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This research demonstrates how quantum walks can accelerate sampling for dynamic Schrödinger bridge problems, reducing query complexity from quadratic to linear relative to the time horizon. By adapting quantum Gibbs samplers and Newton methods, the study offers significant algorithmic improvements over classical counterparts for optimizing potentials in finite state spaces. The work represents a concrete step toward leveraging quantum algorithms for complex stochastic control and optimization tasks.
Quantum-Enhanced Sampling of Schr\"odinger Bridges
arXiv:2609.27103v1 Announce Type: new Abstract: We consider the dynamic Schr\"odinger bridge problem on a finite state space and exploit its Markov structure to decompose the problem into an endpoint coupling and a collection of conditional Markov bridges. To sample from the conditional bridges, we develop a quantum Gibbs sampler based on quantum walks on the bridge path space. Under uniformly positive transition probabilities, we derive a spectral-gap bound and provide an explicit procedure for preparing the initial state. Compared with the update bounds of the corresponding classical Gibbs sampler, our quantum walk-query complexity improves the dependence on the time horizon $T$ from quadratic to linear. For the endpoint coupling, we adapt a quantum box-constrained Newton method to compute the Schr\"odinger potentials, improving the matrix-scaling complexity, at fixed accuracy, from quadratic to $N^{3/2}$ in the state-space size $N$. Finally, we show that exponential reweighting extends both the Schr\"odinger bridge formulation and the Gibbs spectral-gap estimate to models with additive running costs.
Abstract:We consider the dynamic Schrödinger bridge problem on a finite state space and exploit its Markov structure to decompose the problem into an endpoint coupling and a collection of conditional Markov bridges. To sample from the conditional bridges, we develop a quantum Gibbs sampler based on quantum walks on the bridge path space. Under uniformly positive transition probabilities, we derive a spectral-gap bound and provide an explicit procedure for preparing the initial state. Compared with the update bounds of the corresponding classical Gibbs sampler, our quantum walk-query complexity improves the dependence on the time horizon $T$ from quadratic to linear. For the endpoint coupling, we adapt a quantum box-constrained Newton method to compute the Schrödinger potentials, improving the matrix-scaling complexity, at fixed accuracy, from quadratic to $N^{3/2}$ in the state-space size $N$. Finally, we show that exponential reweighting extends both the Schrödinger bridge formulation and the Gibbs spectral-gap estimate to models with additive running costs.
Quantum Physics (quant-ph); Optimization and Control (math.OC); Probability (math.PR)
arXiv:2609.27103 [quant-ph]
(or arXiv:2609.27103v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.27103 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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