This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with postselection are as powerful as classical computers with postselection (PostBQP = PostBPP), (ii) any one of several quantum sampling experiments (BosonSampling, IQP, DQC1) can be approximately performed by a classical computer (contingent on existing assumptions). This last result implies that if any of these experiment’s hardness conjectures hold, then quantum computers can implement functions classical computers cannot (FBQP≠ FBPP) unless the polynomial hierarchy collapses to its 2nd level. These results are an improvement over previous work which either achieved a collapse to the third level or were concerned with exact sampling, a physically impractical case. The workhorse of these results is a new technical complexity-theoretic result which we believe could have value beyond quantum computation. In particular, we prove that if there exists an equivalence between problems solvable with an exact counting oracle and problems solvable with an approximate counting oracle, then the polynomial hierarchy collapses to its second level, indeed to ZPP^NP.

We define and study a variant of QMA (Quantum Merlin Arthur) in which Arthur can make multiple non-collapsing measurements to Merlin’s witness state, in addition to ordinary collapsing measurements. By analogy to the class PDQP defined by Aaronson, Bouland, Fitzsimons, and Lee (2014), we call this class PDQMA. Our main result is that PDQMA = NEXP; this result builds on the PCP theorem and complements the result of Aaronson (2018) that PDQP/qpoly = ALL. While the result has little to do with quantum mechanics, we also show a more "quantum" result: namely, that QMA with the ability to inspect the entire history of a hidden variable is equal to NEXP, under mild assumptions on the hidden-variable theory. We also observe that a quantum computer, augmented with quantum advice and the ability to inspect the history of a hidden variable, can solve any decision problem in polynomial time.

With the advent of real-world quantum computing, the idea that parametrized quantum computations can be used as hypothesis families in a quantum-classical machine learning system is gaining increasing traction. Such hybrid systems have already shown the potential to tackle real-world tasks in supervised and generative learning, and recent works have established their provable advantages in special artificial tasks. Yet, in the case of reinforcement learning, which is arguably most challenging and where learning boosts would be extremely valuable, no proposal has been successful in solving even standard benchmarking tasks, nor in showing a theoretical learning advantage over classical algorithms. In this work, we achieve both. We propose a hybrid quantum-classical reinforcement learning model using very few qubits, which we show can be effectively trained to solve several standard benchmarking environments. Moreover, we demonstrate, and formally prove, the ability of parametrized quantum circuits to solve certain learning tasks that are intractable to classical models, including current state-of-art deep neural networks, under the widely-believed classical hardness of the discrete logarithm problem.