Quantum-Compatible Dictionary Learning via Doubly Sparse Models

TL;DR

This work addresses the misalignment between classical dictionary learning and near-term quantum hardware by proposing a structurally compatible reformulation called doubly sparse dictionary learning (DSDL), where the dictionary is constrained as D = Φ A with a fixed basis Φ and sparse coefficients A. The main approach replaces greedy and dense optimization with projection-based randomized Kaczmarz iterations, using quantum inner product estimates to accelerate these linear solves in a hybrid quantum–classical loop, without relying on QRAM. Key contributions include the conceptual shift to projection-centric learning, the design of a simple hybrid algorithm, and an open-source reference implementation that demonstrates feasibility on near-term devices. The work provides a principled path toward quantum-compatible representation learning, clarifying when quantum acceleration is plausible and how to structure learning problems to leverage quantum primitives, even without exponential speedups.

Abstract

Dictionary learning (DL) is a core tool in signal processing and machine learning for discovering sparse representations of data. In contrast with classical successes, there is currently no practical quantum dictionary learning algorithm. We argue that this absence stems from structural mismatches between classical DL formulations and the operational constraints of quantum computing. We identify the fundamental bottlenecks that prevent efficient quantum realization of classical DL and show how a structurally restricted model, doubly sparse dictionary learning (DSDL), naturally avoids these problems. We present a simple, hybrid quantum-classical algorithm based on projection-based randomized Kaczmarz iterations with Qiskit-compatible quantum inner products. We outline practical considerations and share an open-source implementation at https://github.com/AngshulMajumdar/quantum-dsdl-kaczmarz. The goal is not to claim exponential speedups, but to realign dictionary learning with the realities of near-term quantum devices.