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

TL;DR

This work considers the problems of learning junta distributions, their quantum counterparts (quantum junta states) and $\mathsf{QAC}^0$ circuits, which are shown to be close to juntas and shows that the Choi states of those circuits are close to be juntas.

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 $p:\{-1,1\}^n\to \mathbb [0,1]$ 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 $O(2^k\log(n)/\varepsilon^2)$ 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 $Ω((4^k+\log (n))/\varepsilon^2)$ copies. Additionally, we show that, for constant $k$, $\tildeΘ(2^n/\varepsilon^2)$ copies are necessary and sufficient to test whether a state is $\varepsilon$-close or $7\varepsilon$-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.

Theorems & Definitions (28)

• Theorem 1
• Definition 2: Junta state
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• Lemma 7: Hoeffding bound
• Lemma 8: Bernstein inequality
• Theorem 8
• Remark 9