# The Feedback Hamiltonian is the Score Function: A Diffusion-Model Framework for Quantum Trajectory Reversal

arXiv: https://arxiv.org/abs/2604.21210 · HTML: https://mlsanity.com/paper/2604.21210/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 13 references that are in the map.
- First posted on arXiv 23 Apr 2026.

## Builds on (most cited first, 2 of 2)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)

## Abstract (arXiv)

In continuously monitored quantum systems, the feedback protocol of Garc'ia-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian Hmeas = r A / τ applied with gain X tilts the distribution of measurement trajectories, with X < -2 producing statistically time-reversed outcomes. Why this specific Hamiltonian achieves reversal, and how the mechanism relates to score-based diffusion models in machine learning, has remained unexplained. We compute the functional derivative of the log path probability of the quantum trajectory distribution directly in density-matrix space. Combining Girsanov's theorem applied to the measurement record, Fr'echet differentiation on the Banach space of trace-class operators, and K"ahler geometry on the pure-state projective manifold, we prove that δ log PF / δ ρ = r A / τ = Hmeas. The Garc'ia-Pintos feedback Hamiltonian is the score function of the quantum trajectory distribution -- exactly the object Anderson's reverse-time diffusion theorem requires for trajectory reversal. The identification extends to multi-qubit systems with independent measurement channels, where the score is a sum of local operators. Two consequences follow. First, the feedback gain X generates a continuous one-parameter family of path measures (for feedback-active Hamiltonians with [H, A] neq 0), with X = -2 recovering the backward process in leading-order linearization -- a structure absent from classical diffusion, where reversal is binary. Second, the score identification enables machine learning (ML) score estimation methods -- denoising score matching, sliced score matching -- to replace the analytic formula when its idealizations (unit efficiency, zero delay, Gaussian noise) fail in real experiments.