Microstate Distinguishability, Quantum Complexity, and the Eigenstate Thermalization Hypothesis

• URL: http://arxiv.org/abs/2009.00632v3
• Date: Wed, 31 Mar 2021 20:46:22 GMT
• Title: Microstate Distinguishability, Quantum Complexity, and the Eigenstate Thermalization Hypothesis
• Authors: Ning Bao, Jason Pollack, David Wakeham, Elizabeth Wildenhain
• Abstract summary: We quantify the difficulty of distinguishing eigenstates obeying the Eigenstate Thermalization Hypothesis (ETH) We show that an exponential hardness of distinguishing between states implies ETH-like matrix elements.
• Score: 0.2294014185517203
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we use quantum complexity theory to quantify the difficulty of distinguishing eigenstates obeying the Eigenstate Thermalization Hypothesis (ETH). After identifying simple operators with an algebra of low-energy observables and tracing out the complementary high-energy Hilbert space, the ETH leads to an exponential suppression of trace distance between the coarse-grained eigenstates. Conversely, we show that an exponential hardness of distinguishing between states implies ETH-like matrix elements. The BBBV lower bound on the query complexity of Grover search then translates directly into a complexity-theoretic statement lower bounding the hardness of distinguishing these reduced states.

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