A Distribution Testing Oracle Separating QMA and QCMA

Authors Anand Natarajan , Chinmay Nirkhe

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Author Details

Anand Natarajan
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Chinmay Nirkhe
  • IBM Quantum, Cambridge, MA, USA


We thank Srinivasan Arunachalam, Andrew Childs, Elizabeth Crosson, Yi-Kai Liu, Aram Harrow, Zhiyang He, Robin Kothari, William Kretschmer, Yupan Liu, Kunal Marwaha, Mehdi Soleimanifar, Umesh Vazirani, and Elizabeth Yang for helpful discussions. Some of the early ideas of this result were developed while Chinmay Nirkhe was at the University of California, Berkeley. This work was partially completed while both authors were participants in the Simons Institute for the Theory of Computing program The Quantum Wave in Computing: Extended Reunion.

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Anand Natarajan and Chinmay Nirkhe. A Distribution Testing Oracle Separating QMA and QCMA. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 22:1-22:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


It is a long-standing open question in quantum complexity theory whether the definition of non-deterministic quantum computation requires quantum witnesses (QMA) or if classical witnesses suffice (QCMA). We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson and Kuperberg, 2007; Bill Fefferman and Shelby Kimmel, 2018] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over n-bit boolean functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • quantum non-determinism
  • complexity theory


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