Misconceptions/Circuits/Circuit expressibility
AdvancedCircuits9 minInteractive

The myth

A bigger circuit is always more expressive

01

Why people believe this

More gates means more operations means more things the circuit can compute. Adding gates to a circuit should always increase what states it can reach.

02

The correction

Expressibility — the ability to reach a diverse set of states — does not increase monotonically with circuit size. H followed by H is the identity. CNOT followed by CNOT is the identity. Adding redundant gates reduces expressibility per gate while increasing depth and noise. Furthermore, for parametrized circuits, high expressibility correlates with barren plateaus — circuits that can reach any state have exponentially flat loss landscapes and cannot be trained. The optimal circuit is the smallest one that can express the target state, not the largest.

03

Try it in the simulator

What to do

Build two circuits of the same depth — one with only X and Z gates, one mixing H, Rx, CNOT. Run both and compare the state vectors. The second reaches states the first cannot. But try H+H on the same qubit — they cancel completely. More gates does not always mean more expressiveness.

Open in simulator
04

Research notes

Tags

#expressibility#circuit design#ansatz#trainability

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