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Whether such near-crossings occur in typical large samples, and how small their gaps become, decides whether annealing is exponentially slow even after the critical point is passed.","posed_since":"","precise":"For the transverse-field Edwards-Anderson model in $d = 2\\ \\text{and}\\ 3$, and its infinite-range (Sherrington-Kirkpatrick) limit, at $0 < \\Gamma < \\Gamma_c$, determine the probability that the instantaneous ground state has an avoided crossing as $\\Gamma$ decreases to 0, and the scaling of the minimum gap $\\Delta_{\\min}$ with N: $\\exp(-c N^a)$ with the value of a, or a power law. An answer gives the typical $\\Delta_{\\min}(N)$ and its distribution over disorder samples.","problem_ref":"d0aca997503a38c1677ab58d857405770438a2aa5adbb8b0b4a93aea8c7982a0","references":"","settled_by":"Exact diagonalization and parity-resolved quantum Monte Carlo or tensor-network tracking of the two lowest levels along $\\Gamma$ for $N$ up to a few hundred spins over many disorder samples.","status_note":"Knysh (Nature Communications 2016) argued that such bottlenecks are generic in the spin-glass phase and give gaps exponentially small in N; tests in finite dimensions remain limited.","title":"Exponentially small gaps inside the quantum spin-glass 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