Barren plateaus in quantum tensor network optimization
1Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom
2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
| Published: | 2023-04-13, volume 7, page 974 |
| Eprint: | arXiv:2209.00292v3 |
| Doi: | https://doi.org/10.22331/q-2023-04-13-974 |
| Citation: | Quantum 7, 974 (2023). |
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Abstract
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.
Featured image: We consider the qMERA (all gates shown; top light green gates connect to bottom ones), the qTTN (dark red gates) and the qMPS (dark red gates in shaded area). We show that the gradient variance w.r.t. random parameters decreases exponentially with the distance of the cost function's observable from the canonical centre.
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