Henry Yuen | LIP6 - Équipe QI

A complexity theory for non-local quantum computation

Thu, 01 Jan 2026 00:00:00 +0000

Non-local quantum computation (NLQC) replaces a local interaction between two systems with a single round of communication and shared entanglement. Despite many partial results, it is known that a characterization of entanglement cost in at least certain NLQC tasks would imply significant breakthroughs in complexity theory. Here, we avoid these obstructions and take an indirect approach to understanding resource requirements in NLQC, which mimics the approach used by complexity theorists: we study the relative hardness of different NLQC tasks by identifying resource efficient reductions between them. Most significantly, we prove that $f$-measure and $f$-route, the two best studied NLQC tasks, are in fact equivalent under $O(1)$ overhead reductions. This result simplifies many existing proofs in the literature and extends several new properties to $f$-measure. For instance, we obtain sub-exponential upper bounds on $f$-measure for all functions, and efficient protocols for functions in the complexity class $\mathsf{Mod}_k\mathsf{L}$. Beyond this, we study a number of other examples of NLQC tasks and their relationships.

A complexity theory for non-local quantum computation

Tue, 03 Jun 2025 00:00:00 +0000

Quantum statistical query learning

Mon, 07 Dec 2020 00:00:00 +0000

We propose a learning model called the quantum statistical learning QSQ model, which extends the SQ learning model introduced by Kearns to the quantum setting. Our model can be also seen as a restriction of the quantum PAC learning model: here, the learner does not have direct access to quantum examples, but can only obtain estimates of measurement statistics on them. Theoretically, this model provides a simple yet expressive setting to explore the power of quantum examples in machine learning. From a practical perspective, since simpler operations are required, learning algorithms in the QSQ model are more feasible for implementation on near-term quantum devices. We prove a number of results about the QSQ learning model. We first show that parity functions, (log n)-juntas and polynomial-sized DNF formulas are efficiently learnable in the QSQ model, in contrast to the classical setting where these problems are provably hard. This implies that many of the advantages of quantum PAC learning can be realized even in the more restricted quantum SQ learning model. It is well-known that weak statistical query dimension, denoted by WSQDIM(C), characterizes the complexity of learning a concept class C in the classical SQ model. We show that log(WSQDIM(C)) is a lower bound on the complexity of QSQ learning, and furthermore it is tight for certain concept classes C. Additionally, we show that this quantity provides strong lower bounds for the small-bias quantum communication model under product distributions. Finally, we introduce the notion of private quantum PAC learning, in which a quantum PAC learner is required to be differentially private. We show that learnability in the QSQ model implies learnability in the quantum private PAC model. Additionally, we show that in the private PAC learning setting, the classical and quantum sample complexities are equal, up to constant factors.