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 $\hat{f}$ of a function $f$ is another function such that $f(x)$ can be recovered from $\hat{f}(x)$, and nothing except for $f(x)$ is leaked from $\hat{f}(x)$. Its quantum version, quantum randomized encoding, has been introduced recently [Brakerski and Yuen, arXiv:2006.01085]. Intuitively, quantum randomized encoding $\hat{F}$ of a quantum operation $F$ is another quantum operation such that, for any quantum state $\rho$, $F(\rho)$ can be recovered from $\hat{F}(\rho)$, and nothing except for $F(\rho)$ is leaked from $\hat{F}(\rho)$. In this paper, we show three results. First, 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. Our second result is, however, to show that too good quantum randomized encoding is impossible: if quantum randomized encoding for the generation of even simple states (such as BB84 states) is possible with a classical encoding operation, then the no-cloning is violated. Finally, we 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.

On Gap-Based Lower Bounding Techniques for Best-Arm Identification

In this paper, we consider techniques for establishing lower bounds on the number of arm pulls for best-arm identification in the multi-armed bandit problem. While a recent divergence-based approach was shown to provide improvements over an older gap-based approach, we show that the latter can be refined to match the former (up to constant factors) in many cases of interest under Bernoulli rewards, including the case that the rewards are bounded away from zero and one. Together with existing upper bounds, this indicates that the divergence-based and gap-based approaches are both effective for establishing sample complexity lower bounds for best-arm identification.

Robust Bell inequalities from communication complexity

The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.