# Devising "structured initial guesses" for random parametrized quantum circuits to avoid getting stuck in a flat plateau

The recent McClean et al. paper Barren plateaus in quantum neural network training landscapes shows that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits.

This seems to affect Noisy Intermediate-Scale Quantum (NISQ) programs (as proposed by e.g. John Preskill) since they involve hybrid quantum-classical algorithms, ie training a parameterized quantum circuit with a classical optimization loop.

My question: How do you avoid getting stranded on those barren plateaus? Concretely, how would one go about building one's Ansatz Haar states to avoid getting stuck in those plateaus? The paper proposes but does not elaborate:

One approach to avoid these landscapes in the quantum setting is to use structured initial guesses, such as those adopted in quantum simulation.

• Using Hamiltonian Morphing: this method is based on the Adiabatic theorem. If you are trying to solve a linear system $$Ax=b$$ then you can "evolve" the matrix $$A$$ during the computation as $$A(t)=(1-t/T)I+t/T \cdot A$$. Where T is a fixed time and t is the time variable of the simulation. I think "Variational algorithms for linear algebra" explains it much better than I do.