# Complexity-theoretic limitations on blind delegated quantum computation

## TL;DR

This paper investigates the fundamental complexity-theoretic limitations of blind delegated quantum computation protocols, showing that certain secure protocols are unlikely to exist under standard conjectures and exploring their implications.

## Contribution

It provides complexity-theoretic constraints on the existence of information-theoretically secure blind quantum delegation protocols, highlighting their limitations and implications for quantum computational complexity.

## Key findings

- Classical communication protocols imply unlikely complexity class containments.
- Protocols for quantum sampling problems relate to non-uniform circuit sizes.
- One-round quantum-classical protocols constrain the functions that can be delegated.

## 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 information-theoretic security. 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 $\sf BQP$ 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 ITS-BQC protocol exists with polynomial classical communication and which allows the client to delegate quantum sampling problems, then there exist non-uniform circuits of size $2^{n - \mathsf{\Omega}(n/log(n))}$, making polynomially-sized queries to an $\sf NP^{NP}$ oracle, for computing the permanent of an $n \times n$ 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, namely $\mathsf{QCMA/qpoly \cap coQCMA/qpoly}$. Then, we show that having such a protocol for delegating $\mathsf{NP}$-hard functions implies $\mathsf{coNP^{NP^{NP}}} \subseteq \mathsf{NP^{NP^{PromiseQMA}}}$.