Improved Quantum Query Complexity on Easier Inputs

Noel T. Anderson1, Jay-U Chung1, Shelby Kimmel1, Da-Yeon Koh2, and Xiaohan Ye1,3

1Middlebury College, Middlebury, VT, USA
2Williams College, Williamstown, MA, USA
3Brown University, Providence, RI, USA

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Abstract

Quantum span program algorithms for function evaluation sometimes have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these improvements persist even without a promise ahead of time, and we extend this approach to the more general problem of state conversion. As an application, we prove exponential and superpolynomial quantum advantages in average query complexity for several search problems, generalizing Montanaro's Search with Advice [Montanaro, TQC 2010].

We expect that quantum algorithms, like classical algorithms, should run faster on easier inputs. For example, if you are searching for an item in an unordered list, and there are many copies of that item, we would expect the quantum algorithm should run faster in this situation compared to when if there is only one marked item, even without knowing the number of target items ahead of time. Indeed, for the problem of search, it is known how to get such an advantage on easier inputs. However, generalizing this idea to problems beyond search is challenging when there is not an obvious way to flag when the computation has run for long enough. We modify several popular algorithmic frameworks in the query model to create flags that alert us to whether the computation has run for long enough, allowing us to end the algorithm early on easier inputs, without knowing ahead of time if the instance is easy or hard. As an application, given a distribution of both easy and hard inputs to a problem, we can analyze the average query complexity. We show that certain distributions of inputs to search problems yield large average quantum query advantages over classical algorithms.

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Cited by

[1] Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita, "Quantum Algorithm for Path-Edge Sampling", arXiv:2303.03319, (2023).

[2] Michael Czekanski, Shelby Kimmel, and R. Teal Witter, "Robust and Space-Efficient Dual Adversary Quantum Query Algorithms", arXiv:2306.15040, (2023).

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