Average-case quantum complexity from glassiness

Abstract

Glassiness -- a phenomenon in physics characterized by a rough free-energy landscape -- implies hardness for stable classical algorithms. For example, it can obstruct constant-time Langevin dynamics and message-passing in random $k$-SAT and max-cut instances. We provide an analogous framework for average-case quantum complexity showing that a natural family of quantum algorithms (e.g., Lindbladian evolution) fails for natural Hamiltonian ensembles (e.g., random 3-local Hamiltonians). Specifically, we prove that the standard notion of quantum glassiness based on replica symmetry breaking obstructs stable quantum algorithms for Gibbs sampling, which we define by a Lipschitz temperature dependence in quantum Wasserstein complexity. Our proof relies on showing that such algorithms fail to capture a structural phase transition in the Gibbs state, where glassiness causes the Gibbs state to decompose into clusters extensively separated in quantum Wasserstein distance. This yields average-case lower bounds for constant-time local Lindbladian evolution and shallow variational circuits. Unlike mixing time lower bounds, our results hold even when dynamics are initialized from the maximally mixed state. We apply these lower bounds to non-commuting, non-stoquastic Hamiltonians by showing a glass transition via the replica trick. We find that the ensemble of all 3-local Pauli strings with independent Gaussian coefficients is average-case hard, while providing analytical evidence that the general $p$-local Pauli ensemble is non-glassy for sufficiently large constant $p$, in contrast to its classical (Ising $p$-spin, always glassy) and fermionic (SYK, never glassy) counterparts.