Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond

Hybrid quantum-classical algorithms are central to much of the current research in quantum computing, particularly when considering the noisy intermediate-scale quantum (NISQ) era, with a number of experimental demonstrations having already been performed. In this perspective, we discuss in a very broad sense what it means for an algorithm to be hybrid quantum-classical. We first explore this concept very directly, by building a definition based on previous work in abstraction/representation theory, arguing that what makes an algorithm hybrid is not directly how it is run (or how many classical resources it consumes), but whether classical components are crucial to an underlying model of the computation. We then take a broader view of this question, reviewing a number of hybrid algorithms and discussing what makes them hybrid, as well as the history of how they emerged, and considerations related to hardware. This leads into a natural discussion of what the future holds for these algorithms. To answer this question, we turn to the use of specialized processors in classical computing.The classical trend is not for new technology to completely replace the old, but to augment it. We argue that the evolution of quantum computing is unlikely to be different: hybrid algorithms are likely here to stay well past the NISQ era and even into full fault-tolerance, with the quantum processors augmenting the already powerful classical processors which exist by performing specialized tasks.