# Quantum randomized encoding, verification of quantum computing, no-cloning, and blind quantum computing

Randomized encoding is a powerful cryptographic primitive with various applications such as secure multiparty computation, verifiable computation, parallel cryptography, and complexity lower-bounds. Intuitively, randomized encoding f̂ of a function f is another function such that f(x) can be recovered from f̂(x), and nothing except for f(x) is leaked from f̂(x). Its quantum version, quantum randomized encoding, has been introduced recently [Brakerski and Yuen, arXiv:2006.01085]. Intuitively, quantum randomized encoding F̂ of a quantum operation F is another quantum operation such that, for any quantum state ρ, F(ρ) can be recovered from F̂(ρ), and nothing except for F(ρ) is leaked from F̂(ρ). In this paper, we show that if quantum randomized encoding of BB84 state generations is possible with an encoding operation E, then a two-round verification of quantum computing is possible with a classical verifier who can additionally do the operation E. One of the most important goals in the field of the verification of quantum computing is to construct a verification protocol with a verifier as classical as possible. This result therefore demonstrates a potential application of quantum randomized encoding to the verification of quantum computing: if we can find a good quantum randomized encoding (in terms of the encoding complexity), then we can construct a good verification protocol of quantum computing. We, however, also show that too good quantum randomized encoding is impossible: if quantum randomized encoding with a classical encoding operation is possible, then the no-cloning is violated. We finally consider a natural modification of blind quantum computing protocols in such a way that the server gets the output like quantum randomized encoding. We show that the modified protocol is not secure.

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