Quantum Metric Encoder (QME)

• Quantum Metric Encoder (QME) is a data-driven, trainable embedding method that maps classical or quantum data into Hilbert space for geometric and metric analysis.
• It leverages approaches such as Hermitian operator encoding, parameterized quantum circuits, and amplitude encoding to enable efficient metric learning and dimensionality control.
• QME demonstrates practical applications in supervised classification, reinforcement learning, and topological diagnostics while ensuring resource-efficient hybrid quantum computation.

The Quantum Metric Encoder (QME) is a family of data-driven, trainable quantum or quantum-inspired embedding modules designed to map classical or quantum data into Hilbert space representations that admit tractable quantum metrics, enable global geometric analysis, and facilitate downstream machine learning tasks such as supervised classification or reinforcement learning. Originating in diverse frameworks—including the Quantum Cognition Machine Learning (QCML) approach (Abanov et al., 22 Jul 2025), quantum circuit-based metric learning for classification (Lloyd et al., 2020), and recent quantum-inspired approaches to offline reinforcement learning (Lv et al., 13 Nov 2025)—QMEs provide a versatile interface between geometric data encoding, metric learning, and resource-efficient quantum (or classical/quantum-hybrid) computation.

1. Mathematical Foundations and Model Architecture

1.1 QCML QME: Hermitian Operator Encoding

In the QCML framework, the QME consists of a set of learned Hermitian operators $\{H_\mu\}$ for each of the $D$ features or coordinates $x^\mu$ of the data. A displacement Hamiltonian is constructed as

$$H(x) = \frac{1}{2} \sum_{\mu=1}^D \left( H_\mu - x^\mu \, \mathbb{I} \right)^2,$$

and each data point $x \in \mathbb{R}^D$ is mapped to the unique lowest-energy eigenvector (ground state) $|\psi(x)\rangle$ of $H(x)$. This quantum encoding is formally represented as $|\psi(x)\rangle = U(x)|0\rangle$ for some data-dependent unitary $U(x)$ that diagonalizes $H(x)$ (Abanov et al., 22 Jul 2025).

1.2 Quantum Circuit-Based QME for Machine Learning

Alternatively, the QME can take the form of a parameterized quantum circuit built from layers of data-encoding rotations, trainable local rotations, and entangling gates. For an $n$-qubit circuit, classical features $x_k$ are encoded via rotations $R_x(x_k)$, combined with layers of trainable $R_y(\theta_k)$ and $U_{ZZ}(\phi_k)$ gates, yielding a unitary $U(x;\theta)$ that acts on an initial state $|0^{\otimes n}\rangle$: $|x;\theta\rangle = U(x;\theta)|0\cdots 0\rangle$ (Lloyd et al., 2020).

1.3 Quantum-Inspired Autoencoder QME

For reinforcement learning, classical states $s \in \mathbb{R}^n$ are amplitude encoded to $q$-qubit states: $$|s\rangle = \frac{1}{\|s\|_2} \sum_{i=0}^{2^q-1} s_i\, |i\rangle.$$ The unitary circuit $$U(\boldsymbol\theta) = U_L(\theta_L) \cdots U_1(\theta_1)$$, split into encoder, trash disposer, and decoder subcircuits, enables both metric learning and reward decoding. The embedding $f_\theta(s)$ is extracted as a classical vector from the latent qubits after discarding 'trash' and 'reward' qubits (Lv et al., 13 Nov 2025).

2. Quantum Metric and Geometric Analysis

2.1 Quantum Metric Tensor

Given a manifold of ground states $|\psi(x)\rangle$, the QME framework supports the explicit computation of a quantum metric tensor via the Fubini–Study pullback: $$g_{\mu\nu}(x) = \Re \left[ \langle \partial_\mu \psi(x) | (1 - |\psi(x)\rangle\langle\psi(x)|) | \partial_\nu \psi(x) \rangle \right ]$$ or equivalently, using the quantum Fisher information in terms of the symmetric logarithmic derivative $L_\mu$ (Abanov et al., 22 Jul 2025).

2.2 Explicit Expansion via Learned Hermitians

The metric tensor components can be directly related to the trained Hermitian operators: $$g_{\mu\nu} = \sum_{n>0} \frac{\langle 0 | H_\mu - x^\mu I | n \rangle \langle n | H_\nu - x^\nu I | 0 \rangle}{(E_n - E_0)^2}$$ where $|n\rangle$ and $E_n$ are excited states and eigenvalues of $H(x)$.

2.3 Berry Curvature and Topological Structure

The antisymmetric component, the Berry curvature

$$F_{\mu\nu}(x) = 2\, \Im \langle \partial_\mu \psi(x) | \partial_\nu \psi(x) \rangle$$

encodes geometric phase properties of the learned quantum manifold.

2.4 Intrinsic (Quantum) Dimension

The spectrum of $g_{\mu\nu}(x)$ measures local distinguishability; a spectral gap after the first $d$ eigenvalues signals an intrinsic dimension $d$. The Laplacian

$\Delta = \sum_\mu [H_\mu, [H_\mu, \cdot]]$

can also be analyzed spectrally to recover $d$ via Weyl’s law

$$N(\Lambda) \sim \Lambda^{d/2} \text{ as } \Lambda \to \infty.$$

2.5 Hyperbolicity and State-Space Geometry

In reinforcement learning applications, the $\Delta$-hyperbolicity of the QME-embedded state space—measured using the Gromov 4-point criterion—drops from $0.5$–$0.6$ (original states) to $0.1$–$0.2$ after quantum metric encoding, indicating a more 'tree-like' and efficiently navigable geometry (Lv et al., 13 Nov 2025).

