The integration of quantum computing into robotics requires mathematical representations that preserve the geometric structure and configuration dependence of robot motion. This paper introduces a structure-preserving quantum representation of rigid-body transformations and serial-manipulator kinematics, establishing a correspondence between classical kinematic models and parameterized quantum circuits. Building on the Bloch-sphere representation, spatial directions are encoded into single-qubit states, while quantum rotations reproduce the ordered composition of reference-frame transformations. Translational contributions are combined through quantum-state aggregation, with their metric scales retained classically, enabling the reconstruction of end-effector position and orientation from Pauli expectation values. The formulation preserves the original joint variables and accommodates different kinematic conventions and joint types. Beyond forward kinematics, we establish that the manipulator geometric Jacobian can be reconstructed from quantum-observable derivatives using the parameter-shift rule, with analytical equivalence to its classical counterpart under ideal readout. To demonstrate the applicability of this representation to robot-level algorithms, we introduce Quantum Closed Loop Inverse Kinematics, a hybrid closed-loop method that exploits quantum-derived pose and Jacobian information. The proposed framework provides a geometrically interpretable foundation for developing and investigating quantum and hybrid computational methods for robotic kinematics and motion.

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