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arXiv · 2504.02115

Quantum algorithms through graph composition

Arjan Cornelissen
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Abstract

In this work, we unify several quantum algorithmic frameworks for boolean functions that are based on the quantum adversary bound. First, we show that the $st$-connectivity framework subsumes the (adaptive/extended) learning graph framework, and the weighted-decision-tree framework. Additionally, we show that every randomized algorithm can be turned into an $st$-connectivity problem as well, with the same complexity up to constants. This situates the $st$-connectivity framework in between randomized and quantum algorithms, indicating that it's an intermediate computational model between classical and quantum. We also introduce a generalization of the $st$-connectivity framework, the \textit{graph composition framework}, and show that it subsumes part of the quantum divide and conquer framework, and it is itself subsumed by the multidimensional quantum walk framework. Second, we investigate these frameworks' power by investigating the most efficient algorithms they can produce, in terms of the number of queries to the input. We show that the weighted-decision-tree framework's power is polynomially related to deterministic query complexity, showing that the quantum speed-ups that can be obtained with this framework are at most quadratic. Finally, we turn our attention to time-efficient implementations of the algorithms constructed through the $st$-connectivity and graph composition frameworks. To that end, we convert instances to the two-subspace phase estimation framework, and we show how we can implement these as transducers. This has the added benefit of removing the effective spectral gap lemma from the construction, significantly simplifying the analysis. We showcase the techniques developed in this work to give improved algorithms for various string search problems.

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Arjan Cornelissen. 2025-04-02. Quantum algorithms through graph composition. https://arxiv.org/abs/2504.02115

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