Generalised coherent states for variational quantum circuits

One challenge with designing variational quantum algorithms is choosing a good ansatz for the parameterised quantum circuit. One needs to balance multiple competing requirements: 

1. The circuit needs to be complex enough to be able to express a good solution to the problem at hand.

2. The parameterisation should be simple enough to admit analytical calculation of gradients, so that the solution can be found efficiently using gradient descent.

3. The circuit should generate non-trivial quantum entanglement, which is a necessary (but not sufficient) condition for a classical computer to be inefficient at obtaining the same solution.

The most common choices for parameterised circuits either do not meet all three requirements, or give rise to very deep circuits that cannot be run on current quantum processors.

Last week I saw an interesting preprint posted to arXiv:

A Variational Ansatz for the Ground State of the Quantum Sherrington-Kirkpatrick Model

This work builds on an earlier paper by the same group: Generalization of group-theoretic coherent states for variational calculations.

The authors consider variational quantum states based on generalised coherent states, which are formed by applying a single qubit and two qubit (entangling) rotations. Crucially, the form of the ansatz enables expectation values of Pauli operators (required to obtained energies and gradients) to be measured efficiently, with the required number of measurements scaling polynomially in the system size. At the same time, the non-trivial entanglement of the generalised coherent states means that evaluating the overlap between different parameterised states is a classically hard task.

These properties seem quite promising for quantum kernel methods: the analytical gradients would allow one to optimise key properties of the quantum feature map, while the model output (quantum state overlaps with the training data) remains hard to simulate classically.

It is interesting to speculate on whether these parameterised quantum states could be adapted form the basis for a quantum perceptron; they are simple and efficiently trainable, but I am not sure whether they are also modular and can be stacked to form a deep quantum neural network.