3. Training Procedures and Optimization

3.1 QCML Training Objective

Parameters $\{H_\mu\}$ are optimized by minimizing the loss

$$L[H] = \sum_t \left( \| \langle\psi(x^t)|H_\mu|\psi(x^t)\rangle - x^t_\mu \|^2 + w \left\langle \left(H_\mu - \langle H_\mu \rangle \right)^2 \right\rangle \right )$$

where $w > 0$ modulates the tradeoff between data fidelity and quantum localization. Gradient-based optimization (e.g., Adam) is performed, typically using automatic differentiation through eigenvector computation (Abanov et al., 22 Jul 2025).

3.2 Quantum Circuit Metric Learning

For classification tasks, the objective is to maximize separation between class-averaged quantum states $\rho$ and $\sigma$ using metrics such as the trace distance $D_{\mathrm{tr}}(\rho, \sigma)$ or Hilbert–Schmidt distance $D_{\mathrm{hs}}(\rho, \sigma)$. The loss to minimize under the $\ell_2$ metric is

$$J_{\mathrm{hs}}(\theta) = 1 - \frac{1}{2} D_{\mathrm{hs}}(\rho(\theta),\sigma(\theta)) = 1-\frac{1}{2}\left[\mathrm{Tr}\,\rho^2+\mathrm{Tr}\,\sigma^2-2\mathrm{Tr} (\rho\sigma)\right]$$

with gradients readily obtained via the parameter-shift rule (Lloyd et al., 2020).

3.3 Quantum Autoencoder Loss for RL

In reinforcement learning, the loss per sample is

$$L_i = (1-\delta)\,[1-\langle Z_0\rangle_i] + \frac{\delta}{n_{\text{trash}}} \sum_{s=1}^{n_{\text{trash}}}[1-\langle Z_s\rangle_i]$$

where $\langle Z_0 \rangle_i$ is the expectation on the reward qubit. The optimizer can be COBYLA or standard gradient-based methods (Lv et al., 13 Nov 2025).

3.4 Avoidance of Curse of Dimensionality

Capacity control is effected by choosing the Hilbert space dimension $N$ just exceeding the intrinsic data dimension; typical examples employ $N \sim 8$–$32$ in synthetic/real datasets (Abanov et al., 22 Jul 2025).

4. Measurement and Inference

4.1 Optimal Measurement for Classification

Once the embedding parameters are trained, the measurement minimizing linear classification loss is analytically determined:

• Helstrom measurement for $\ell_1$ (trace) distance: measure $\Delta = \rho-\sigma$, separate positive/negative eigenspaces and apply the two-outcome POVM.
• Overlap (fidelity) measurement for $\ell_2$ (Hilbert–Schmidt) distance: estimate expectation $$\langle x;\theta | \rho - \sigma | x;\theta \rangle$$ via SWAP or inversion tests (Lloyd et al., 2020).

This closed-form determination eliminates the need for a variational measurement circuit at inference.

4.2 Reward Decoding in RL

For the QME autoencoder, the decoded reward $r_q$ is extracted from the measured $p_0$ of the reward qubit: $$r_{\text{q}} = r_{\min} + p_0 (r_{\max} - r_{\min}).$$

5. Empirical Results and Applications

5.1 Geometric and Topological Structure

• Synthetic sphere data ($S^2 \subset \mathbb{R}^3$): QME recovers the canonical round-sphere metric, Laplacian spectrum, and Berry monopole charge consistent with theoretical predictions (Abanov et al., 22 Jul 2025).
• Wisconsin Breast Cancer dataset: Intrinsic quantum dimension determined as $d=2$ via both metric gap and Laplacian spectrum; eigenmap analysis relates abstract coordinates to prominent data features.

5.2 Reinforcement Learning Performance

On three D4RL robotics datasets, offline RL agents trained on QME-embedded data achieve $116.2\%$ (SAC) and $117.6\%$ (IQL) average improvement over baseline RL. Normalization alone yields moderate improvement; CNN and QNN decoders fail to match QME's gains. Ablation indicates QME's contributions are statistically significant (Lv et al., 13 Nov 2025).

5.3 Circuit Complexity and Resource Estimates

• For $n\approx 50$ qubits and $L\approx10$ layers at $10$ MHz, up to $10^{10}$ classical bits can be encoded within coherence times on current NISQ devices (Lloyd et al., 2020).
• Amplitude encoding typically requires $q=\lceil \log_2 (\dim(s)) \rceil$ qubits for state $s$ (Lv et al., 13 Nov 2025).

5.4 Geometric Diagnostics

After quantum metric encoding, $\Delta$-hyperbolicity of the state space approaches $0.1$–$0.2$, correlating strongly with empirical RL performance and highlighting altered underlying geometry (Lv et al., 13 Nov 2025).

6. Limitations, Open Questions, and Prospects

• QME methods, in existing formulations, require reward supervision; unsupervised generalization is unresolved (Lv et al., 13 Nov 2025).
• Theoretical understanding of generalization from few samples remains open.
• While quantum-inspired encodings outperform classical and quantum neural nets on tested RL benchmarks, the existence of equally performant classical architectures is undetermined.
• The connection between low $\Delta$-hyperbolicity and inductive bias/goodness of geometric embedding merits deeper investigation, especially in relation to hyperbolic neural architectures (Lv et al., 13 Nov 2025).
• QME provides analytic tractability for both metric and measurement, efficient circuit depth for NISQ hardware, and avoids local overfitting typical of classical high-dimensional encoders (Lloyd et al., 2020, Abanov et al., 22 Jul 2025).

In summary, QME represents a convergence of quantum geometric analysis, metric learning, and practical circuit realizability, with demonstrated benefits across both synthetic geometric datasets and reinforcement learning applications, while presenting fertile ground for further theoretical and empirical investigation.