Hamiltonian variational ansatz without barren plateaus

Chae-Yeun Park and Nathan Killoran

Xanadu, Toronto, ON, M5G 2C8, Canada

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Abstract

Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed $\textit{barren plateaus}$, has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.

Variational quantum algorithms (VQAs) solve a target problem by optimizing the parameters of a quantum circuit. While VQAs are one of the most promising applications of a near-term quantum computer, the practical usefulness of VQAs is often questioned. One of the central issues is that quantum circuits with random parameters often have exponentially small gradients, limiting the trainability of the circuits. This problem, dubbed barren plateaus, has gained lots of interest recently, but a general solution is still unavailable. This work proposes a solution to the barren plateaus problem for the Hamiltonian variational ansatz, a type of quantum circuit ansatz widely studied for solving quantum many-body problems.

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