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.
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.
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.
Understanding the dynamics of interacting quantum systems is a central goal of quantum simulation. For geometrically local Hamiltonians, locality suggests that digital simulation should retain much of the parallelism of the underlying dynamics. For a target accuracy ε, we construct a nearest-neighbor quantum circuit that simulates the dynamics of an n-qubit, time-independent, finite-range Hamiltonian with normalized local interaction strength on a fixed-dimensional lattice for time T≥1, with circuit depth O(Tlog(nT/ε)) and gate count O(nTlog(nT/ε)). This reduces the polylogarithmic overhead of Haah, Hastings, Kothari, and Low to a single logarithm. On logarithmic-scale blocks from the Lieb--Robinson decomposition, a shallow but highly accurate implementation is essential for reducing the circuit depth. Our key observation is that spatial localization also controls the cost of high precision. In each block, the error of a constant-order product formula is compensated for by a correction, yielding exponential accuracy. The correction can be implemented with depth proportional to the block diameter, introducing no additional asymptotic depth scale beyond that required by spatial localization.
Furthermore, we prove an unconditional circuit-depth lower bound for uniform XXZ Hamiltonians: at fixed evolution time T in the stated regime and sufficiently small fixed error ε, any nearest-neighbor quantum circuit simulating these dynamics requires depth Ω(logn/loglogn). This matches our O(logn) upper bound up to a doubly logarithmic factor, establishing near-optimal circuit depth for lattice Hamiltonian simulation in this regime. Together, these results clarify the role of locality and precision in determining the circuit complexity of quantum dynamics and suggest broader principles for spatially constrained quantum computation.
The quantum lattice Boltzmann scheme of Succi and Dellar advances a four-component Dirac spinor on a lattice by a fixed sequence of local, exactly norm-preserving operations: a basis rotation, a collision, a streaming shift, and the inverse rotation. While this correspondence has been noted qualitatively, it has remained unclear whether it can be realized exactly and unitarily as a gate-model quantum circuit. This unitarity is a structural property of the scheme, which should, in principle, enable an operator-by-operator port onto quantum hardware. We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme: the fixed rotation gates, the collision gate, the streaming shift as a controlled increment on a position register, the position-dependent potential as a phase oracle, and periodic and reflecting boundary conditions as unitary circuits. Together, these constructions provide the first complete demonstration that the scheme is exactly representable in the gate model without any approximation beyond the classical method itself. We then compose them into single-axis, two- and three-dimensional time steps. On a state-vector emulator the resulting circuits reproduce the classical QLB solver to machine precision, thus verifying our quantum implementation. Our aim is representability and correctness rather than computational speedup; questions of state preparation, asymptotic cost, and hardware noise are identified as open avenues for future work. Appendices restore physical units, derive the scheme's Zitterbewegung in closed form and compare it with a trapped-ion simulation, and execute a reduced two-site instance on a Rigetti superconducting processor, where choosing the qubits from the live calibration lowers the state-preparation error. All operators, circuits, and figures are reproducible from the open-source quantumKineticMethods library.
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.
We introduce Quantum Orthogonal Separable Physics-Informed Neural Networks (QO-SPINNs), which replace selected dense hidden transformations in physics-informed PDE models with quantum-compatible orthogonal maps. A pyramidal reconfigurable-beamsplitter circuit realizes these maps with Θ(d) circuit depth. Under the stated signed-readout access model and fixed per-call failure probability, a width-d map has sequential access-cost upper bound O(dlogd/ε2), which is asymptotically smaller than the Θ(d2) cost of conventional direct dense multiplication when ε(d)−2logd=o(d). Because unary state preparation discards magnitude, we retain each hidden-state norm classically and restore it after readout, yielding the exact map Wh under ideal readout. We study this construction in two complementary settings: forward PDE regression and a norm-controlled residual model for representation geometry and uncertainty quantification. In the norm-controlled model, structural orthogonality yields explicit bi-Lipschitz bounds, while finite-readout analysis relates sufficient reconstruction precision to a prescribed physical-coordinate resolution through trained-model constants. This makes the access-cost condition assessable for trained checkpoints, although favorable fixed-resolution scaling across widths is not established. A post-hoc Bayesian-linear covariance provides uncertainty from the same frozen predictor, and cross-PDE transfer shows that strong distance-awareness can persist even when absolute coverage calibration fails. All Quantum Orthogonal (QO) layers are evaluated by classical simulation: the results therefore characterize a quantum-compatible structured neural primitive rather than demonstrating a practical quantum-hardware advantage.
In this work, we develop quantum reversible counterparts of the logical AND, NAND, OR, NOR, and XOR gates within a single architectural template: a quantum Fourier transform (QFT)–based adder that emulates a digital half‐adder by computing the sum and carry of two input bits. Each member of the gate family is obtained from the same adder by inserting only X and SWAP gates as pre‐ and postprocessing steps, with the required Boolean identities supplied by De Morgan’s laws. We prove that every proposed circuit is unitary prior to readout, so the input data remain recoverable from the output state. The construction generalizes to N ‐input gates through an in‐place population‐count (Hamming‐weight) register, and we derive explicitly that the resulting qubit requirement is N + log 2 N , compared with the 2 N − 1 qubits of the standard Toffoli‐ladder synthesis; the derivation is validated by exhaustive simulation for N ≤ 8. We provide a comprehensive resource analysis‐qubit count, gate count, circuit depth, CX count, T count, transpiled depth, and success probability‐against Toffoli‐, CNOT‐, and X‐gate‐based reversible circuits as well as three multicontrolled‐NOT decompositions with different ancilla‐depth trade‐offs. The analysis shows that the qubit saving is obtained at the price of a larger two‐qubit‐gate count, and identifies the regimes in which the QFT‐based design is advantageous. All two‐input gates and a four‐input NAND instance are executed for every input combination on IBM quantum hardware and on a calibrated noise‐model simulator, and the measured outputs consistently reproduce the corresponding truth tables, with hardware success probabilities of 0.97–0.99 for the two‐input gates and 0.88–0.93 for the four‐input NAND.