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Robust and Space-Efficient Dual Adversary Quantum Query Algorithms

Authors Michael Czekanski, Shelby Kimmel, R. Teal Witter

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Author Details

Michael Czekanski
  • Department of Statistics and Data Science, Cornell University, Ithaca, NY, USA
Shelby Kimmel
  • Department of Computer Science, Middlebury College, VT, USA
R. Teal Witter
  • Department of Computer Science and Engineering, New York University, Brooklyn, NY, USA

Funding

  • Kimmel, Shelby: sponsored by the U.S. Army Research Office and this work was accomplished under Grant Number W911NF-20-1-0327. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
  • Witter, R. Teal: supported by NSF No. 1922658 and 1916505.

Acknowledgements

We thank Stacey Jeffery for elucidating discussions.

Cite AsGet BibTex

Michael Czekanski, Shelby Kimmel, and R. Teal Witter. Robust and Space-Efficient Dual Adversary Quantum Query Algorithms. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.36

Abstract

The general adversary dual is a powerful tool in quantum computing because it gives a query-optimal bounded-error quantum algorithm for deciding any Boolean function. Unfortunately, the algorithm uses linear qubits in the worst case, and only works if the constraints of the general adversary dual are exactly satisfied. The challenge of improving the algorithm is that it is brittle to arbitrarily small errors since it relies on a reflection over a span of vectors. We overcome this challenge and build a robust dual adversary algorithm that can handle approximately satisfied constraints. As one application of our robust algorithm, we prove that for any Boolean function with polynomially many 1-valued inputs (or in fact a slightly weaker condition) there is a query-optimal algorithm that uses logarithmic qubits. As another application, we prove that numerically derived, approximate solutions to the general adversary dual give a bounded-error quantum algorithm under certain conditions. Further, we show that these conditions empirically hold with reasonable iterations for Boolean functions with small domains. We also develop several tools that may be of independent interest, including a robust approximate spectral gap lemma, a method to compress a general adversary dual solution using the Johnson-Lindenstrauss lemma, and open-source code to find solutions to the general adversary dual.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Quantum query complexity
Keywords
  • Quantum Computing
  • Robust Quantum Algorithms
  • Johnson-Lindenstrauss Lemma
  • Span Programs
  • Query Complexity
  • Space Complexity

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