Quantum Latent Compression with Adaptive Observables

• The paper presents a novel hybrid quantum-classical diffusion model that integrates amplitude encoding for latent compression with adaptive observables for global feature extraction.
• It employs a parameterized quantum circuit and trainable non-local Hermitian observables to capture multiscale correlations and mitigate mode collapse in generative tasks.
• Empirical results on the MNIST dataset demonstrate improved stability and expressibility in multi-class image synthesis under near-term quantum hardware constraints.

Quantum Latent Compression with Adaptive Non-local Observables (ANO) is a methodology developed to address the scalability and expressibility limitations of quantum generative models in high-dimensional and multi-modal data domains. By leveraging amplitude encoding for quantum latent-space compression and extracting trainable, non-local features via adaptive non-local observables within a hybrid quantum-classical U-Net diffusion framework, this approach facilitates the representation of complex structures in generative modeling tasks, such as image synthesis with the full MNIST dataset. Distinctive to this method are the integration of parameterized quantum circuits for latent evolution and the measurement of K trainable non-local Hermitian observables whose expectations inform the final output via a classical decoder and skip-connections, thereby enhancing multi-class generation stability and mitigating mode collapse under near-term quantum hardware restrictions (Jo et al., 3 Feb 2026).

1. Quantum Encoding and Latent-Space Compression

This framework begins with a classical-to-quantum encoder $\psi_{cl}: \mathbb{R}^{256} \to \mathbb{C}^{N}$, where the original 256-dimensional image vector is mapped into an $N=16$-dimensional normalized complex latent vector $z$, with $\|z\|_2 = 1$. The encoding process utilizes amplitude encoding, implemented by a unitary $U_{enc}$ such that $|0\rangle^{\otimes n}$ is mapped to $$|\psi(z)\rangle = U_{enc}|0\rangle^{\otimes n} = \sum_{i=0}^{N-1} z_i |i\rangle$$. The state preparation is achieved using a tree of multi-controlled $R_y$ and phase gates, or an equivalent state-preparation circuit, with depth scaling as $O(N)$.

The reduction from 256 to 16 dimensions yields a compression ratio $r = \frac{256}{16} = 16$, providing significant latent-space compression that preserves global informational content through quantum superposition.

2. Adaptive Non-Local Observables: Definition and Measurement

Following evolution of the quantum state by a parameterized quantum circuit $U(\theta)$, the extraction of features is performed by measuring a set of $K$ trainable Hermitian observables $\{H_k(\phi_k)\}_{k=1}^K$, each defined as $$H_k(\phi_k) = \frac{1}{2}[M_k(\phi_k) + M_k(\phi_k)^\dagger]$$ for arbitrary complex matrices $M_k(\phi_k)\in\mathbb{C}^{N \times N}$. Alternatively, $H_k(\phi_k)$ can be expanded in the $n$-qubit Pauli basis as $$H_k(\phi_k) = \sum_\alpha \theta_{k,\alpha}P_\alpha$$, capturing non-local multi-qubit correlations through higher-weight Pauli strings.

For a quantum state $|\psi(\theta)\rangle = U(\theta)U_{enc}|0\rangle$, the measurement protocol computes $$y_k = \langle\psi(\theta)|H_k(\phi_k)|\psi(\theta)\rangle = \mathrm{Tr}[\rho H_k(\phi_k)]$$, for $k=1,\dots,K$, where $\rho = |\psi(\theta)\rangle\langle\psi(\theta)|$. The resulting $\mathbf{y}$ vectors, $y\in\mathbb{R}^K$, embody the trainable non-local features.

3. Integration into the Hybrid Quantum-Classical U-Net Diffusion Model

At each reverse diffusion step $t$, the process involves several stages:

• A noised image $x_t\in\mathbb{R}^{256}$ serves as input.
• A classical encoder produces $z_t = \psi_{cl}(x_t)\in\mathbb{C}^{16}$.
• Quantum state preparation and dynamic evolution yield $|\psi_t\rangle = U(\theta_t)|\psi_0\rangle$.
• Measurement of $K$ adaptive non-local observables gives feature vector $y^{(t)}$.
• Intermediate classical features $s_t$ are produced via a skip-connection: $s_t = \mathrm{skip}(x_t)$.
• The combined feature $v_t = [s_t; y^{(t)}]\in\mathbb{R}^{d_s+K}$ is passed into the classical decoder.
• The decoder outputs the predicted noise $\epsilon_\theta(x_t, t)$, or equivalently, the mean $\mu_\theta(x_t, t)$ of $p_\theta(x_{t-1}|x_t)$.

