🤖 AI Summary
This work addresses the challenge of learning permutation-variant functions—i.e., functions whose inputs are sets or sequences of variable length—in quantum machine learning. Methodologically, it introduces (1) a permutation-invariant quantum set representation, achieved via quantum state averaging to model unordered collections; and (2) a quantum sequence model based on optimal coherentization of triply random tensors, where ordered structure is encoded through matrix product density operators. The framework unifies support for classification, regression, and density estimation. Empirical evaluation on synthetic benchmarks demonstrates superior expressive power and generalization performance compared to classical deep set models. By enabling principled quantum modeling of variable-length data, this work significantly expands the capability frontier of quantum machine learning for non-fixed-length structured inputs.
📝 Abstract
This paper introduces the quantum deep sets model, expanding the quantum machine learning tool-box by enabling the possibility of learning variadic functions using quantum systems. A couple of variants are presented for this model. The first one focuses on mapping sets to quantum systems through state vector averaging: each element of the set is mapped to a quantum state, and the quantum state of the set is the average of the corresponding quantum states of its elements. This approach allows the definition of a permutation-invariant variadic model. The second variant is useful for ordered sets, i.e., sequences, and relies on optimal coherification of tristochastic tensors that implement products of mixed states: each element of the set is mapped to a density matrix, and the quantum state of the set is the product of the corresponding density matrices of its elements. Such variant can be relevant in tasks such as natural language processing. The resulting quantum state in any of the variants is then processed to realise a function that solves a machine learning task such as classification, regression or density estimation. Through synthetic problem examples, the efficacy and versatility of quantum deep sets and sequences (QDSs) is demonstrated.