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In this work, we use quantum complexity theory to quantify the difficulty ofdistinguishing eigenstates obeying the Eigenstate Thermalization Hypothesis(ETH). After identifying simple operators with an algebra of low-energyobservables and tracing out the complementary high-energy Hilbert space, theETH leads to an exponential suppression of trace distance between thecoarse-grained eigenstates. Conversely, we show that an exponential hardness ofdistinguishing between states implies ETH-like matrix elements. The BBBV lowerbound on the query complexity of Grover search then translates directly into acomplexity-theoretic statement lower bounding the hardness of distinguishingthese reduced states.

Microstate Distinguishability, Quantum Complexity, and the Eigenstate Thermalization Hypothesis