This mechanism embeds the quantum circuit (with ANO readout) as the network’s bottleneck, introducing global quantum correlations to the denominator of the generative model.

4. Training, Loss Function, and Gradient Estimation

The primary training objective is the simplified Denoising Diffusion Probabilistic Model (DDPM) loss:

$$L(\theta, \phi, \psi_{cl}) = \mathbb{E}_{t, x_0, \epsilon}\left[\|\epsilon - \epsilon_\theta(x_t, t)\|^2\right],$$

where $$x_t = \sqrt{\bar{\alpha}_t} x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon$$ and $\epsilon\sim\mathcal{N}(0, I)$. The U-Net, equipped with the quantum bottleneck, outputs the estimate $\epsilon_\theta(x_t, t)$.

Optimization is conducted in a hybrid fashion:

• Classical parameters $(\psi_{cl}, \text{decoder})$ are updated via backpropagation through the concatenated classical-quantum feature space.
• Quantum parameters $(\theta, \phi)$ are updated using the parameter-shift rule for unbiased gradient estimation:

$$\partial_{\theta_i}\langle H_k\rangle = \frac{\langle H_k\rangle_{\theta_i+\pi/2} - \langle H_k\rangle_{\theta_i-\pi/2}}{2},$$

with analogous treatment for $\phi$.

5. Forward and Reverse Diffusion, Feature Roles

The forward diffusion process is purely classical and Gaussian:

$$q(x_t|x_{t-1}) = \mathcal{N}\left(\sqrt{1-\beta_t} x_{t-1}, \beta_t I\right), \quad x_t = \sqrt{\bar{\alpha}_t} x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon.$$

The reverse process integrates the quantum bottleneck:

$$p_\theta(x_{t-1}|x_t) = \mathcal{N}(\mu_\theta(x_t, t), \Sigma_t),$$

$$x_{t-1} = \frac{1}{\sqrt{\alpha_t}}\left(x_t - \frac{1-\alpha_t}{\sqrt{1-\bar{\alpha}_t}}\epsilon_\theta(x_t,t)\right) + \sigma_t z.$$

The quantum latent, with its 256→16 dimensionality reduction, enables the PQC $U(\theta_t)$ to efficiently learn a compact kernel for reverse diffusion, while the ANO measurements yield non-local features unavailable to purely classical or local-measurement hybrid models. This richer feature set improves generative fidelity and supports stable multi-class output.

6. Empirical Performance and Limitations

On the full MNIST dataset (digits 0–9), structurally coherent and class-recognizable images were generated for all classes using this hybrid architecture. Hardware constraints limited the image resolution, but evidence supports the method’s potential for advancing generative capabilities on current and near-term quantum devices ("NISQ era"). The architecture’s adaptive measurement layer mitigated mode collapse and enhanced the expressibility of learned distributions, outperforming purely classical or local-measurement hybrids in multi-modal generative settings (Jo et al., 3 Feb 2026).

7. Pipeline Summary and Significance

The central innovation is the full hybrid quantum-classical diffusion pipeline, structured as:

• Classical encoder $\to$ amplitude-encoded quantum latent $\to$ parameterized dynamics $U(\theta_t)$
• $\to$ adaptive non-local observables $\{H_k(\phi_k)\}$ measurement $\to$ classical decoder with skip-connection
• $\to$ reverse-step prediction $\to$ training via hybrid loss, leveraging parameter-shift gradient estimation for quantum parameters.

This configuration demonstrates a practical route to incorporating quantum latent-space compression and globally correlated feature extraction within large-scale generative models, particularly in constrained NISQ hardware regimes. The explicit combination of skip connections and adaptive trainable quantum measurements enables information preservation and expressive generation not accessible to prior architectures lacking these elements (Jo et al., 3 Feb 2026).