology, noncommutative algebra, and quantum computation.
Standard quantum information is formulated over complex Hilbert spaces, where the pure state space of a single qubit is geometrically encoded by the first Hopf fibration S3→S2. Beyond this familiar setting, the normed division algebras C, H and O provide a hierarchy of increasingly rich algebraic and topological structures that has motivated several extensions of quantum theory and quantum computation. This review synthesizes the literature connecting quaternionic and octonionic frameworks, Clifford algebras, spinors, projective spaces and Hopf fibrations. We emphasize a central conceptual point that is often blurred in the literature: the second and third Hopf fibrations play two distinct roles, namely, (i) as kinematical descriptions of hypercomplex single-particle state spaces such as the quaternionic projective line HP1≃S4, and (ii) as entanglement-sensitive descriptions of multi-qubit complex systems, especially two- and three-qubit Hilbert spaces. On the algebraic side, Clifford and geometric algebras provide a natural language for rotations, spinors, and gate synthesis, while quaternionic Hilbert modules furnish a mathematically consistent extension of standard qubit kinematics and dynamics. By contrast, octonionic models face major obstructions due to non-associativity, which affects inner products, tensor products, spectral theory and circuit composition. We therefore distinguish carefully between robust results, partial constructions and speculative directions. The outcome is a unified geometric review of hypercomplex quantum information, together with a map of open problems at the interface of topology, noncommutative algebra, and quantum computation.
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 H is polynomially equivalent to the standard complex quantum circuit model and hence captures the same complexity class BQP up to polynomial reductions. Over H, we construct a complete model—quaternionic qubits on right 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 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, G2-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.
Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of describing the state space for two-qubit quantum states. In this research, we have studied the geometry of quantum states for a two-level quantum system, along with the correspondence between complex Hopf bundle and quaternionic Hopf bundle. In the first part of our study, we investigate the behaviour of local coordinates for both Hopf bundles under different degree of entanglement such as entangled states and non-entangled states. Fubini-Study metric for complex projective space is also obtained. Its form suggests that for the intermediate entangled states, complex projective space CP3 can be described as a set of flat three-tori parametrized by a three-sphere. The local inhomogeneous coordinate of CP3 (base space of complex Hopf fibration) is found to carry the description of both subsystems A and B, whereas in the case of maximally entangled state, basis elements of local coordinates CP3 is not linearly independent of each other. Next, we construct a base space map between CP3 and S4, which is denoted as η map. After the mapping, we obtained phases in the base space manifold, different sections and coordinate charts are related by transition functions. We found that there is an inherent symmetry of coordinate transformation corresponds to different sections of CP3, which is expressed in terms of transition functions having the U(1) group structure. Also under η map, phases and transition function in S4 is doubled over that of CP3, indicating subtle symmetric changes after the mapping. The base space coordinates of quaternionic Hopf bundle are consist of two parts, whereby the first part is invariant to the coordinate transformation in CP3 but sensitive to the coordinate transformation in S4.
Geometric and topological methods play an increasingly important role in quantum information science and quantum computation. Beyond the conventional Hilbert space formalism, a variety of mathematical frameworks, including group representations, mapping class groups, modular tensor categories, and character varieties, have been proposed to describe quantum states and quantum gates in a structurally robust manner. This review surveys the development of topological and geometric approaches to quantum information, with particular emphasis on representations of fundamental groups into SL(2,C), their associated character varieties, and the algebraic surfaces arising from trace coordinates, such as Fricke and Cayley cubic surfaces. These structures provide a geometric encoding of quantum degrees of freedom and offer alternative perspectives on topological quantum computing beyond anyon-based models. We also examine connections with integrable systems and isomonodromic deformations, where Painlevé equations and monodromy data supply a dynamical viewpoint on quantum state evolution. A critical comparison is provided with other geometric and topological approaches to quantum information, including geometric quantum mechanics, information geometry, tensor network geometry, and category-theoretic formulations. By synthesising results from topology, algebraic geometry, and mathematical physics, this review aims to clarify the conceptual landscape of topological quantum information geometry and to identify open problems and emerging directions in the field.
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that case. Generalization of formal complex plane to an an arbitrary plane in three dimensions and of usual Hopf fibration to the map generated by an arbitrary unit value element of even sub-algebra of the three-dimensional Geometric Algebra are resulting in more profound description of qubits compared to quantum mechanical Hilbert space formalism.
In this thesis, the complex Clifford algebra and its representation are introduced as a foundation of our geometrical description of quantum states. An inner product and a metric are defined on the matrix representation in a natural way, so that it has the structure of a Hilbert space and a Riemannian manifold. In particular, the set of D x D pure density operators which de-scribe n-qubit pure states, where D = 2\\(^n\\) and n is a natural number, is characterised as the set of D x D positive operators whose associated quadratic forms satisfy the Fierz identities and the normalisation condition. Then this set is, in fact, nothing but the complex projective space ‚Äövëvá‚Äövëv¥\\(^{D-1}\\). Further-more, the fibre bundle S\\(^{2D-1}\\) ‚Äövúv¿ ‚Äövëvá‚Äövëv¥\\(^{D-1}\\) is derived from the construction of the space of D x D pure density operators, and its bundle projection provides the natural correspondence between the two formulations of quantum mechanics, namely, the state vector formulation and the density operator formulation. The single qubit state case and the two-qubit pure state case are ex-plored intensively as examples. For the single qubit case, both pure and mixed states are discussed explicitly in terms of the Clifford algebra description along with the `mixedness' of a single qubit state, and the Riemannian structure of the space of single qubit pure density operators is examined. For the two-qubit case, the Clifford algebra description is discussed explicitly in relation to the Fierz identities, and, by employing the reduced density operators, it is shown that the concurrence of a two-qubit pure density operator coincides with the 'mixedness' of the reduced density operators. In addition, from this viewpoint, the \EPR paradox\" is examined as a more geometrically precise illustrated example."
The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.