Efficient Non-Adaptive Quantum Algorithms for Tolerant Junta Testing
Zongbo Bao, Yuxuan Liu, Penghui Yao, Zekun Ye, Jialin Zhang
TL;DR
The paper tackles tolerant quantum junta testing for both $n$-qubit unitaries and $n$-variable Boolean functions, defining $(\varepsilon_1, \varepsilon_2)$-tolerance and focusing on non-adaptive quantum testers. It introduces a Coordinate-Extractor that uses Fourier- and Pauli-based analysis to reduce the problem to a small set of high-influence coordinates and then presents three testers: (i) a constant-gap $O(k\log k)$-query tester for unitaries, (ii) a constant-gap $O(k\log k)$-query tester for Boolean functions, and (iii) a gapless, arbitrary-gap tester for unitaries with $2^{\tilde{O}(k)}$ queries. The first two testers can be implemented using only single-qubit operations in experiment-friendly variants, yielding exponential quantum advantages over classical approaches. The results generalize tolerant junta testing to quantum channels with a robust framework based on Pauli analysis and nuclear-norm estimation, offering practical pathways for near-term quantum experiments and advancing quantum property testing theory.
Abstract
We consider the problem of deciding whether an $n$-qubit unitary (or $n$-bit Boolean function) is $\varepsilon_1$-close to some $k$-junta or $\varepsilon_2$-far from every $k$-junta, where $k$-junta unitaries act non-trivially on at most $k$ qubits and as the identity on the rest, and $k$-junta Boolean functions depend on at most $k$ variables. For constant numbers $\varepsilon_1,\varepsilon_2$ such that $0 < \varepsilon_1 < \varepsilon_2 < 1$, we show the following. (1) A non-adaptive $O(k\log k)$-query tolerant $(\varepsilon_1,\varepsilon_2)$-tester for $k$-junta unitaries when $2\sqrt{2}\varepsilon_1 < \varepsilon_2$. (2) A non-adaptive tolerant $(\varepsilon_1,\varepsilon_2)$-tester for Boolean functions with $O(k \log k)$ quantum queries when $4\varepsilon_1 < \varepsilon_2$. (3) A $2^{\widetilde{O}(k)}$-query tolerant $(\varepsilon_1,\varepsilon_2)$-tester for $k$-junta unitaries for any $\varepsilon_1,\varepsilon_2$. The first algorithm provides an exponential improvement over the best-known quantum algorithms. The second algorithm shows an exponential quantum advantage over any non-adaptive classical algorithm. The third tester gives the first tolerant junta unitary testing result for an arbitrary gap. Besides, we adapt the first two quantum algorithms to be implemented using only single-qubit operations, thereby enhancing experimental feasibility, with a slightly more stringent requirement for the parameter gap.