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Quantum Sabotage Complexity

Authors Arjan Cornelissen, Nikhil S. Mande , Subhasree Patro

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ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Quantum complexity theory
Keywords
  • Sabotage complexity
  • quantum query complexity
  • Boolean functions
  • fractional block sensitivity

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Abstract

Given a Boolean function f : {0,1}ⁿ → {0,1}, the goal in the usual query model is to compute f on an unknown input x ∈ {0,1}ⁿ while minimizing the number of queries to x. One can also consider a "distinguishing" problem denoted by f_sab: given an input x ∈ f^{-1}(0) and an input y ∈ f^{-1}(1), either all differing bits are replaced by a *, or all differing bits are replaced by †, and an algorithm’s goal is to identify which of these is the case while minimizing the number of queries.
Ben-David and Kothari [ToC'18] introduced the notion of randomized sabotage complexity of a Boolean function to be the zero-error randomized query complexity of f_sab. A natural follow-up question is to understand the 𝖰(f_sab), the quantum query complexity of f_sab. In this paper, we initiate a systematic study of this. The following are our main results for all Boolean functions f : {0,1}ⁿ → {0,1}.  
- If we have additional query access to x and y, then 𝖰(f_sab) = O(min{𝖰(f),√n}). 
- If an algorithm is also required to output a differing index of a 0-input and a 1-input, then 𝖰(f_sab) = O(min{𝖰(f)^{1.5}, √n}). 
- 𝖰(f_sab) = Ω(√{fbs(f)}), where fbs(f) denotes the fractional block sensitivity of f. By known results, along with the results in the previous bullets, this implies that 𝖰(f_sab) is polynomially related to 𝖰(f). 
- The bound above is easily seen to be tight for standard functions such as And, Or, Majority and Parity. We show that when f is the Indexing function, 𝖰(f_sab) = Θ(fbs(f)), ruling out the possibility that 𝖰(f_sab) = Θ(√{fbs(f)}) for all f.

Cite As Get BibTex

Arjan Cornelissen, Nikhil S. Mande, and Subhasree Patro. Quantum Sabotage Complexity. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.19

Author Details

Arjan Cornelissen
  • Simons Institute for the Theory of Computing, University of California, Berkeley, CA, USA
  • IRIF - CNRS, Paris, France
Nikhil S. Mande
  • University of Liverpool, UK
Subhasree Patro
  • Technische Universiteit Eindhoven, The Netherlands
  • Centrum Wiskunde en Informatica (QuSoft), Amsterdam, The Netherlands

Funding

  • Patro, Subhasree: This work is co-funded by the European Union (ERC, ASC-Q, 101040624).

Acknowledgements

We thank the anonymous reviewers for useful comments and pointers to related results that we had missed.

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