QOTAM-QAE: Tensor-Network Quantum Amplitude Estimation for Complex Optimal-Transport Scattering Amplitudes *

We propose the _Quantum Optimal Transport Amplitude Method_ (QOTAM), a new framework in which scattering amplitudes arise from a complex optimal-transport problem on on-shell kinematic space. Incoming and outgoing wavepackets define probability distributions, and we introduce a complex kernel $K(q|p)$ whose modulus squared transports $\rho_{\rm in}$ to $\rho_{\rm out}$ while its phase encodes the classical action. The physical amplitude is obtained from the variational principle $\delta\mathcal{S}[K]=0$ , where $\mathcal{S}[K]$ includes a transport cost, an $iS_{\rm cl}/\hbar$ phase term, and regularizers enforcing locality and factorization. Tree amplitudes emerge from linear response of the optimal kernel, and the formulation admits efficient numerical realization via entropically regularized optimal transport. QOTAM offers a geometric–variational approach to scattering with potential applications to high-multiplicity and non-perturbative regimes.We propose the _Quantum Optimal Transport Amplitude Method_ (QOTAM), a new framework in which scattering amplitudes arise from a complex optimal-transport problem on on-shell kinematic space. Incoming and outgoing wavepackets define probability distributions, and we introduce a complex kernel $K(q|p)$ whose modulus squared transports $\rho_{\rm in}$ to $\rho_{\rm out}$ while its phase encodes the classical action. The physical amplitude is obtained from the variational principle $\delta\mathcal{S}[K]=0$, where $\mathcal{S}[K]$ includes a transport cost, an $iS_{\rm cl}/\hbar$ phase term, and regularizers enforcing locality and factorization. Tree amplitudes emerge from linear response of the optimal kernel, and the formulation admits efficient numerical realization via entropically regularized optimal transport. QOTAM offers a geometric–variational approach to scattering with potential applications to high-multiplicity and non-perturbative regimes.

QeiosNov 21, 20251 citation

Selected topics in quantum scattering theory and emerging optimal-transport methods

Fidele Twagirayezu

Quantum scattering connects microscopic dynamics to amplitudes, phase shifts, cross sections, transport coefficients, and resonances. This selective Review organizes the Lippmann–Schwinger equation, resolvents, transfer and scattering matrices, nonequilibrium Green’s functions, and R-matrix theory as related representations of a common scattering problem. Born, Wentzel–Kramers–Brillouin, and eikonal error tests accompany comparisons of numerical scaling, stability, and method selection. Inverse scattering is examined through the effects of phase loss, incomplete aperture, Jacobian null spaces, singular-value decay, model discrepancy, and noise, together with regularized, Bayesian, and learned approaches for stabilizing reconstruction. The Review then discusses probability-level optimal transport and its extension to emerging amplitude-based formulations, using the Quantum Optimal Transport Amplitude Method (QOTAM) as a concrete example. Its Euclidean and real-time variational constructions are summarized and tested across resonant barriers, multichannel and angular scattering, three-dimensional targets, and disordered waveguides. Independent reference calculations verify complex-amplitude closure, unitarity, flux conservation, and optical-theorem relations to benchmark-dependent numerical precision. In inverse problems, QOTAM is found to be competitive with correctly specified phase-sensitive likelihood methods while showing useful robustness in selected model-mismatch tests. These examples illustrate the promise of optimal-transport-based scattering methods. More broadly, the Review emphasizes how physical constraints, uncertainty quantification, identifiability, and computational scalability jointly shape reliable forward and inverse-scattering analyses.Quantum scattering connects microscopic dynamics to amplitudes, phase shifts, cross sections, transport coefficients, and resonances. This selective Review organizes the Lippmann–Schwinger equation, resolvents, transfer and scattering matrices, nonequilibrium Green’s functions, and R-matrix theory as related representations of a common scattering problem. Born, Wentzel–Kramers–Brillouin, and eikonal error tests accompany comparisons of numerical scaling, stability, and method selection. Inverse scattering is examined through the effects of phase loss, incomplete aperture, Jacobian null spaces, singular-value decay, model discrepancy, and noise, together with regularized, Bayesian, and learned approaches for stabilizing reconstruction. The Review then discusses probability-level optimal transport and its extension to emerging amplitude-based formulations, using the Quantum Optimal Transport Amplitude Method (QOTAM) as a concrete example. Its Euclidean and real-time variational constructions are summarized and tested across resonant barriers, multichannel and angular scattering, three-dimensional targets, and disordered waveguides. Independent reference calculations verify complex-amplitude closure, unitarity, flux conservation, and optical-theorem relations to benchmark-dependent numerical precision. In inverse problems, QOTAM is found to be competitive with correctly specified phase-sensitive likelihood methods while showing useful robustness in selected model-mismatch tests. These examples illustrate the promise of optimal-transport-based scattering methods. More broadly, the Review emphasizes how physical constraints, uncertainty quantification, identifiability, and computational scalability jointly shape reliable forward and inverse-scattering analyses.

