Quantum circuits are inherently parametric and optimization of the parameters is required. While this might be explicit as with a Variational-Quantum-Algorithm (VQA), or implicit as with the Quantum-Singular-Value-Transform, the optimal choice of parameters depends not only on the desired algorithm, but also hardware characteristics such as the set of basis gates. As quantum circuits are typically only part of a larger problem setting, and it is desirable to optimize circuits in terms of runtimes, fidelities, or other desired quantities, it can be helpful to also consider the context of the larger problem as in the end we are often interested in the overall time-to-solution, and not just the time-to-readout of the quantum subroutine.
This opportunity arises, for instance, in machine learning tasks as the outer level of the problem scope. For example, a quantum neural network (QNN) requires the optimization of weights, often represented by parameters such as rotation angles of quantum gates, or the training of Gaussian process (GP) emulators. Traditionally, GPs have mostly been used for classification and regression tasks. Their generalization into GP emulators aims at substituting computationally expensive models, e.g., simulation models, via a cheap-to-evaluate, uncertainty-informed alternative. In particular, their ability to quantify uncertainty makes them a powerful tool for high-throughput tasks [1,2,3], as it enables the separation of surrogate-induced uncertainty from other sources of input uncertainty. Training a GP emulator requires inverting a potentially dense matrix, which has O(N^3) computational complexity and therefore limits its applicability to a moderate number of dimensions in the design parameter space.
While quantum algorithms have been proposed to tackle this computational bottleneck, any real-world speedups would be lost in an attempt to read the inverted matrix from the quantum computer due to the readout-problem. However, the main task in GP emulator training is formulated as a hyper-parameter tuning, where the hyper-parameters determine how to populate the covariance matrix to invert. It is therefore possible to not only optimize the quantum circuit in a black-box manner, but also with respect to this hyper-parameter tuning.
