$$\operatorname{Var}\!\left[\partial_{\theta_i} E(\theta)\right] \sim O\!\left(2^{-n}\right)$$

Equation 9 · Evolutionary Physics

Variational Evolution: How Quantum Computers Learn Their Answers

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The result, now known throughout the field as the barren-plateau problem, is precise and unwelcome. For a parametrized circuit that forms an approximate unitary 2-design — informally, a circuit random enough, or deep enough, that its output statistics resemble those of a genuinely random unitary — the variance of the gradient of the cost function with respect to almost any parameter shrinks exponentially as the number of qubits n grows: $\operatorname{Var}\!\left[\partial_{\theta_i} E(\theta)\right] \sim O\!\left(2^{-n}\right)$. The mean gradient is essentially zero and its variance collapses just as fast, so a classical optimizer sampling that gradient from a finite number of circuit measurements sees something statistically indistinguishable from flat,…

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