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Circuit Extraction for ZX-Diagrams Can Be #P-Hard

Authors Niel de Beaudrap , Aleks Kissinger , John van de Wetering



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Author Details

Niel de Beaudrap
  • University of Sussex, UK
Aleks Kissinger
  • University of Oxford, UK
John van de Wetering
  • Radboud University Nijmegen, The Netherlands
  • University of Oxford, UK

Acknowledgements

The authors wish to thank the anonymous reviewers for their valuable suggestions regarding the presentation of this paper.

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Niel de Beaudrap, Aleks Kissinger, and John van de Wetering. Circuit Extraction for ZX-Diagrams Can Be #P-Hard. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 119:1-119:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.119

Abstract

The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from m to n qubits for any m,n ≥ 0. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been conjectured that the general problem of circuit extraction is hard. That is, that it should not be possible to efficiently convert an arbitrary ZX-diagram describing a unitary linear map into an equivalent quantum circuit. In this paper we prove this conjecture by showing that the circuit extraction problem is #P-hard, and so is itself at least as hard as strong simulation of quantum circuits. In addition to our main hardness result, which relies specifically on the circuit representation, we give a representation-agnostic hardness result. Namely, we show that any oracle that takes as input a ZX-diagram description of a unitary and produces samples of the output of the associated quantum computation enables efficient probabilistic solutions to NP-complete problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • ZX-calculus
  • circuit extraction
  • quantum circuits
  • #P

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