Abstract
Parametrized quantum circuits initialized with random initial parameter values are characterized by barren plateaus where the gradient becomes exponentially small in the number of qubits. In this technical note we theoretically motivate and empirically validate an initialization strategy which can resolve the barren plateau problem for practical applications. The technique involves randomly selecting some of the initial parameter values, then choosing the remaining values so that the circuit is a sequence of shallow blocks that each evaluates to the identity. This initialization limits the effective depth of the circuits used to calculate the first parameter update so that they cannot be stuck in a barren plateau at the start of training. In turn, this makes some of the most compact ansätze usable in practice, which was not possible before even for rather basic problems. We show empirically that variational quantum eigensolvers and quantum neural networks initialized using this strategy can be trained using a gradient based method.
Highlights
Parametrized quantum circuits have recently been shown to suffer from gradients that vanish exponentially towards zero as a function of the number of qubits
This is known as the ‘barren plateau’ problem and has been demonstrated analytically and numerically [1]. The implication of this result is that for a wide class of circuits, random initialization will cause gradient-based optimization methods to fail. Resolving this issue is critical to the scalability of algorithms such as the variational quantum eigensolver (VQE) [2, 3] and quantum neural networks (QNNs) [4, 5, 6]
In Ref. [7] the author shows that the barren plateau problem is not an issue of the chosen parametrization, but rather extends the result to any direction in the tangent space of a point in the unitary group
Summary
Parametrized quantum circuits have recently been shown to suffer from gradients that vanish exponentially towards zero as a function of the number of qubits This is known as the ‘barren plateau’ problem and has been demonstrated analytically and numerically [1]. The variance of the gradient decreases exponentially with the number of qubits This does not preclude the existence of a parametrization that would allow for efficient gradient-based optimization. [1], the barren plateau problem affects traditional ansätze such as the unitary coupled cluster, when initialized randomly, even for a small number of orbitals The authors leave it as an open question whether the problem can be solved by employing alternative hardware-efficient ansätze. This limits the effective depth of the circuits used to calculate the gradient at the first iteration and allows us to efficiently train a variety of parametrized quantum circuits
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