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Quantum Theory and Application of Contextual Optimal Transport
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Proposes QontOT, a quantum circuit ansatz that amortizes prediction of contextual optimal transport plans and doubly stochastic matrices via a unitary-to-DSM link, validated on synthetic/SciPlex data and a 24-qubit IBM hardware run.
Problem
Learning a global, context-conditioned optimal transport map that predicts transport plans for unseen contexts (e.g., drug dosage) and unseen initial distributions, where existing neural OT methods rely on Brenier's theorem and squared Euclidean cost.
Approach
Exploits the link between unitary operators and doubly stochastic matrices (U ⊙ conj(U) is doubly stochastic) to build a parametric quantum circuit (Operator-Schmidt-based ansatz with auxiliary qubits and projective measurements) that outputs context-dependent DSMs/transport plans, which are cost-agnostic and rescaled to arbitrary initial marginals.
Key results
- On SciPlex mocetinostat, QontOT-L_T reaches SAE 0.78 and Frobenius 0.61 versus CondOT's 0.45 and 0.40, while QontOT-L_M achieves L2 0.16 and R^2 0.57, on par or better than CondOT on marginal metrics.
- On the 24-qubit IBM Sherbrooke hardware run for contextual DSM prediction, QontOT achieves a better Frobenius distance than all 15 tested NeuCOT classical configurations.
- On synthetic linear perturbations with L2 cost, QontOT-L_T reduces SAE to 0.97 from 1.07 for the Average baseline and 1.50 for Identity.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| train QontOT to predict transport plans with L_T instead of using the Average baseline | QontOT · Synthetic linear perturbation, L2 cost | SAE: 1.07 -> 0.97 | Helped |
| optimize marginal distributions with L_M rather than transport plans with L_T (ablation) | QontOT · Synthetic linear perturbation, L2 cost | L2: 0.45 -> 0.41 | Helped |
| optimize marginal distributions with L_M rather than transport plans with L_T (ablation) | QontOT · Synthetic linear perturbation, L2 cost | Frob.: 0.70 -> 0.79 | Hurt |
| use L_M instead of L_T training (ablation) | QontOT · Synthetic linear perturbation, cosine cost | SAE: 0.97 -> 1.10 | Hurt |
| use L_M instead of L_T training (ablation) | QontOT · Synthetic nonlinear perturbation, L2 cost | SAE: 0.86 -> 0.97 | Hurt |
| replace QontOT with CondOT | QontOT · Mocetinostat, SciPlex | Frob.: $0.61$ -> $0.40$ | Helped |
| replace QontOT-L_T with QontOT-L_M (ablation) | QontOT · Mocetinostat, SciPlex | R2: $0.49$ -> $0.57$ | Helped |
| replace QontOT with CellOT | QontOT · Pracinostat, SciPlex | SAE: $0.82$ -> $0.44$ | Helped |
| replace QontOT with NeuCOT | QontOT · Synthetic contextual relaxed assignment problem | Frobenius distance | Hurt |
| replace COBYLA with Nevergrad (ablation) | QontOT · Synthetic circuit ablations | performance | No clear effect |
| increase ansatz depth or add auxiliary qubits (ablation) | QontOT · Synthetic circuit ablations | performance | Mixed |
| replace the checkerboard ansatz or atop aggregation (ablation) | QontOT · Synthetic circuit ablations | performance | No clear effect |
Surprising findings
- The 24-qubit hardware run outperformed the 18-qubit noiseless-style simulation of the same circuit, despite device noise and no error mitigation.
- The quantum model with gradient-free COBYLA optimization outperforms NeuCOT, a classical model with backpropagation and up to 5M parameters, on the DSM prediction task.
Limitations
- On SciPlex, CellOT and CondOT achieve better transport-plan metrics (SAE, Frobenius) than QontOT.
- Gradient-free optimization limits performance; BFGS improves results in simulation but is not feasible on hardware.
- Large-scale single-cell problems require clustering into few cells (8 or 16 clusters) because qubit and shot counts scale prohibitively; the 24-qubit hardware circuit is classically simulable.
Experimental setup
Datasets: Splatter-based synthetic dosage perturbation data SciPlex (mocetinostat, pracinostat)
Benchmarks: Synthetic dosage perturbation data SciPlex mocetinostat SciPlex pracinostat Contextual relaxed assignment (synthetic DSMs)
Metrics: SAE relative Frobenius norm L2 norm R^2
Baselines: Identity Average CellOT CondOT NeuCOT
Builds on: Neural OT CellOT CondOT Monge Gap Sinkhorn algorithm
Closest papers by meaning
- Amortized Optimal Transport from Sliced Potentials (NeurIPS 2026 · similarity 0.86)
- One More Time: Revisiting Neural Quantum States from a Reinforcement Learning Perspective (NeurIPS 2026 · similarity 0.85)
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- Winning Lottery Tickets in Neural Networks via a Quantum-Inspired Classical Algorithm (NeurIPS 2026 · similarity 0.84)
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- Expected Sliced Transport Plans (ICLR 2025 · similarity 0.83)
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