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We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the "quantum difference methods"). For the heat and linear hyperbolic equations we study the impact of explicit and implicit time discretizations on quantum advantages over the classical difference method. For the multiscale problem, we find the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as O (1 / ε) , where ε is the scaling parameter, while the scaling for the multiscale Asymptotic-Preserving (AP) schemes does not depend on ε. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing. • New quantum algorithm is proposed for a prototype multiscale problem. • Asymptotic-preserving schemes are shown to be important in quantum computation. • There is no difference in the cost of the quantum algorithm for the heat equation when using the forward-Euler compared to the Crank-Nicolson scheme. [ABSTRACT FROM AUTHOR]

Quantum computation requires large classical datasets to be embedded into quantum states in order to exploit quantum parallelism. However, this embedding requires considerable resources in general. It would therefore be desirable to avoid it, if possible, for noisy intermediate-scale quantum (NISQ) implementation. Accordingly, we consider a classical-quantum hybrid architecture, which allows large classical input data, with a relatively small-scale quantum system. This hybrid architecture is used to implement a sampling oracle. It is shown that in the presence of noise in the hybrid oracle, the effects of internal noise can cancel each other out and thereby improve the query success rate. It is also shown that such an immunity of the hybrid oracle to noise directly and tangibly reduces the sample complexity in the framework of computational learning theory. This NISQ-compatible learning advantage is attributed to the oracle's ability to handle large input features. [ABSTRACT FROM AUTHOR]

Continuous-variable quantum information, encoded into infinite-dimensional quantum systems, is a promising platform for the realization of many quantum information protocols, including quantum computation, quantum metrology, quantum cryptography, and quantum communication. To successfully demonstrate these protocols, an essential step is the certification of multimode continuous-variable quantum states and quantum devices. This problem is well studied under the assumption that multiple uses of the same device result in identical and independently distributed (i.i.d.) operations. However, in realistic scenarios, identical and independent state preparation and calls to the quantum devices cannot be generally guaranteed. Important instances include adversarial scenarios and instances of time-dependent and correlated noise. In this Letter, we propose the first set of reliable protocols for verifying multimode continuous-variable entangled states and devices in these non-i.i.d scenarios. Although not fully universal, these protocols are applicable to Gaussian quantum states, non-Gaussian hypergraph states, as well as amplification, attenuation, and purification of noisy coherent states. [ABSTRACT FROM AUTHOR]