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Variational quantum computing schemes have received considerable attentiondue to their high versatility and potential to make practical use of near-termquantum devices. At their core, these models train a loss function by sendingan initial state through a parametrized quantum circuit, and evaluating theexpectation value of some operator at the circuit's output. Despite theirpromise, the trainablity of these algorithms is hindered by barren plateausinduced by the expressiveness of the parametrized quantum circuit, theentanglement of the input data, the locality of the observable, or the presenceof hardware noise. Up to this point, these sources of barren plateaus have beenregarded as independent and have been studied only for specific circuitarchitectures. In this work, we present a general Lie algebraic theory thatprovides an exact expression for the variance of the loss function ofsufficiently deep parametrized quantum circuits, even in the presence ofcertain noise models. Our results unify under one single framework allaforementioned sources of barren plateaus by leveraging generalized (andsubsystem independent) notions of entanglement and operator locality, as wellas generalized notions of algebraic decoherence due to noise. This theoreticalleap resolves a standing conjecture about a connection between lossconcentration and the dimension of the Lie algebra of the generators of theparametrized circuit.

A Unified Theory of Barren Plateaus for Deep Parametrized Quantum Circuits