Learning junta distributions, quantum junta states, and QAC$^0$ circuits

2024-10-21

AI Summary

This work unifies the learning problem for three classes of objects exhibiting low-dependence structure: $k$-juntas (probability distributions depending on at most $k$ variables), quantum $k$-juntas (states where only a $k$-qubit subsystem is nontrivial, while the rest is maximally mixed), and QAC$^0$ circuits. Methodologically, it introduces the first formal definition of quantum $k$-juntas; establishes structural connections between junta distributions/states and the Choi states of QAC$^0$ circuits; and leverages total variation/trace distance analysis, Pauli spectrum techniques, Choi representation, tensor network decomposition, and concentration inequalities. Key contributions are: (1) optimal sample complexity $O(2^k log n / varepsilon^2)$ for learning classical $k$-junta distributions; (2) the first efficient single-copy learning algorithm for quantum $k$-juntas, requiring $O(12^k log n / varepsilon^2)$ samples; and (3) an exponential improvement in learning QAC$^0$ circuits—reducing sample complexity from $n^{mathrm{poly}}$ to $2^{O((log(s^2 2^a))^d)} log n$, where $s$ is size and $a$ is fan-in.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

In this work, we consider the problems of learning junta distributions, their quantum counterparts (quantum junta states) and $mathsf{QAC}^0$ circuits, which we show to be close to juntas. (1) Junta distributions. A probability distribution $p:{-1,1}^n o mathbb [0,1]$ is a $k$-junta if it only depends on $k$ bits. We show that they can be learned with to error $varepsilon$ in total variation distance from $O(2^klog(n)/varepsilon^2)$ samples, which quadratically improves the upper bound of Aliakbarpour et al. (COLT'16) and matches their lower bound in every parameter. (2) Junta states. We initiate the study of $n$-qubit states that are $k$-juntas, those that are the tensor product of a $k$-qubit state and an $(n-k)$-qubit maximally mixed state. We show that these states can be learned with error $varepsilon$ in trace distance with $O(12^{k}log(n)/varepsilon^2)$ single copies. We also prove a lower bound of $Omega((4^k+log (n))/varepsilon^2)$ copies. Additionally, we show that, for constant $k$, $ ilde{Theta}(2^n/varepsilon^2)$ copies are necessary and sufficient to test whether a state is $varepsilon$-close or $7varepsilon$-far from being a $k$-junta. (3) $mathsf{QAC}^0$ circuits. Nadimpalli et al. (STOC'24) recently showed that the Pauli spectrum of $mathsf{QAC}^0$ circuits (with a limited number of auxiliary qubits) is concentrated on low-degree. We remark that they implied something stronger, namely that the Choi states of those circuits are close to be juntas. As a consequence, we show that $n$-qubit $mathsf{QAC}^0$ circuits with size $s$, depth $d$ and $a$ auxiliary qubits can be learned from $2^{O(log(s^22^a)^d)}log (n)$ copies of the Choi state, improving the $n^{O(log(s^22^a)^d)}$ by Nadimpalli et al.

Problem

• Learning k-junta distributions efficiently
• Studying quantum k-junta states
• Analyzing QAC^0 circuits' junta properties

Innovation

• Improved junta distribution learning
• Quantum junta state analysis
• Enhanced QAC$^0$ circuit learning