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Span Programs and Quantum Time Complexity

Authors Arjan Cornelissen, Stacey Jeffery, Maris Ozols, Alvaro Piedrafita

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Author Details

Arjan Cornelissen
  • QuSoft, Amsterdam, The Nehterlands
  • University of Amsterdam, The Netherlands
Stacey Jeffery
  • QuSoft, Amsterdam, The Netherlands
  • CWI, Amsterdam, The Netherlands
Maris Ozols
  • QuSoft, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands
Alvaro Piedrafita
  • QuSoft, Amsterdam, The Netherlands
  • CWI, Amsterdam, The Netherlands

Funding

  • Jeffery, Stacey: Supported by an NWO Veni Innovational Research Grant under project number 639.021.752, an NWO WISE Grant, and QuantERA project QuantAlgo 680-91-03. SJ is a CIFAR Fellow in the Quantum Information Science Program.
  • Ozols, Maris: Supported by an NWO Vidi grant VI.Vidi.192.109.

Acknowledgements

Stacey Jeffery thanks Tsuyoshi Ito for illuminating discussions on the topic of span programs and time complexity.

Cite As Get BibTex

Arjan Cornelissen, Stacey Jeffery, Maris Ozols, and Alvaro Piedrafita. Span Programs and Quantum Time Complexity. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.26

Abstract

Span programs are an important model of quantum computation due to their correspondence with quantum query and space complexity. While the query complexity of quantum algorithms obtained from span programs is well-understood, it is not generally clear how to implement certain query-independent operations in a time-efficient manner. In this work, we prove an analogous connection for quantum time complexity. In particular, we show how to convert a sufficiently-structured quantum algorithm for f with time complexity T into a span program for f such that it compiles back into a quantum algorithm for f with time complexity 𝒪̃(T). This shows that for span programs derived from algorithms with a time-efficient implementation, we can preserve the time efficiency when implementing the span program, which means that span programs capture time, query and space complexities and are a complete model of quantum algorithms.
One practical advantage of being able to convert quantum algorithms to span programs in a way that preserves time complexity is that span programs compose very nicely. We demonstrate this by improving Ambainis’s variable-time quantum search result using our construction through a span program composition for the OR function.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum query algorithms
  • span programs
  • variable-time quantum search

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References

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