Quantum Search with Generalized Wildcards Achieves Near-Tight Bounds of Cost and Lower Bound of

The challenge of searching for hidden information forms the basis of many computational problems, and researchers continually seek more efficient methods to tackle them. Arjan Cornelissen, Nikhil Mande from University of Liverpool, Subhasree Patro from Eindhoven University of Technology, Nithish Raja from Eindhoven University of Technology, and Swagato Sanyal from University of Sheffield, now present a significant advance in this field by exploring a generalized form of wildcard search. Their work investigates the complexity of finding a hidden bit-string when algorithms can test equality against subsets of that string, and establishes near-optimal bounds for various types of these subsets, including bounded-size sets, contiguous blocks, and prefixes. Crucially, the team develops a novel framework and employs a direct optimization program to achieve these upper bounds, representing a departure from traditional methods and offering new insights into the limits of efficient search algorithms.

Quantum Search With Wildcard Generalizations

Scientists are exploring quantum algorithms for searching hidden information, specifically addressing the “search with wildcards” problem where the goal is to identify an unknown sequence of bits. A quantum algorithm now achieves a cost of O(√n log2 n) for a generalized search, representing an improvement over existing methods. This algorithm utilizes a novel technique, constructing a specific operator that efficiently encodes information about the wildcard queries. Researchers demonstrate that this approach not only reduces the search cost but also provides a deeper understanding of the search space’s underlying structure. Furthermore, the team establishes a lower bound of Ω(√n), confirming that their algorithm is optimal to within a logarithmic factor.

Query Complexity Bounds Using Adversarial Optimization

Researchers have developed a new framework for analyzing how many questions are needed to learn hidden information, focusing on scenarios where algorithms can only ask about specific parts of the hidden sequence. This study pioneers a method for establishing tight limits on query complexity when algorithms are restricted to querying subsets of the sequence, including bounded-size sets, contiguous blocks, and prefixes. The framework characterizes query complexity using an optimization program that maximizes a ratio dependent on the function’s maximum value and standard deviation on a subcube. Scientists uniquely employed a technique to derive upper bounds on query complexity without relying on standard mathematical duality techniques.

To demonstrate its power, the team investigated the case of contiguous blocks, where queries are limited to consecutive variables. They established an upper bound of n queries by observing that single-bit queries are always permissible and applying their general framework. A lower bound was proven using a function constructed from disjoint blocks, ensuring that restrictions to contiguous variables only affect a limited number of blocks. For prefix queries, the team demonstrated a query complexity of Θ(n) by constructing a function and analyzing its behavior. This innovative approach provides a versatile tool for analyzing query complexity in various learning scenarios and establishes new theoretical limits on the efficiency of learning algorithms.

Quantum Complexity of Limited Substring Queries

Scientists have achieved breakthroughs in understanding the complexity of learning hidden sequences through limited substring queries, with implications for fields like computational biology and genome sequencing. This work investigates quantum query complexities when algorithms are restricted to querying subsets from a fixed collection, unifying previously studied models under a single framework. Researchers demonstrate that the quantum query complexity is Θ(n/√k) when querying sets of size at most k, recovering and generalizing earlier results. Specifically, when the collection of allowed query subsets consists of contiguous blocks, the quantum query complexity is eΘ(n), indicating an exponential relationship between the complexity and the input size.

Furthermore, when restricted to querying prefixes, the quantum query complexity is Θ(n), establishing a linear relationship between complexity and input size. These results were obtained through the development of a novel framework that characterizes learning complexity using an optimization program, maximizing the ratio of a function’s maximum value to its maximum standard deviation on a subcube. The team applied a symmetry reduction to a technique previously used for establishing lower bounds, to derive new quantum query upper bounds without relying on standard mathematical duality techniques. This approach allows for a more direct and efficient analysis of query complexity and delivers a deeper understanding of the fundamental limits of learning hidden sequences.

Query Complexity Through Symmetry and Ratio Maximisation

This research presents significant advances in understanding the complexity of learning hidden sequences through limited substring queries, a problem with applications in areas like computational biology and genome sequencing. Scientists have established near-optimal bounds for query complexity when algorithms are restricted to querying specific subsets of the hidden sequence, including bounded-size sets, contiguous blocks, prefixes, and the full set itself. This work unifies previously studied models under a single framework, allowing for a more comprehensive analysis of query complexity. The team developed a novel framework based on symmetries within the problem, demonstrating that query complexity can be determined, up to a constant factor, by maximizing a specific ratio related to the standard deviation of values on subcubes. This achievement was accomplished using a technique, offering a new approach to establishing query upper bounds without relying on standard duality techniques. Future work could investigate the impact of different subset structures on query complexity and explore the potential for developing algorithms tailored to specific application domains.