Complexitytheoretic limitations on blind delegated quantum computation
Abstract
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 that blind delegation protocols can achieve informationtheoretic security. In this paper we prove, provided certain complexitytheoretic conjectures are true, that the power of informationtheoretically secure blind delegation protocols for quantum computation (ITSBQC protocols) is in a number of ways constrained. In the first part of our paper we provide some indication that ITSBQC protocols for delegating $\sf BQP$ computations in which the client and the server interact only classically are unlikely to exist. We first show that having such a protocol with $O(n^d)$ bits of classical communication implies that $\mathsf{BQP} \subset \mathsf{MA/O(n^d)}$. We conjecture that this containment is unlikely by providing an oracle relative to which $\mathsf{BQP} \not\subset \mathsf{MA/O(n^d)}$. We then show that if an ITSBQC protocol exists with polynomial classical communication and which allows the client to delegate quantum sampling problems, then there exist nonuniform circuits of size $2^{n  \mathsf{\Omega}(n/log(n))}$, making polynomiallysized queries to an $\sf NP^{NP}$ oracle, for computing the permanent of an $n \times n$ matrix. The second part of our paper concerns ITSBQC 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 complexitytheoretic upper bound on the types of functions that could be delegated in such a protocol, namely $\mathsf{QCMA/qpoly \cap coQCMA/qpoly}$. Then, we show that having such a protocol for delegating $\mathsf{NP}$hard functions implies $\mathsf{coNP^{NP^{NP}}} \subseteq \mathsf{NP^{NP^{PromiseQMA}}}$.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 arXiv:
 arXiv:1704.08482
 Bibcode:
 2017arXiv170408482A
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 Improves upon, supersedes and corrects our earlier submission, which previously included an error in one of the main theorems