Quantum Speedup for Tarski Fixed-Point Finding

Determine whether quantum algorithms can achieve a speedup over classical algorithms for finding a fixed point of a monotone function in Tarski(n,k), particularly in the high-dimensional regime.

Background

The paper establishes an Ω(k log n) bounded-error quantum query lower bound for Tarski(n,k), matching the classical complexity in the regimes n=2 and k=1 but leaving a logarithmic-factor gap when n≥k. The authors note that prior work indicates that quantum algorithms do not readily overcome the nested binary-search structure in low dimensions, while the paper's lower bound suggests that quadratic-speedup techniques such as wildcard search may not apply directly.

The unresolved issue is whether quantum computation can nevertheless improve on the best classical algorithms, potentially through a coherent analogue of the decomposition lemmas used in classical Tarski fixed-point algorithms.