  Every year, substantial theoretical and experimental progress is made towards
the realisation of a genuinely new computational paradigm in the construction
of a quantum computer. But progress is fractal; to make headway is to unearth
the next set of obstacles. Decades of work has so far overcome physical,
mathematical, engineering, and information theoretic obstacles to produce the
remarkable high-fidelity devices we see today. But these devices must be near
perfect to be useful. Indeed, advancements so far have precipitated sensitivity
to a host of complex dynamical and control-based effects. Chief among these
today are non-Markovian memory effects, where interactions between a quantum
system and its surrounding environment can give rise to multi-time
correlations. In this thesis, we address this issue and formally present a
generalisation of quantum process tomography, called process tensor tomography
(PTT). This establishes the ability to rigorously and systematically
characterise non-Markovian open quantum systems, resolving many long-standing
issues in the field. In the first part of this work, we present an original
review of the literature and motivate the problem at hand. In the second, we
develop the framework of PTT, including experiment design, post-processing
algorithms, and both simulated and near-term device demonstrations. In
particular, we demonstrate this as a tool for obtaining in-depth diagnostics
about the nature and origin of temporal quantum correlations. Lastly, we
dedicate our efforts to efficiency and self-consistency. To this effect, we
explore theoretically processes with sparse memory structures. We then leverage
this to develop various efficient estimation techniques tailored for different
settings. The result is a robust and lightweight framework capable of both
reconstructing and optimally controlling any non-Markovian open quantum
dynamics.
