Both hypercomplex numbers and qubits are mathematical frameworks that are employed in various branches of mathematics and physics. In fact, hypercomplex numbers are a generalization of complex numbers that involve more than one imaginary unit. They include quaternions, which involve three imaginary units, and octonions, which involve seven imaginary units. Quaternions, in particular, can be used to represent rotations in three-dimensional space. Similarly, a single qubit state can be represented as a point on the surface of a three-dimensional sphere(Bloch sphere), and this point can be represented using a unit quaternion.

To see how this works, consider a single qubit state that can be written as a superposition of the basis states |0⟩ and |1⟩:

|ψ⟩ = α|0⟩ + β|1⟩,

where α and β are complex numbers, and |α|2 + |β|2 = 1 (to ensure normalization of the state). This state can be represented on the Bloch sphere as a vector whose length is 1 and whose direction is determined by the complex numbers α and β.

To represent this qubit state using a quaternion, we can define a quaternion q as:

q = α + iβ,

where i is the imaginary unit of the quaternion algebra. The norm of this quaternion is:

||q||2 = |α|2 + |β|2 = 1,

which ensures that it represents a unit quaternion. The vector part of this quaternion corresponds to the direction of the qubit state on the Bloch sphere.

The robust correspondence between quaternions and qubits indicates that quaternions can serve as a feasible approach for depicting the state of a qubit.