Information-Theoretic Limits of Quantum Learning via Data Compression

TL;DR

This work develops an information-theoretic framework linking quantum data, lossy compression, and PAC learning to bound the power of quantum data in practical learning scenarios. By deriving a PAC Nayak-type bound and applying it to Zipf-distributed data, it shows that quantum sample complexity for learning arbitrary functions under Zipf scales as $m \ge \frac{(1-\delta)(1-H(\beta))N - H(\delta)}{n}$ with $N=2^n$, while classical sample complexity remains $O\left(\frac{N}{\epsilon}\log\frac{N}{\delta}\right)$, indicating at most polynomial quantum advantage. It also proves an Ω(n) qubit lower bound for PAC learning linear functions, establishing Θ(n) qubits as necessary and sufficient, and extends the framework to tighten security bounds for MBQC-based delegated quantum computation, showing that sublinear communication cannot reveal large portions of the target computation (e.g., $\varepsilon>0.08$). Overall, the paper unifies rate-distortion concepts with learning and secure delegated computation, with implications for realistic data distributions and cloud-based quantum hardware.

Abstract

Understanding the power of quantum data in machine learning is central to many proposed applications of quantum technologies. While access to quantum data can offer exponential advantages for carefully designed learning tasks and often under strong assumptions on the data distribution, it remains an open question whether such advantages persist in less structured settings and under more realistic, naturally occurring distributions. Motivated by these practical concerns, we introduce a systematic framework based on quantum lossy data compression to bound the power of quantum data in the context of probably approximately correct (PAC) learning. Specifically, we provide lower bounds on the sample complexity of quantum learners for arbitrary functions when data is drawn from Zipf's distribution, a widely used model for the empirical distributions of real-world data. We also establish lower bounds on the size of quantum input data required to learn linear functions, thereby proving the optimality of previous positive results. Beyond learning theory, we show that our framework has applications in secure delegated quantum computation within the measurement-based quantum computation (MBQC) model. In particular, we constrain the amount of private information the server can infer, strengthening the security guarantees of the delegation protocol proposed in (Mantri et al., PRX, 2017).

Figures (1)

• Figure 1: Quantum lossy data compression and applications. An overview of the quantum lossy data compression framework and the applications we study in this work. The PAC Nayak bound we derive places bounds on the receiver in a source coding protocol decoding the correct binary string, encoded in less qubits than information-theoretically required to perform a perfect decoding. The first application is in the quantum sample complexity of supervised learning under the $\mathsf{Zipf}$ distribution. Here, PAC source coding can be used to derive a sample complexity for quantum learners which may only be polynomially better than using classical samples. The second is in proving bounds on the ability of an adversary in a delegated quantum computation protocol (in the measurement-based framework of quantum computing (MBQC)) to guess the client's chosen computation.

Theorems & Definitions (19)

• Lemma 1.1: Nayak's bound
• Lemma 1.2: PAC Nayak bound
• proof : Proof of Lemma \ref{['lem:PACNayak']}
• Lemma 1.3: Learning a string with quantum data
• proof
• Theorem 1.4
• proof
• Theorem 1.5
• proof
• Lemma 1.6: Approximate Recovery of Linear Functions