Barren plateaus in quantum tensor network optimization

Enrique Cervero Martín1,2, Kirill Plekhanov1, and Michael Lubasch1

1Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom
2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543

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We analyze the barren plateau phenomenon in the variational optimization of quantum circuits inspired by matrix product states (qMPS), tree tensor networks (qTTN), and the multiscale entanglement renormalization ansatz (qMERA). We consider as the cost function the expectation value of a Hamiltonian that is a sum of local terms. For randomly chosen variational parameters we show that the variance of the cost function gradient decreases exponentially with the distance of a Hamiltonian term from the canonical centre in the quantum tensor network. Therefore, as a function of qubit count, for qMPS most gradient variances decrease exponentially and for qTTN as well as qMERA they decrease polynomially. We also show that the calculation of these gradients is exponentially more efficient on a classical computer than on a quantum computer.

Quantum tensor networks (qTNs) are parameterized quantum circuits (PQCs) that are inspired by and can have advantages over classical tensor networks. We address the question of whether the variational qTN optimization can suffer from barren plateau problems. We consider PQCs inspired by matrix product states (qMPS), tree tensor networks (qTTN) and the multiscale entanglement renormalization ansatz (qMERA). We show that, for a local cost function, the gradient variance w.r.t. random variational parameters decreases exponentially with the distance from the cost function’s observable to the qTN’s canonical centre. Specifically, for qMPS gradient variances can decrease exponentially with qubit count, while for qTTN and qMERA they decrease polynomially.

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