Structure-Preserving Quantum Circuit Architectures for Robot Kinematics

I. INTRODUCTION

• The mapping decomposes each translation into a classical magnitude and signed Bloch-vector direction, propagates directions through parameterized rotations, and encodes terminal-frame axes for orientation readout.It specializes the representation to standard and Craig modified-DH kinematics while preserving the joint-variable semantics and ordered frame propagation.Source: §I · p. 2
• A selector register weights directional branches by contribution magnitudes, so the position qubit encodes their normalized sum and the retained scale reconstructs metric position.The position estimate is obtained from measured Pauli expectation values.Source: §I · p. 2
• IBSS and SCP produce the same ideal reduced position state, while distributing resources differently across item qubits, selector preparation, controlled operations, depth, and routing.IBSS routes a prepared bank of directional states using controlled SWAPs; SCP directly prepares the selected direction through controlled rotations.Source: §I · p. 2
• IWR, tree-factorized selectors, and parallel XYZ readout trade circuit width, control depth, measurement settings, executions, and classical reconstruction in different ways.IWR uses shallower circuits and fewer multi-qubit controls but more measurements; parallel XYZ readout uses more qubits and replicated preparation to reduce measurement settings.Source: §I · p. 2
• The evaluation spans ideal and finite-shot simulations, noisy simulation, IBM hardware, and physical-robot pose comparison, and makes no claim of quantum speedup or computational advantage.Resource analysis distinguishes logical, decomposed, transpiled, and executed circuits, including gate, measurement, shot, and classical-processing costs.Source: §I · p. 2

II. RELATED WORK

• Prior quantum-robotics research has addressed tasks such as manipulation, navigation, planning, coordination, learning, and optimization, but generally not continuous robot kinematics within the circuit.The related-work discussion focuses on approaches encoding geometric or kinematic quantities; Table 1 compares representative quantum representations of robotic geometry and kinematics.Source: §II · pp. 2–3, Table 1 · p. 3

A. QUANTUM REPRESENTATIONS OF ROBOTIC GEOMETRY

• Zioui et al. encode orientations with single-qubit states, while Fazilat et al. use quantum rotations but assemble translation classically for one ABB manipulator.Neither approach supplies a general circuit construction for aggregating weighted translational directions.Source: A. QUANTUM REPRESENTATIONS OF ROBOTIC… · p. 3
• Otani et al. weight link-direction Pauli expectations by classical lengths, without deriving circuit parameters from joint variables or aggregating directions into a compact position state.Nigatu et al. instead use a learned surrogate for forward kinematics, rather than deriving operations analytically from the robot model.Source: A. QUANTUM REPRESENTATIONS OF ROBOTIC… · p. 3
• The reviewed literature lacks a unified model-derived framework combining compact and item-wise circuits for position and orientation with separate classical magnitudes and quantum directions.This is the identified gap in prior work.Source: A. QUANTUM REPRESENTATIONS OF ROBOTIC… · p. 3

B. POSITIONING OF THE PRESENT WORK

• The present work derives parameterized circuits from rigid-body and Denavit–Hartenberg data while separating classical metric quantities from quantum directional states.It compares compact selector-based architectures with NISQ-oriented item-wise reconstruction, tree-factorized controls, and parallel Cartesian readout.Source: B. POSITIONING OF THE PRESENT WORK · p. 3
• The architecture comparison covers pose accuracy, logical width, depth, controlled operations, measurement settings, repeated executions, and classical post-processing.The circuits apply established primitives including probability-amplitude preparation, uniformly controlled rotations, quantum multiplexing, and general circuit synthesis –.Source: B. POSITIONING OF THE PRESENT WORK · p. 3
• The proposed architectures are evaluated at logical, decomposed, transpiled, finite-shot, noisy, and hardware-execution levels.The paper notes that coherence limits, imperfect gates, restricted connectivity, and finite sampling can increase costs after decomposition, routing, and transpilation,.Source: B. POSITIONING OF THE PRESENT WORK · p. 3

III. METHODOLOGICAL BACKGROUND

• Rigid-body transformations and the directional decomposition of serial-manipulator kinematics provide the mathematical foundations for the architectures.The section first defines transformation representation and composition, then connects them to directional decomposition.Source: §III · pp. 3–4
• Directional qubit states and their evolution under single-qubit rotations establish how geometric directions are represented and transformed.These concepts are introduced alongside selector-controlled operations and reduced-state readout.Source: §III · p. 4
• The background also covers selector-controlled operations, reduced-state readout, and finite-shot Pauli estimation.These concepts supply the control, measurement, and estimation context for the proposed circuits.Source: §III · p. 4

