Efficiency of tensor-network evaluation at intermediate circuit depths

Determine whether a tensor-network algorithm for evaluating the trace filter function on geometrically local one-dimensional Clifford circuits remains efficient through the depth regime in which the circuit approaches a uniformly random global stabilizer state, particularly for depths of order $O(\log n)$.

Background

The paper studies the classical cost of evaluating the interpolating filter functions used in trace-filtered randomized benchmarking. For Clifford circuits, direct evaluation can require exponentially many terms in the worst case and is #P-hard for general parameter values, although product circuits can be evaluated in linear time and globally random Clifford circuits have an expected constant number of nonzero terms. For geometrically local one-dimensional circuits, tensor-network methods provide efficiency that is polynomial in the number of qubits but exponential in circuit depth.

The authors note that circuits of depth $O(\log n)$ are expected to approximate uniformly random global stabilizer states, suggesting that tensor-network evaluation might remain efficient until direct evaluation becomes efficient in expectation. They leave the rigorous analysis of this efficiency transition unresolved.