DAPS++: Decoupled Inference & Quantum Protocols

• DAPS++ is a decoupled framework that reorders diffusion-based prior sampling and measurement-driven refinement to efficiently solve Bayesian inverse problems.
• The framework significantly reduces neural function evaluations while achieving state-of-the-art results in super-resolution, inpainting, and other imaging tasks.
• In quantum processing, DAPS++ leverages NV-center charge qubits with dipole-dipole coupling to achieve high readout fidelity and scalable operations.

DAPS++ refers to two distinct frameworks in contemporary research: (1) Decoupled Annealing Posterior Sampling++ (DAPS++), a paradigm for efficient Bayesian inference in diffusion-based inverse problems, and (2) a protocol for quantum information processing using donor-acceptor-pair-like charge states in shallow nitrogen-vacancy (NV) centers in diamond. Both exploit the "decoupled" structure in their respective domains—posterior annealing for diffusion inverse problems and dipole-dipole coupled charge qubits for quantum computing.

1. Bayesian Inverse Problem Solving with DAPS++

DAPS++ in the context of inverse problems formalizes a two-stage, expectation–maximization (EM)-style approach that decouples measurement-driven refinement from diffusion-based prior sampling. The starting point is the classical Bayesian inverse problem

$$\mathbf{y} = \mathcal{A}\,\mathbf{x}_0 + \boldsymbol{\epsilon}, \quad \boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \gamma^2 \mathbf{I}),$$

seeking to recover $\mathbf{x}_0$ given measurements $\mathbf{y}$ and a linear or nonlinear forward operator $\mathcal{A}$. The posterior is given by

$$p(\mathbf{x}_0 \mid \mathbf{y}) \propto p(\mathbf{y} \mid \mathbf{x}_0) p(\mathbf{x}_0).$$

Diffusion models leverage learned priors $p(\mathbf{x}_0)$ and perform sampling via a forward noising SDE and its reverse, characterized by the score network $s_\theta(\mathbf{x}_t, \sigma_t)$ that estimates $\nabla_{\mathbf{x}_t} \log p_t(\mathbf{x}_t)$. Existing score-based Bayesian solvers inject likelihood gradients into each diffusion step, but these are dominated by the measurement term, limiting insight and efficiency.

DAPS++ reinterprets this as a decoupled EM-like procedure: a pure diffusion (prior-initialized) step provides a denoised estimate, followed by a measurement-driven Langevin refinement that directly enforces fidelity with observations. This separation leads to significant reductions in neural function evaluations (NFEs) and wall-clock runtime, without sacrificing reconstruction quality (Chen et al., 21 Nov 2025).

2. DAPS++ Algorithmic Details

The DAPS++ algorithm alternates between diffusion-based E-steps and measurement-driven M-steps, fully decoupling prior and likelihood operations. The E-step produces an unconditional sample or its expectation via Tweedie’s formula: $$\hat{\mathbf{x}}_0 \approx \mathbf{x}_t + \sigma_t^2 s_\theta(\mathbf{x}_t, \sigma_t),$$ with high-order ODE integration for $\sigma_t$ below a threshold $\bar\sigma$ for fine detail.

The subsequent M-step optimizes the likelihood via unadjusted Langevin algorithm (ULA) updates: $$\mathbf{x}_0^{(j+1)} = \mathbf{x}_0^{(j)} - \frac{\eta}{\gamma^2} \nabla_{\mathbf{x}_0^{(j)}} \|\mathbf{y} - \mathcal{A}(\mathbf{x}_0^{(j)})\|^2 + \sqrt{2\eta} \boldsymbol{\epsilon}_j,$$ where noise $\boldsymbol{\epsilon}_j$ maintains exploration of the posterior.

Schedule design is crucial. DAPS++ employs a polynomial time discretization,

$$t_i = t_{\min} + (t_1 - t_{\min}) \Bigl(\frac{i}{N-1}\Bigr)^\rho,$$

with negative $\rho$ to allocate more steps at low noise. Empirically, $\bar\sigma = 10\gamma$ balances speed and detail.

The decoupled structure reduces repetitive neural network evaluations typical in previous approaches (e.g., DPS, DAPS), as all likelihood optimization is performed in post-diffusion refinement rather than after every annealing step (Chen et al., 21 Nov 2025).

3. Theoretical Insights and Performance Analysis

Lipschitz-based analysis substantiates that, under high noise ($\sigma_t \gg \gamma$), the prior score scales as $O(1/\sigma_t)$, while the likelihood gradient grows as $\sigma_t \|\mathbf{r}_t\| / \gamma^2$. Thus, the measurement-consistency term quickly dominates and justifies the DAPS++ decoupling.