A. RIGID-BODY TRANSFORMATIONS AND DIRECTIONAL DECOMPOSITION

• Serial-chain poses compose rotations by matrix multiplication and translations by adding each displacement after expressing it in the preceding reference frame.For revolute joints θ_i varies with q_i while d_i is fixed; for prismatic joints d_i varies while θ_i is fixed.Source: A. RIGID-BODY TRANSFORMATIONS AND… · p. 4
• Both DH conventions express end-effector translation as a sum of item contributions ℓ_k(q)u_k(q), with each direction propagated into the base frame.Each item pairs a nonnegative magnitude with a signed local unit axis and a rotation prefix; items with zero magnitude contribute zero.Source: A. RIGID-BODY TRANSFORMATIONS AND… · pp. 4–5
• A nonzero fixed base translation p_0 is included as a configuration-independent translational item, and zero-magnitude items remain in the item set but contribute nothing.The sign of each original DH coefficient is carried by its unit direction rather than its nonnegative magnitude.Source: A. RIGID-BODY TRANSFORMATIONS AND… · pp. 4–5
• The terminal orientation is represented by two propagated frame axes, with the remaining axis determined by the right-handed cross-product constraint.Finite-shot axis estimates need not be normalized or mutually orthogonal; reconstruction and any projection onto SO(3) are specified in the experimental methodology.Source: A. RIGID-BODY TRANSFORMATIONS AND… · p. 5

B. SINGLE-QUBIT STATES AND SPATIAL ROTATIONS

• The Bloch-vector coordinates are the ideal expectation values of the Pauli observables σx, σy, and σz.Each coordinate is the expectation value of its corresponding Pauli operator.Source: B. SINGLE-QUBIT STATES AND SPATIAL… · p. 5
• Pure single-qubit states correspond to unit vectors on the Bloch sphere, while mixed states lie inside the Bloch ball.This three-dimensional representation applies to individual qubits, not general multi-qubit states.Source: B. SINGLE-QUBIT STATES AND SPATIAL… · p. 5
• Single-qubit unitary rotations induce corresponding right-handed spatial rotations of the Bloch vector.The induced Bloch-vector transformation is represented by a classical rotation matrix in SO(3).Source: B. SINGLE-QUBIT STATES AND SPATIAL… · p. 5
• Ordered rotations act rightmost first and generally do not commute; the SU(2) representatives UA and −UA induce the same spatial rotation.The map from SU(2) to SO(3) is two-to-one.Source: B. SINGLE-QUBIT STATES AND SPATIAL… · p. 5

C. REDUCED STATES AND CONVEX AGGREGATION OF QUBIT OBSERVABLES

• Tracing out unobserved registers leaves a readout-qubit state containing all locally accessible measurement statistics.The construction applies when measurements are performed only on Q.Source: C. REDUCED STATES AND CONVEX… · pp. 5–6
• A selector register assigns orthogonal basis labels to branches and prepares them with probabilities λ_k, correlating each label with its branch’s readout state.Unassigned computational-basis states have zero amplitude; branch-dependent workspace states may also be correlated with the labels.Source: C. REDUCED STATES AND CONVEX… · p. 6
• Tracing out the selector and other registers yields a convex mixture of branch readout states, even when the global state is coherent and entangled.The same reduced-state and observable-linearity relations hold for mixed branch states.Source: C. REDUCED STATES AND CONVEX… · p. 6
• The readout Bloch vector is the convex combination of branch Bloch vectors; this is a mixture of states and expectations, not a coherent sum of readout state vectors.Even pure branch states can produce a mixed reduced state.Source: C. REDUCED STATES AND CONVEX… · p. 6

D. PAULI READOUT AND FINITE-SHOT ESTIMATION

• Pauli measurements yield outcomes ±1 with probabilities set by the corresponding Bloch component; σx and σy require basis rotations before computational-basis measurement.σz needs no basis transformation; σx uses H, while σy uses S† followed by H.Source: D. PAULI READOUT AND FINITE-SHOT… · p. 6
• Under independent, identically distributed shots, the finite-shot estimator of each Bloch component is unbiased, but a complete vector requires three measurement settings and 3Nshots executions under equal allocation.Equivalent readout-state copies can instead be measured in different bases within one execution, trading additional qubits and state-preparation operations for fewer measurement settings.Source: D. PAULI READOUT AND FINITE-SHOT… · p. 6
• Independent component estimates need not form a physical Bloch vector or preserve frame-direction normalization and orthogonality, so any geometric post-processing must be stated explicitly.In particular, the estimated vector is not guaranteed to have norm at most one, even when the ideal state is pure.Source: D. PAULI READOUT AND FINITE-SHOT… · pp. 6–7

