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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. Fig 1 from the paper

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.

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I am not an expert but I read a few papers and here is what I have found. Similarly to NN, people found strategies to avoid this issue with the gradients.

Basically, for some problems, you can use ansatzes that are inspired by the physics of the problem itself. For example, in quantum chemistry, people use something called unitary coupled clusters. See Quantum computational chemistry and The theory of variational hybrid quantum-classical algorithms.

However, for many problems, you don't have physical insights. Then, the solution I found in the literature are the following:

  • Gradually adding layers: basically you start with a low depth ansatz, optimize this one to its local minimum. Then add a layer, optimize again, add another layer etc. The idea is that lower depth ansatz is less prone to this issue. Check "Variational quantum state diagonalization"
  • Introducing a local version of the cost function: I haven't fully understood this one, but here is the paper you can check and find more info: "Variational Quantum Linear Solver: A Hybrid Algorithm for Linear Systems"
  • 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.
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  • $\begingroup$ Thank you very much Enrico for the time you took to assemble these helpful pointers, B"H" they will guide me well. Appreciate! $\endgroup$ – Daniel Bilar Apr 12 at 1:31

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