Journal of Applied PhysicsOct 28, 20260 citations

Quantum Amplitude Estimation for Travel Time Estimation in Stochastic Vehicle Routing Problems

Xingyue Wang, Monika Filipovska

Solving the Vehicle Routing Problem (VRP) in Stochastic Transportation Networks (STNs), a core task in Intelligent Transportation Systems (ITS), introduces estimation challenges for stochastic path travel times and the resulting VRP objective function. These challenges have typically been addressed through computationally expensive sampling-based techniques such as Monte Carlo simulation, whose performance depends on sample size, the sampling strategy, and the underlying travel time distributions. To address these issues, this study proposes and validates a quantum computing technique, Quantum Amplitude Estimation (QAE) for path-level travel time estimation in STNs. Without relying on sampling or prior assumptions of the travel time distribution, the proposed framework encodes all feasible travel time realizations into a quantum superposition, enabling a theoretical quadratic speed-up over Monte Carlo simulation. Four QAE variants are implemented in IBM's Qiskit framework, namely Canonical AE (CAE), Iterative AE (IAE), Maximum Likelihood AE (MLAE), and Faster AE (FAE), together with four rotation-angle scaling strategies for handling different discrete travel time distributions. Experiments on a small-scale STN show that the choice of scaling method and rotation-angle range significantly affects estimation accuracy, while the four QAE variants produce comparable estimates across all tested conditions, with IAE exhibiting the most stable overall performance. The results provide practical guidance on parameter selection for future hybrid quantum-classical optimization frameworks in ITS applications.Solving the Vehicle Routing Problem (VRP) in Stochastic Transportation Networks (STNs), a core task in Intelligent Transportation Systems (ITS), introduces estimation challenges for stochastic path travel times and the resulting VRP objective function. These challenges have typically been addressed through computationally expensive sampling-based techniques such as Monte Carlo simulation, whose performance depends on sample size, the sampling strategy, and the underlying travel time distributions. To address these issues, this study proposes and validates a quantum computing technique, Quantum Amplitude Estimation (QAE) for path-level travel time estimation in STNs. Without relying on sampling or prior assumptions of the travel time distribution, the proposed framework encodes all feasible travel time realizations into a quantum superposition, enabling a theoretical quadratic speed-up over Monte Carlo simulation. Four QAE variants are implemented in IBM's Qiskit framework, namely Canonical AE (CAE), Iterative AE (IAE), Maximum Likelihood AE (MLAE), and Faster AE (FAE), together with four rotation-angle scaling strategies for handling different discrete travel time distributions. Experiments on a small-scale STN show that the choice of scaling method and rotation-angle range significantly affects estimation accuracy, while the four QAE variants produce comparable estimates across all tested conditions, with IAE exhibiting the most stable overall performance. The results provide practical guidance on parameter selection for future hybrid quantum-classical optimization frameworks in ITS applications.

quant-pharXiv:2608.06145Aug 7, 20260 citations

Tensor networks for complex quantum systems

Román Orús

arXiv:1812.04011Nature Reviews PhysicsAug 5, 2019628 citations

Computations in quantum tensor networks

Thomas Huckle, Konrad Waldherr, Thomas Schulte‐Herbrüggen

arXiv:1212.5005Linear Algebra and its ApplicationsJan 27, 201257 citations

K(q|p)

whose modulus squared transports

\rho_{\rm in}

\rho_{\rm out}

while its phase encodes the classical action. The physical amplitude is obtained from the variational principle

\delta\mathcal{S}[K]=0

, where

\mathcal{S}[K]

includes a transport cost, an

iS_{\rm cl}/\hbar

phase term, and regularizers enforcing locality and factorization. Tree amplitudes emerge from linear response of the optimal kernel, and the formulation admits efficient numerical realization via entropically regularized optimal transport. QOTAM offers a geometric–variational approach to scattering with potential applications to high-multiplicity and non-perturbative regimes.