Markovian Nonunital Anticoncentration Obstruction

Claim/Theorem

Fefferman, Ghosh, Gullans, Kuroiwa, and Sharma prove that random quantum circuits subject to the paper's practical constant-rate nonunital noise models exhibit lack of anticoncentration at every circuit depth in the paper's strong second-moment sense (Definition 10). In particular, this remains true when amplitude-damping noise is combined with unital depolarizing noise in either order, so the output distribution never resembles a maximally entropic distribution at any depth. The paper also gives a broader parameter-range generalization to arbitrary channels with a nonunital component.

Thus, in the Markovian constant-rate setting treated by the paper, realistic entropy-reducing noise creates a depth-uniform obstruction to the flat output statistics required by standard random-circuit-sampling hardness arguments.

Dependencies

• None.

Conflicts/Gaps

• The result is a Markovian constant-rate theorem, whereas Conjecture 1 asks for temporally correlated process-tensor noise with locality radius, memory scale, and contraction assumptions.
• The paper establishes failure of anticoncentration, but Conjecture 1 is phrased through a specific second-moment lower bound and then a stronger Porter-Thomas separation.
• The theorem is for practical nonunital noise models, not yet for a general finite-memory causal process tensor.

Sources

• 10.1103/PRXQuantum.5.030317 (abstract; Theorems 6-8 and Definition 10)