Intermediate-depth complexity of local Clifford filter evaluation

Characterize the intermediate-depth computational complexity of evaluating the trace filter function for geometrically local one-dimensional Clifford circuits.

Background

The trace filter function can be computationally difficult to evaluate even when all circuit gates are Clifford. The paper proves worst-case exponential complexity and #P-hardness for general Clifford circuits, while identifying efficient special regimes: product circuits admit factorized linear-time evaluation, globally random Clifford circuits are typically efficient in expectation, and tensor-network methods are efficient with exponential dependence on circuit depth.

Between these regimes lies the intermediate-depth setting for local Clifford circuits. The paper does not provide a complete complexity characterization there and explicitly identifies that complexity as unresolved.

References

For the Clifford circuits, stabilizer methods make the computation feasible in the relevant regimes, but we showed the worst-case evaluation problem is hard and the intermediate-depth complexity of local Clifford circuits is not yet fully understood.

— Quickly extracting fidelity decay rates in random circuit benchmarking experiments  (2610.11540 - Köhne et al., 8 Oct 2026) in Section "Discussion"