Noise-induced shallow circuits and the absence of barren plateaus

Without a successful implementation of fault-tolerant quantum error correction, calculations on quantum computers are subject to noise that limits their capabilities. Here, motivated by realistic near-term hardware considerations, we study the impact of uncorrected local noise on logical quantum circuits. We first show that, in the task of estimating observable expectation values, any noise effectively truncates most quantum circuits to logarithmic depth. We then prove that quantum circuits under any non-unital noise do not exhibit barren plateaus for cost functions composed of local observables. However, by using the effective shallowness, we also design an efficient classical algorithm to estimate observable expectation values within any constant additive accuracy, with high probability over the choice of the circuit, in any circuit architecture. Taken together, our results establish that, unless we carefully engineer quantum circuits to take advantage of the noise, noisy quantum circuits are unlikely to offer an advantage over shallow ones for algorithms that output observable expectation value estimates, such as many variational quantum machine learning proposals.