Circuit lower bounds for low-energy states of quantum code Hamiltonians
summary
Proves a concrete circuit-depth lower bound for every low-energy state of Hamiltonians built from stabilizer codes, giving depth at least Θ(log D) and a second term depending on the code rate (so high-rate or large-distance codes force nontrivial depth). Techniques combine light-cone arguments, local indistinguishability of code/eigenspaces, syndrome measurements with gentle measurement and entropy counting to turn low-energy constraints into a K-vs-entropy contradiction. Applies to LDPC stabilizer families; e.g., hypergraph-product codes yield depth ≥ c log n for all states with energy ≤ n^0.99.
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chapters (13)
- 0:00Introduction
- 0:24Overview
- 1:50No low energy trivial states
- 4:06Technical statement
- 7:30Example codes
- 20:00Proof
- 25:05Error Correcting Codes
- 28:43Low Energy States
- 31:13Low Distance States
- 32:54Theorem
- 37:57LDPC stabilizer codes
- 40:53Gentle measurement
- 44:22Entropy
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