Quantum impurity models: easy at equilibrium, universal in motion

Abstract:A quantum impurity model describes a small interacting subsystem embedded into a large bath of free fermions. Here we study the computational complexity of calculating the ground energy, thermal equilibrium, and dynamical properties of these models. Our work reveals a sharp contrast: equilibrium properties can be efficiently approximated by classical means, whereas time evolution can implement a universal quantum computation. More precisely, let $H$ be the Hamiltonian of an impurity model with $n$ fermionic modes and a constant-size impurity. We show that: (1) the ground energy of $H$ can be approximated to additive error $\varepsilon$ by a classical algorithm with runtime $\textsf{poly}(n,1/\varepsilon)$, improving on the quasi-polynomial runtime of the best previously known algorithm; (2) at inverse temperature $\beta$, the Helmholtz free energy and a classical description of the thermofield double state can be computed to precision $\varepsilon$ in time $\textsf{poly}(n,\beta,1/\varepsilon)$; (3) simulating the time evolution $e^{-iHt}$ is $\textsf{BQP}$-complete, for $H$ that is time-independent and has a fixed, constant impurity size. Our algorithms exploit exponential suppression of multi-particle bath excitations in a basis organized by energy scale and Krylov depth. Our universality construction realizes a stationary quantum processor whose program arrives in a stream of freely propagating fermions.

arXiv:2610.02130 [quant-ph]

(or arXiv:2610.02130v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.02130 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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