Empirical benchmarks on FFHQ-256 and ImageNet-256 datasets demonstrate leading or near-leading performance across super-resolution, inpainting, and deblurring tasks, as well as nonlinear inverse problems such as phase retrieval and HDR imaging. DAPS++ achieves state-of-the-art or second-best scores in SSIM, LPIPS, and FID, often with $10\times$ fewer NFEs than unified likelihood-prior methods (Chen et al., 21 Nov 2025).

Table: Sampling complexity and runtime comparison

Method	Diffusion Steps	M-Steps per σ	Total NFE	Time (s/image)
DPS	--	1	1000	40.90
DAPS-100	2×50	100	100	6.26
DAPS++-20	1×20	2	40	0.89
DAPS++-50	1×50	2	100	2.31

This efficiency is attributed to performing all high-cost diffusion inference in a single initialization, with only minimal network calls.

4. Limitations and Open Questions in DAPS++

DAPS++ has documented limitations:

• For highly nonlinear forward operators (e.g., phase retrieval), convergence can fail or stall.
• The framework primarily targets Gaussian measurement noise; extension to other noise models (Poisson, speckle, adversarial) is not addressed.
• While the EM analogy guides the decoupling, a provable convergence analysis of the interleaved two-stage process is not established.
• Some settings may benefit from finer adaptive or non-monotonic annealing schedules and model-based refinement of the initialization threshold $\bar\sigma$ and polynomial schedule exponent $\rho$.

Future work includes deriving posterior contraction guarantees, incorporating alternative noise models, and optimizing resource allocation in the annealing schedule (Chen et al., 21 Nov 2025, Zhang et al., 1 Jul 2024).

5. Relationship to DAPS and Theoretical Implications

DAPS++ substantially extends DAPS (Zhang et al., 1 Jul 2024) by:

• Replacing the coupled stepwise likelihood-prior updates with a clear two-stage procedure, clarifying the algorithmic role of each.
• Avoiding the irrecoverable errors associated with the Gaussian posterior approximation in the DAPS Langevin step via direct measurement-driven optimization.
• Enabling faster inference by minimizing expensive calls to the score network.

A plausible implication is that DAPS++ unifies empirical observations regarding the dominance of fidelity terms in joint SDEs with a theoretically principled EM decomposition. By decoupling regularization and measurement inversion, DAPS++ provides a more interpretable and robust computational scheme for inverse problems in imaging and related domains (Chen et al., 21 Nov 2025).

6. DAPS++ in Solid-State Quantum Information Processing

DAPS++ also denotes an integrated protocol for quantum computation using NV-center charge states in diamond (Defo et al., 26 Feb 2024). In this setting:

• Each shallow NV center is treated as a charge qubit, with electric dipole operator $\mu_i$.
• Centers interact via the static dipole-dipole Hamiltonian,

$$H_{dd} = \frac{1}{4\pi\epsilon_0 R^3}[\mu_d \cdot \mu_a - 3(\mu_d \cdot \hat{R})(\mu_a \cdot \hat{R})],$$

with $\Delta\mu \approx 1$–$4$ Debye, yielding coupling strengths $J_{ij}/h \sim 1$–$100$ MHz for $R = 10$–$20$ nm.

• Initialization exploits dark decay and surface termination to set desired charge-state populations, with surface Fermi-level engineering tuning equilibration rates from $\sim 18$ ns to $\sim$ ms.
• Readout utilizes NV$^0$ electroluminescence: the radiative cycle at 575 nm ($^2E \to ^2A_2 + h\nu$) offers projective measurement analogous to spin-dependent fluorescence in NV$^-$.
• The full DAPS++ scheme: initialize each qubit, perform gate evolution under $H_{dd}$, optionally refocus via local Fermi-level bias, and perform EL-based projective readout.

Reported figures of merit for shallow NV-center DAPS++ include charge-qubit coherence $T_2^{\text{charge}} \gtrsim 1\,\mu$s, readout fidelity $\gtrsim 90\%$, and tunable charge-state lifetimes (via termination) from 10 ns to s scale (Defo et al., 26 Feb 2024).

This framework exploits the scalability and addressability of NV-center systems, with long-range coupling and all-electrical/optical interfaces, constituting a promising solid-state quantum information architecture.

7. Summary and Outlook

DAPS++ encapsulates "decoupled" inference in two disparate frontiers: (1) as a computational inference scheme that reorders diffusion and measurement updates for Bayesian inverse problems, and (2) as a protocol for quantum computation with charge-based qubits in diamond. Both exploit the dissociation of physical or probabilistic dependencies to improve tractability, efficiency, or control. In both settings, further research directions include more sophisticated posterior approximations, noise model generalizations, adaptive algorithms, and theoretical analysis of convergence or error properties (Chen et al., 21 Nov 2025, Zhang et al., 1 Jul 2024, Defo et al., 26 Feb 2024).