Efficiency of “sausage” versus full fermion matrix approaches at large volume

Determine, for sign-problem-free simulations at large spatial volumes (V ≥ 1000) and small inverse temperature (β ≤ 20), whether it is more computationally efficient to use the reduced “sausage” matrix \hat M = 1 + \prod_t M_t or to revert to computations with the full VN_t × VN_t fermion matrix (e.g., via matrix-free pseudo-fermion methods), and characterize how this efficiency depends on parameters such as the temperature.

Background

In the large-volume, high-temperature, sign-problem-free regime, dense treatments of the reduced fermion matrix (“sausage”) \hat M become prohibitively expensive. The authors note that one may need to switch to matrix-free approaches such as pseudo-fermions, which operate with the full fermion matrix through solves and matrix–vector products.

However, the authors explicitly state that it is not completely clear which strategy is more efficient in this regime: continuing with the sausage formalism or reverting to the full fermion matrix. They further suggest that the answer may depend on parameters such as the temperature, highlighting an unresolved efficiency trade-off to be clarified.