Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing (1507.03126v1)

Abstract: In the $k$-junta testing problem, a tester has to efficiently decide whether a given function $f:{0,1}^n\rightarrow {0,1}$ is a $k$-junta (i.e., depends on at most $k$ of its input bits) or is $\epsilon$-far from any $k$-junta. Our main result is a quantum algorithm for this problem with query complexity $\tilde O(\sqrt{k/\epsilon})$ and time complexity $\tilde O(n\sqrt{k/\epsilon})$. This quadratically improves over the query complexity of the previous best quantum junta tester, due to At\i c\i\ 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 $\Omega(k^{1/3})$ queries for junta-testing (for constant $\epsilon$).

• The paper introduces quantum algorithms that significantly reduce query complexity in gapped group testing and junta testing of Boolean functions.
• It achieves up to a quartic speedup in group testing and leverages efficient quantum Fourier transforms for improved junta testing.
• The analysis underscores theoretical advances with practical implications for scalable quantum property testing in modern computer science.

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

This paper explores quantum property testing, focusing on algorithms for the k-junta resistance problem and the gapped group testing problem (QGGT). The key contributions include quantum algorithms offering substantial improvements over classical counterparts and a detailed analysis of their efficiency in both query and time complexity.

Quantum Property Testing

Quantum algorithms provide new avenues for improving computational efficiency in property testing. The paper leverages quantum mechanics' potential to design testers for properties of Boolean functions, focusing on scenarios where the function either possesses a property or is distant from any function with that property. This approach significantly eases many computationally taxing problems.

Gapped Group Testing (GGT) and Quantum Algorithm

The gapped group testing problem seeks to determine whether a hidden set A, accessed via a function, is small or large. The authors introduce a quantum algorithm resolving this with query complexity O(√k/d), achieving up to a quartic improvement over classical algorithms in certain regimes. This showcases quantum computing's power to greatly expedite problem-solving where classical methods are limited by inherent constraints.

The Key Results in Junta Testing

Junta testing involves distinguishing whether a function depends on at most k variables or is ε-far from any such function. The quantum algorithm designed here exhibits query complexity O(√k/ε log k), significantly improving upon previous attempts and rendering any matching classical tester obsolete with current techniques. Notably, the upper bound excludes extremely small influences from individual variables, thus leveraging grouped hypothesis testing. Through constructing an efficient quantum Fourier transform on symmetric groups, the algorithm's implementation achieves time efficiency, ensuring its practical applicability.

Theoretical Implications and Future Directions

The advancements demonstrated within the paper suggest promising future developments in quantum algorithms. It opens dialogue around further improving quantum tester infrastructure, particularly regarding complexities and potential larger class testing problems. Moreover, the adaptability and precision of these quantum approaches may find broad applicability across various domains, emphasizing the practical relevance of quantum computing in modern computer science.

Further exploration of algorithmic design could result in even more effective techniques, possibly extending beyond Boolean functions to richer structures in computer science and applied mathematics. The paper encourages ongoing investigation into both theoretical implications and concrete advancements within the field of quantum algorithmic property testing.

In summary, this paper establishes a crucial milestone in leveraging quantum computation for efficient property testing, offering not only mathematical insights but paving the way for scalable quantum solutions in broader computational problems.