Complexity-Theoretic Limitations on Blind Delegated Quantum Computation

Authors Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu , Elham Kashefi

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Author Details

Scott Aaronson
  • Department of Computer Science, University of Texas at Austin, USA
Alexandru Cojocaru
  • School of Informatics, University of Edinburgh, UK
Alexandru Gheorghiu
  • Department of Computing and Mathematical Sciences, California Institute of Technology, USA
  • School of Informatics, University of Edinburgh, UK
Elham Kashefi
  • School of Informatics, University of Edinburgh, UK
  • CNRS LIP6, Université Pierre et Marie Curie, Paris, France


We would like to thank the following people for useful discussions and comments: Petros Wallden, Matty J Hoban, Kousha Etessami, Marc Kaplan, Ronald de Wolf, Urmila Mahadev, Umesh Vazirani, Chris Heunen, Thomas Vidick, Ashley Montanaro, Tina Zhang and Pia Kullik. A.G. is in particular grateful to Petros Wallden and Matty J Hoban for their patience and for their help in clarifying several technical issues.

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Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, and Elham Kashefi. Complexity-Theoretic Limitations on Blind Delegated Quantum Computation. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Blind delegation protocols allow a client to delegate a computation to a server so that the server learns nothing about the input to the computation apart from its size. For the specific case of quantum computation we know, from work over the past decade, that blind delegation protocols can achieve information-theoretic security (provided the client and the server exchange some amount of quantum information). In this paper we prove, provided certain complexity-theoretic conjectures are true, that the power of information-theoretically secure blind delegation protocols for quantum computation (ITS-BQC protocols) is in a number of ways constrained. In the first part of our paper we provide some indication that ITS-BQC protocols for delegating polynomial-time quantum computations in which the client and the server interact only classically are unlikely to exist. We first show that having such a protocol in which the client and the server exchange O(n^d) bits of communication, implies that BQP subset MA/O(n^d). We conjecture that this containment is unlikely by proving that there exists an oracle relative to which BQP not subset MA/O(n^d). We then show that if an ITS-BQC protocol exists in which the client and the server interact only classically and which allows the client to delegate quantum sampling problems to the server (such as BosonSampling) then there exist non-uniform circuits of size 2^{n - Omega(n/log(n))}, making polynomially-sized queries to an NP^{NP} oracle, for computing the permanent of an n x n matrix. The second part of our paper concerns ITS-BQC protocols in which the client and the server engage in one round of quantum communication and then exchange polynomially many classical messages. First, we provide a complexity-theoretic upper bound on the types of functions that could be delegated in such a protocol by showing that they must be contained in QCMA/qpoly cap coQCMA/qpoly. Then, we show that having such a protocol for delegating NP-hard functions implies coNP^{NP^{NP}} subseteq NP^{NP^{PromiseQMA}}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Quantum complexity theory
  • Quantum cryptography
  • Complexity theory
  • Delegated quantum computation
  • Computing on encrypted data


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