arXiv:2610.33715

Quantum Closed Loop Inverse Kinematics Through Hilbert-Space Encoding of Robot Kinematics

Andrea Morghen, Pierluigi Arpenti, Roberto Schiattarella, Giovanni Acampora, Bruno Siciliano

Published Oct 24, 2026• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2610.33715

Related papers

Structure-Preserving Quantum Circuit Architectures for Robot Kinematics

Andrea Morghen, Pierluigi Arpenti, Roberto Schiattarella +2 more

Structured spatial data require quantum encodings that preserve geometric relations, expose measurable observables, and remain implementable on finite-depth hardware. This work introduces a quantum representation and circuit architecture for rigid-body transformations and specializes it to Denavit--Hartenberg kinematics of serial open-chain manipulators. Each translational contribution is factorized into a classical metric magnitude and a signed unit direction encoded by a single-qubit Bloch vector, while parameterized rotations reproduce the ordered propagation of frame directions. A selector register prepares probabilities proportional to the contribution magnitudes, and the reduced state of a designated readout qubit encodes their normalized weighted sum. The retained classical scale then reconstructs the metric end-effector position. Two additional readout qubits encode terminal-frame axes, providing a compact and geometrically interpretable pose interface. At the ideal expectation-value level, measured Pauli observables reproduce the corresponding classical kinematic quantities. Alternative circuit architectures realize the same representation with different tradeoffs in qubit count, circuit depth, controlled operations, and measurement requirements. Validation on a serial manipulator yields numerically negligible position and orientation reconstruction errors under ideal simulation. Finite-shot simulations, noisy executions, transpilation analysis, and a hardware demonstration further characterize statistical error, noise sensitivity, and implementation overhead without asserting computational advantage.

cs.ROCross-listedarXiv:2609.16089Sep 15, 20260 citations

Quantum computation for robot posture optimization

Takuya Otani, Atsuo Takanishi, Nobuyuki Hara +2 more

Quantum computing has gained attention for its potential to surpass classical computing in large-scale computations. In this study, we propose a method for solving the inverse kinematics of a robot using quantum computing. The approach leverages the ability of qubits to represent points on a sphere in three-dimensional space. Forward kinematics calculations are performed using qubits that encode the posture of each robot link, while inverse kinematics solutions are obtained through iterative optimization on a classical computer. Furthermore, we demonstrate that the robot's end-effector position can be effectively represented using a 2-qubit rotation gate, where the root joint angle influences the tip joint angle, resulting in accelerated convergence during inverse kinematics optimization. The proposed method was validated on an actual quantum computer, confirming its feasibility and efficiency. These findings suggest that hybrid quantum-classical approaches can enhance robotic motion planning and optimization in future quantum computing applications.

Scientific ReportsAug 8, 20253 citations

A novel quantum model of forward kinematics based on quaternion/Pauli gate equivalence: Application to a six-jointed industrial robotic arm

Mehdi Fazilat, Nadjet Zioui, Jonathan St-Arnaud

A novel quantum-based kinematic model for computing the position and the orientation of the six-jointed IRB 140 ABB robotic arm is presented. Position is determined through successive elementary translations and rotations from the end effector down to the base frame, whereas orientation is derived from quaternion representation. The model is based on the equivalence between quaternions and Pauli matrices or gates. Based on comparison to experimental results, the quantum kinematic model is valid. Its main advantage is that it uses only one qubit to model both position and orientation, which represents a major saving in terms of computing resources and time. This model could guide development of other quantum models and simulation software to be used in future quantum computers and could also inform classical computing theory for robotic arm kinematics modeling.

Results in EngineeringMar 26, 202236 citations

Quantum Machine Learning and Grover's Algorithm for Quantum Optimization of Robotic Manipulators

Hassen Nigatu, Gaokun Shi, Jituo Li +3 more

Optimizing high-degree-of-freedom robotic manipulators requires searching complex, high-dimensional configuration spaces, a task that is computationally challenging for classical methods. This paper introduces a quantum-native framework that integrates Quantum Machine Learning (QML) with Grover's algorithm to solve kinematic optimization problems efficiently. A parameterized quantum circuit is trained to approximate the forward kinematics model, which then constructs an oracle to identify optimal configurations. Grover's algorithm leverages this oracle to provide a quadratic reduction in search complexity. Demonstrated on simulated 1-DoF, 2-DoF, and dual-arm manipulator tasks, the method achieves significant speedups-up to 93x over classical optimizers like Nelder-Mead-as problem dimensionality increases. This work establishes a foundational, quantum-native framework for robot kinematic optimization, effectively bridging quantum computing and robotics problems.

arXiv:2509.07216IEEE Robotics and Automation LettersOct 31, 20251 citation

Quantum Computing and the Bloch Sphere: Mathematical Framework, Single-Qubit Transformations, and Circuit Logic

Chinnaraji Annamalai

Quantum computing leverages fundamental principles of quantum mechanics to perform computations far exceeding classical architectures. At the core of single-qubit quantum state manipulation is the Bloch sphere, a geometric representation mapping the state space of two-level quantum systems onto a three-dimensional unit sphere. This paper presents a comprehensive theoretical framework of qubits, starting from single-qubit geometric state vectors and expanding to rigorous multi-qubit mathematical formulations. Furthermore, it analyzes single-qubit quantum gates as unitary rotations on the Bloch sphere, evaluates multi-qubit system scaling, and explicitly proves state normalization conditions using Born's rule across multi-dimensional Hilbert spaces.

Sep 14, 20260 citations

Motion-First Reconstruction of Quantum Kinematics

A. R. Wells

Motion-First Reconstruction of Quantum Kinematics establishes the algebraic conditions under which a distinguishability theory admits Hilbert-space representation, without invoking probabilities, measurement postulates, or dynamical assumptions. The reconstruction begins from a primitive motion-history domain and a distinguishability functional defined on finite-support multiplicity bundles of histories. From explicitly stated structural constraints—most centrally the third-order obstruction principle and signed residual nonnegativity—the distinguishability functional is shown to admit a symmetric quadratic kernel representation. This kernel is proven to be positive semidefinite, allowing application of the Moore–Aronszajn construction to obtain a canonical real Hilbert space in which distinguishability corresponds to squared norm differences. The representation is canonical up to isometric equivalence and preserves the distinguishability structure exactly. Distinguishability-preserving transformations are shown to correspond to orthogonal operators, and tensor-product factorization follows under an explicit subsystem independence condition. Under an additional cyclic distinguishability symmetry, a compatible complex structure emerges, yielding the standard complex Hilbert-space framework used in quantum theory. The results isolate the representational kinematic core of Hilbert-space structure as a consequence of algebraic distinguishability constraints rather than as an independent postulate. The reconstruction clarifies the precise structural boundary separating distinguishability theories that admit Hilbert-space representation from those requiring more general representational geometries. This work is representational in scope. It does not assume or derive measurement postulates, probability rules, observable algebras, or dynamical evolution laws. Instead, it establishes the minimal algebraic conditions under which Hilbert-space structure becomes representationally inevitable. These results provide a structural foundation for subsequent work investigating how physical compositional principles on motion histories may give rise to the distinguishability constraints identified here.

Feb 15, 20260 citations