As the etched lines on computer memory chips have become thinner and the chips’ circuits more complex, both the power of the chips and the electronic devices they drive have vastly increased.
Option elimination
(A) the chips’ circuits more complex, both the power of the chips and the electronic devices they drive have - "they" can refer to "the etched lines, "memory chips," and chip's circuits." How can "the etched lines and chip's circuits" increase the number of electronic devices - illogical? In the 2nd part, after "and" the "have become" is eliminated to avoid repetition (the use of ellipsis) - The important thing about Ellipsis, as "Daagh" pointed out in one of his comments - Ellipsis is resorted more for brevity (conciseness) than for anything else. But the important requirement is that the understood phrase must exist somewhere before "in" the sentence. Here, we have "have become" in the part before "and".
(B) the chips’ circuits more complex, the power of both the chips and the electronic devices they drive has - the expansion is "the chip's circuits have been more complex, the power of both the chips and the electronic devices .....has" - grammatically and logically correct.
(C) the chips’ circuits are more complex, both the power of the chips and the electronic devices they drive has - "both" - "has" - SV issue
(D) their circuits are more complex, the power of both the chips and the electronic devices they drive have - "the power" - "have" - SV issue
(E) their circuits more complex, both the power of the chips and the electronic devices they drive have - Their can refer to "the etched lines," "memory chips," or "chip's circuits" - it doesn't make sense to say "the etched lines circuits more complex." Wrong.