We study the computational limits of Low-Rank Adaptation (LoRA) for finetuning transformer-based models using fine-grained complexity theory. Our key observation is that the existence of low-rank decompositions within the gradient computation of LoRA adaptation leads to possible algorithmic speedup. This allows us to (i) identify a phase transition behavior of efficiency assuming the Strong Exponential Time Hypothesis (SETH), and (ii) prove the existence of almost linear algorithms by controlling the LoRA update computation term by term. For the former, we identify a sharp transition in the efficiency of all possible rank-$r$ LoRA update algorithms for transformers, based on specific norms resulting from the multiplications of the input sequence $X$, pretrained weights ${W^\star}$, and adapter matrices $αB A/r$. Specifically, we derive a shared upper bound threshold for such norms, and show that efficient (sub-quadratic) approximation algorithms of LoRA exist only below this threshold. For the latter, we prove the existence of almost linear approximation algorithms for LoRA adaptation by utilizing the hierarchical low-rank structures of LoRA gradients and approximating the gradients with a series of chained low-rank approximations. To showcase our theory, we consider two practical scenarios: partial (e.g., only $W_V$ and $W_Q$) and full adaptations (e.g., $W_Q$, $W_V$, and $W_K$) of weights in attention heads.