• The architectures target forward kinematics of serial open-chain manipulators described by standard or modified Denavit–Hartenberg parameters.Source: §IV · p. 7
• The construction maps DH-induced frame-direction propagation to parameterized single-qubit operations and reads out position and terminal-frame orientation.Source: §IV · p. 7
• The section develops IBSS and SCP position architectures, plus NISQ-oriented IWR, tree-factorized, and parallel Cartesian variants.Source: §IV · p. 7

A. PROBLEM FORMULATION AND ARCHITECTURE OVERVIEW

• The circuit maps each admissible configuration to parameterized operations whose measurable outputs reproduce forward-kinematics position and orientation while preserving ordered DH dependence.The compact pose interface uses position-readout qubit Qp and orientation-readout qubits Qx and Qz.Source: A. PROBLEM FORMULATION AND… · p. 7
• The metric position is reconstructed as pn(q) = W(q) r(ρp(q)), with the normalized position represented by a valid single-qubit Bloch vector.W(q) is the positive classical metric scale induced by active translational contributions.Source: A. PROBLEM FORMULATION AND… · p. 7
• The two orientation readouts determine the complete terminal frame, and each designated readout qubit is described by its reduced state.The readouts are Qx and Qz; the compact interface specifies outputs at the ideal expectation-value level.Source: A. PROBLEM FORMULATION AND… · p. 7
• IBSS and SCP use different selector-controlled aggregation mechanisms, while IWR, tree-factorized, and parallel Cartesian alternatives trade circuit width, depth, routing, measurements, executions, and classical post-processing.IWR reconstructs position from separately measured directional contributions; parallel Cartesian readout uses additional qubits and replicated state preparation for fewer measurement settings.Source: A. PROBLEM FORMULATION AND… · p. 7

B. DH-DERIVED DIRECTIONAL QUBIT PREPARATION

• Each translational item separates its positive metric magnitude from a single-qubit state whose Bloch vector represents the signed propagated direction.The classical magnitude is retained to weight the contribution in position-aggregation architectures.Source: B. DH-DERIVED DIRECTIONAL QUBIT… · pp. 7–8
• Canonical seed rotations encode DH translational directions, with configuration-dependent signs changing the seed and zero coefficients excluded from active items.For the listed DH-derived items, the seed directions are ±ex or ±ez; prismatic coordinates can affect both the metric magnitude and signed seed direction.Source: B. DH-DERIVED DIRECTIONAL QUBIT… · pp. 7–8
• The propagation unitary preserves the ordered DH rotation prefix, so ideal Pauli expectations reproduce the signed direction’s Cartesian components in the base frame.The representation is independent of which equivalent SU(2) representative or elementary gate decomposition implements the required Bloch-vector action.Source: B. DH-DERIVED DIRECTIONAL QUBIT… · p. 8
• Two orientation-readout qubits encode the propagated terminal-frame x- and z-axes, sufficient to reconstruct the complete orientation independently of the position-readout architecture.They are initialized with seed rotations Uy(π/2) and I2, respectively, then propagated through the complete ordered DH rotation sequence.Source: B. DH-DERIVED DIRECTIONAL QUBIT… · p. 8

V. ITEM-WISE READOUT

• The translational register contains one independently prepared directional qubit per DH-derived item, with local Pauli expectations encoding each item's unit direction.A zero metric coefficient makes that item's contribution zero while the register structure remains fixed across configurations; the readout uses no inter-item quantum operations.Source: §V · pp. 8–9
• At the ideal expectation-value level, classical weighted aggregation of item-level Pauli readouts recovers the exact end-effector position without an additional quantum subsystem for the translational sum.The metric coefficients remain classical; their normalized directional information is combined with the classical scale factor to recover position in physical units.Source: §V · p. 9
• Separate readout qubits Qx and Qz encode two terminal-frame axes, providing the information used to reconstruct end-effector orientation.The complete IWR pose representation combines these orientation qubits with the translational item register.Source: §V · p. 9