QPMeL - Quantum-Aware Classically-Trained Embeddings via Projective Metric Learning

TL;DR

QPMeL introduces a quantum-aware, classically trained embedding framework that maps classical data to a unified spherical feature space via angular encodings, enabling low-depth, hardware-efficient quantum representations on NISQ devices. It replaces quantum circuit training with a classical encoder and a Projective Metric Function (PMeF) that approximates quantum fidelity using real-valued angular coordinates, thereby avoiding barren plateaus. Empirically, QPMeL achieves state-of-the-art or competitive results on standard MNIST and Fashion-MNIST benchmarks, few-shot learning tasks, and multi-modal few-shot learning, while using as few as $20$ qubits and shallow circuits. The work provides a promising direction for quantum embeddings that leverage classical training while preserving quantum geometric structure, with potential cross-modal applications.

Abstract

Deep metric learning has recently shown extremely promising results in the classical data domain, creating well-separated feature spaces. This idea was also adapted to quantum computers via Quantum Metric Learning(QMeL). QMeL consists of a 2-step process with a classical model to compress the data to fit into the limited number of qubits, then train a Parameterized Quantum Circuit(PQC) to create better separation in Hilbert Space. However, on Noisy Intermediate Scale Quantum (NISQ) devices. QMeL solutions result in high circuit width and depth, both of which limit scalability. We propose Quantum Polar Metric Learning (QPMeL) that uses a classical model to learn the parameters of the polar form of a qubit. We then utilize a shallow PQC with $R_y$ and $R_z$ gates to create the state and a trainable layer of $ZZ(θ)$-gates to learn entanglement. The circuit also computes fidelity via a SWAP Test for our proposed Fidelity Triplet Loss function, used to train both classical and quantum components. When compared to QMeL approaches, QPMeL achieves 3X better multi-class separation, while using only 1/2 the number of gates and depth. We also demonstrate that QPMeL outperforms classical networks with similar configurations, presenting a promising avenue for future research on fully classical models with quantum loss functions.

Figures (6)

• Figure 1: QPMeL utilizes the surface of independent unit spheres as a unified feature space between quantum states and classical vectors. This unified space spans all unentangled quantum states. Once trained, a NISQ friendly depth-efficient circuit using only 2 gates per qubit encodes data.
• Figure 2: Main families of quantum data encodings classified based on the relationship between the classical model and quantum circuit.
• Figure 3: QPMeL Encoder: Consists of a standard metric encoder with a dense block appended to it. The final layer of the dense block is used as the input to 2 independent dense layers, which produce the angular coordinates $\vec{\theta}, \vec{\gamma}$. This is termed 'Angle Projection'.
• Figure 4: QPMeL Inference Pipeline: The angular coordinates from the QPMeL model are used to parameterize the quantum gates. The resulting quantum state is then measured to produce the final embedding.
• Figure 5: QPMeL training with prototypical loss: The QPMeL encoder produces a set of angular coordinates for the multi-modal support and query vectors. These angular coordinates are converted to Cartesian form, after which the Complex Kernel Function (CKF) is used to compute similarity to be used with a Metric Loss function. The combination of CKF and metric loss creates a 'Quantum-Aware' loss function. Cartesian Conversion and CKF jointly create the Projective Metric Function (PMeF).

Theorems & Definitions (2)

• Definition 1: Complex Kernel Function (CKF)
• Definition 2: Projective Metric Function (PMeF)