← All parts of this equation

Equation 9 · Part 2 · Variational Evolution: How Quantum Computers Learn Their Answers

Symbol E

Var⁡ ⁣[∂θiE(θ)]∼O ⁣(2−n).\operatorname{Var}\!\left[\partial_{\theta_i} E(\theta)\right] \sim O\!\left(2^{-n}\right) .
EE

What this part means

E is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Its job in the formula

E is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

The passage around this formula

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: Var⁡ ⁣[∂θiE(θ)]∼O ⁣(2−n)\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,…

Read this part in the article →

Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

Open the illustrated functions: inputs become outputs guide →

See this notation across published equations →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.