Inspired by a recent classical distribution-free junta tester by Chen, Liu, Serverdio, Sheng, and Xie (STOC'18), we construct a quantum tester for the same problem with complexity $O(k/\varepsilon)$, which constitutes a quadratic improvement. We also prove that there is no efficient quantum algorithm for this problem using quantum examples as opposed to quantum membership queries. This result was obtained independently from the $\tilde O(k/\varepsilon)$ algorithm for this problem by Bshouty.
Inspired by a recent classical distribution-free junta tester by Chen, Liu, Serverdio, Sheng, and Xie (STOC'18), we construct a quantum tester for the same problem with complexity O(k/ε), which constitutes a quadratic improvement. We also prove that there is no efficient quantum algorithm for this problem using quantum examples as opposed to quantum membership queries. This result was obtained independently from the O~(k/ε) algorithm for this problem by Bshouty.
In the k-junta testing problem, a tester has to efficiently decide whether a given function f:{0,1}n→{0,1} is a k-junta (i.e., depends on at most k of its input bits) or is ε-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity O~(k/ε) and time complexity O~(nk/ε). This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k1/3) queries for junta-testing (for constant ε).
In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0,1}n → {0,1} is a k-junta (i.e., depends on at most k of its input bits) or is ∈-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity Ō(√∈) and time complexity Ō(n√∈). This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atici and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of ω(k1/3) queries for junta-testing (for constant ∈).
We establish the first super-polynomial quantum advantage for the tolerant junta testing problem in the adaptive setting. Specifically, we show that within a certain parameter regime, tolerant k-junta testing with high precision can be solved using poly(k) quantum queries, whereas any classical algorithm requires at least kΩ(logk) queries.
The problem of tolerant k-junta testing is as follows: given parameters (k,ε1,ε2), with 0≤ε1<ε2≤1/2, and black-box access to a Boolean function f (defined on n variables), distinguish whether f is ε1-close to some k-junta or ε2-far from every k-junta.
We show the quantum advantage for a range of parameters close to 1/2, for example, ε1=1/2−1/k and ε2=1/2−1/(2k2). The (non-adaptive) quantum tester we use was given by a recent work of Bao, Liu, Yao, Ye, and Zhang (SOSA 2026). We slightly adapt their analysis to show that it holds in the above parameter regime. On the other hand, our classical lower bound requires substantial new ideas. Inspired by the lower bound techniques of Chen and Patel (FOCS 2023), we introduce a new hard distribution of ``yes'' instances (i.e., instances with distance at most ε1 to k-juntas) that is based on planting an ``approximate-junta'' as follows: we randomly pick k out of n coordinates, and for each fixing of the k coordinates, the 2n−k values in the restricted subcube are drawn randomly except for the set of points in an error-correcting code on which we place the same random bit. We show that this distribution is much closer to k-juntas than the uniform distribution, but on the other hand, they are indistinguishable with respect to any classical algorithm making ko(logk) queries.
Junta testing for Boolean functions has sparked a long line of work over recent decades in theoretical computer science, and recently has also been studied for unitary operators in quantum computing. Tolerant junta testing is more general and challenging than the standard version. While optimal tolerant junta testers have been obtained for Boolean functions, there has been no knowledge about tolerant junta testers for unitary operators, which was thus left as an open problem in [Chen, Nadimpalli, and Yuen, SODA2023]. In this paper, we settle this problem by presenting the first algorithm to decide whether a unitary is ε1-close to some quantum k-junta or is ε2-far from any quantum k-junta, where an n-qubit unitary U is called a quantum k-junta if it only non-trivially acts on just k of the n qubits. More specifically, we present a tolerant tester with ε1=8ρε, ε2=ε, and ρ∈(0,1), and the query complexity is O(ε2ρ(1−ρ)kklogk), which demonstrates a trade-off between the amount of tolerance and the query complexity. Note that our algorithm is non-adaptive which is preferred over its adaptive counterparts, due to its simpler as well as highly parallelizable nature. At the same time, our algorithm does not need access to U†, whereas this is usually required in the literature.
We study the query complexity of testing quantum juntas. Given an unknown n-qubit unitary or quantum channel, the goal is to decide whether it acts nontrivially on at most k qubits or is ε-far from every such operation. For unitaries, we consider the forward-only model, where the tester has access to U, but not to U† or controlled-U. We give tight bounds for both unitary and channel junta testing. For sufficiently large k, n≥2k, and a broad range of ε, both problems have query complexity Θ(k/(ε2logk)). Our upper bounds are achieved by simple nonadaptive testers using product inputs and single-qubit measurements. For junta channels, this closes the gap by Bao and Yao (COLT 23, TPAMI 2025) between an O(k) upper bound and a Ω(k) lower bound at constant ε, establishing the optimal Θ(k/logk) query complexity. At the heart of both bounds is a strong connection between quantum junta testing and classical support-size testing. For the upper bounds, we reduce quantum junta testing to a generalized support-size problem for distributions over random subsets, and develop a new tester with improved sample complexity. For the lower bounds, we proceed in the reverse direction, reducing classical support-size testing to quantum junta testing through a phase-masked hard ensemble and showing that forward quantum queries can be exactly classicalized on average. Together, these reductions identify support-size testing as the classical core of quantum junta testing and yield matching lower bounds. At constant ε, they also give a separation between Θ(k/logk) forward queries and O(k) queries with inverse access.
In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0, 1}n → {0, 1} is a k-junta (i.e., depends on at most k of its input bits) or is ∊-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity and time complexity . This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k1/3) queries for junta-testing (for constant ∊).