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Alzheimer's disease is a disorder of large-scale brain networks, in which pathology propagates along the connectome and degrades it through synaptic and axonal loss. Cognition depends on distributed regions synchronizing their activity, and that synchronization carries a physical cost, set by how many white-matter connections must be engaged, and how strongly, before a network settles into coordinated activity. That cost measures what a damaged network can still coordinate, and not only which connections it has lost. How that cost changes across the disease continuum remains unquantified. Here, we apply QUIET, an edge-centric network-control framework, to the Alzheimer's Disease Neuroimaging Initiative cohort spanning cognitively unimpaired, prodromal, and dementia stages. QUIET integrates the structural controllability of individual white-matter connections with the mutual information between functional timeseries to quantify the control energy required to synchronize a brain network. We found that the QUIET-derived control energy followed an inverted-U along the amyloid--tau axis, rising under early genetic and amyloid risk, peaking at the amyloid-positive, pre-tau stage, and falling as tau accumulated. APOE-$\epsilon$4 carriage raised control energy before any detectable pathology, an elevation carried almost entirely by assigned female at birth (AFAB) individuals. Entorhinal tau marked the descending phase, and a large, network-specific hemispheric asymmetry persisted from cognitively normal to Alzheimer's brains. QUIET-derived control energy correlates with the established markers of Alzheimer's disease (cortical amyloid, entorhinal tau, and APOE-$\epsilon$4 carriage) in a stage- and sex-dependent manner.