Problem
Research questions and friction points this paper is trying to address.
Quantum PAC Learning
Sample Complexity
State-Preparation Unitaries
Forward-Only Access
Query Complexity
Advantage of Sample Complexity in Quantum PAC Learning Requires Inverse Access to State-Preparation Unitaries
🤖 AI Summary
This study addresses whether forward-only access to state preparation unitaries can reduce sample complexity in quantum PAC learning. To this end, it proposes a novel lower-bound analysis framework based on state copies that integrates Haar-averaged output approximation, VC dimension analysis, and worst-case quantum query complexity theory. The work establishes that the optimal query complexities under forward-only access are Θ((d+log(1/δ))/ε) and Θ((d+log(1/δ))/ε²) for the realizable and agnostic settings, respectively. These bounds strictly match the known sample complexity limits for both classical and quantum data copies. Consequently, this result demonstrates that forward-only access yields no asymptotic advantage and reveals that inverse access is essential for enhancing learning precision.
📝 Abstract
Whether quantum computation can reduce the amount of data sampled from an unknown probability distribution required to learn a prediction rule is a fundamental question in quantum machine learning. Quantum PAC learning studies this question using quantum data as a quantum state whose squared amplitudes encode the unknown distribution from which classical learning data are sampled. With only copies of such quantum data, the optimal worst-case sample complexity asymptotically matches that of classical PAC learning. In contrast, access to both a state-preparation unitary for this state and its inverse can improve the query-complexity dependence on the accuracy parameter in realizable learning. However, it has remained unclear whether forward-only access allows such an improvement.
In this work, taking the worst case over compatible state-preparation unitaries and their finite ambient dimensions, we show that the optimal forward-only query complexities of realizable and agnostic learning are, respectively, $Θ((d+\log(1/δ))/\varepsilon)$ and $Θ((d+\log(1/δ))/\varepsilon^2)$, where $d$ is the VC dimension of the concept class, $\varepsilon$ the accuracy parameter, and $δ$ the failure probability. These bounds match the optimal sample complexities with classical data or quantum data copies. To prove them, we establish a reduction using $q$ copies of the prepared state to approximate the Haar-averaged output of any $q$-query forward-only algorithm.
These results show that forward-only access cannot provide an asymptotic query-complexity advantage over learning from classical data or quantum data copies in this worst-case setting, and establish the essential role of inverse access in the known realizable-setting improvement. Our reduction also provides a new framework for analyzing limitations of forward state-preparation access via state-copy lower bounds.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantum PAC Learning
Sample Complexity
State-Preparation Unitaries
Inverse Access
Query Complexity
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