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arXiv:2610.02130

Quantum impurity models: easy at equilibrium, universal in motion

Srinivasan Arunachalam, Sergey Bravyi, Anirban Chowdhury, Arkopal Dutt, Alexandru Gheorghiu, Zhi Li

quant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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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 $β$, the Helmholtz free energy and a classical description of the thermofield double state can be computed to precision $\varepsilon$ in time $\textsf{poly}(n,β,1/\varepsilon)$; (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.

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81 pages, 2 figures

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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 HHH be the Hamiltonian of an impurity model with nnn fermionic modes and a constant-size impurity. We show that: (1) the ground energy of HHH can be approximated to additive error ε\varepsilonε by a classical algorithm with runtime poly(n,1/ε)\textsf{poly}(n,1/\varepsilon)poly(n,1/ε), improving on the quasi-polynomial runtime of the best previously known algorithm; (2) at inverse temperature βββ, the Helmholtz free energy and a classical description of the thermofield double state can be computed to precision ε\varepsilonε in time poly(n,β,1/ε)\textsf{poly}(n,β,1/\varepsilon)poly(n,β,1/ε); (3) simulating the time evolution e−iHte^{-iHt}e−iHt is BQP\textsf{BQP}BQP-complete, for HHH 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.
Quantum impurity models are paradigmatic models of interacting quantum matter, as well as key computational primitives for modern electronic-structure methods. They describe a small subsystem of interacting fermions coupled to a large, noninteracting bath. We perform a comprehensive study of the computational complexity of simulating impurity models, delineating the boundary between classical and quantum tractability for this class of problems. Our main finding is that static properties of quantum impurity models can be calculated efficiently on a classical computer. Specifically, we give classical algorithms that (1) estimate the ground-state energy to additive precision δδδ in time poly(n,δ−1)\mathrm{poly}(n,δ^{-1})poly(n,δ−1), and (2) estimate the partition function at inverse temperature βββ to relative precision δδδ in time poly(n,β,δ−1)\mathrm{poly}(n,β,δ^{-1})poly(n,β,δ−1), where nnn is the system size. These results improve the previous best-known complexity for ground-state energy estimation from quasipolynomial to polynomial time, while establishing for the first time rigorous polynomial-time guarantees for simulating impurity models in thermal equilibrium. On the other hand, we find that simulating dynamical properties of impurity models is hard for classical computers but easy on a quantum computer. As a canonical example, we show that computing their nonequilibrium Green's functions captures the full power of quantum computation, even at finite temperature. Taken together, our results rule out superpolynomial quantum speedups for computing static properties, but provide an avenue for quantum advantage in simulating impurity physics out of equilibrium.
Impurity Hamiltonians are systems of NNN fermionic modes where O(1)O(1)O(1) of them interact among themselves via quartic (or higher order) fermion terms, while coupling quadratically with O(N)O(N)O(N) bath modes. Without the quartic interactions, these systems are classically simulable with O(N3)O(N^3)O(N3) resources. It was proved that the time-dependent evolution of these systems can perform universal quantum computation. The question of whether or not this remains true for time-independent evolution remains open. Here, we prove that the time evolution of generic time-independent impurity Hamiltonians on O(N)O(N)O(N) qubits is universal on NNN qubits if the input state is a product state of fermions in any single particle basis. In our proof we find that for a computation of depth SSS, the size of the impurity scales as O(Slog⁡S)O(S\log S)O(SlogS).
Out-of-equilibrium fermionic quantum impurity models (QIM), describing a small interacting system coupled to a continuous fermionic bath, play an important role in condensed matter physics. Solving such models is a computationally demanding task, and a variety of computational approaches are based on finding approximate representations of the bath by a finite number of modes. In this paper, we formulate the problem of finding efficient bath representations as that of approximating a kernel of the bath's Feynman-Vernon influence functional by a sum of complex exponentials, with each term defining a fermionic pseudomode. Under mild assumptions on the analytic properties of the bath spectral density, we provide an analytic construction of pseudomodes, and prove that their number scales polylogarithmically with the maximum evolution time TTT and the approximation error $\ensuremath{\varepsilon}$. We then demonstrate that the number of pseudomodes can be significantly reduced by an interpolative matrix decomposition (ID). Furthermore, we present a complementary approach, based on constructing rational approximations of the bath's spectral density using the ``AAA'' algorithm, followed by compression with ID. The combination of two approaches yields a pseudomode count scaling as ${N}_{\text{ID}}\ensuremath{\sim}ln(T)ln(1/\ensuremath{\varepsilon})$, and the agreement between the two approaches suggests that the result is close to optimal. Finally, to relate our findings to QIM, we derive an explicit Liouvillian that describes the time evolution of the combined impurity-pseudomodes system. These results establish bounds on the computational resources required for solving out-of-equilibrium QIMs, providing an efficient starting point for tensor-network methods for QIMs.
Simulating noninteracting fermion systems is a common task in computational many-body physics. In the absence of translational symmetries, modeling free fermions on N modes usually requires poly ( N ) computational resources. While often moderate, these costs can be prohibitive in practice when large systems are considered. We present several free fermion problems that can be solved by a quantum algorithm with substantially reduced computational costs. The memory costs are exponentially improved, poly log ( N ) . The run-time improvement, compared to the best known classical algorithms, is either exponential or significantly polynomial, depending on the geometry of the problem. The simulation of free fermion dynamics belongs to the BQP-hard complexity class (i.e., as hard as any (decision) problem that can be solved on a quantum computer). This implies (under standard assumptions) that our algorithm yields an exponential speedup for classical algorithm at least for some geometries. The key technique in our algorithm is the block encoding of objects such as correlation matrices and Green's functions into a unitary. We demonstrate how such unitaries can be efficiently realized as quantum circuits, in the context of dynamics and thermal states of tight-binding Hamiltonians. The special cases of disordered and inhomogeneous lattices, as well as large nonlattice graphs, are presented in detail. Finally, we show that our simulation algorithm generalizes to other promising targets, including free-boson systems.
Approximating the ground state of many-body systems is a key computational bottleneck underlying important applications in physics and chemistry. The most widely known quantum algorithm for ground state approximation, quantum phase estimation, is out of reach of current quantum processors due to its high circuit-depths. Subspace-based quantum diagonalization methods offer a viable alternative for pre- and early-fault-tolerant quantum computers. Here, we introduce a quantum diagonalization algorithm which combines two key ideas on quantum subspaces: a classical diagonalization based on quantum samples, and subspaces constructed with quantum Krylov states. We prove that our algorithm converges in polynomial time under the working assumptions of Krylov quantum diagonalization and sparseness of the ground state. We then demonstrate the scalability of our approach by performing the largest ground-state quantum simulation of impurity models using a Heron quantum processors and the Frontier supercomputer. We consider both the single-impurity Anderson model with 41 bath sites, and a system with 4 impurities and 7 bath sites per impurity. Our results are in excellent agreement with Density Matrix Renormalization Group calculations.
Recently, there have been several advancements in quantum algorithms for Gibbs sampling. These algorithms simulate the dynamics generated by an artificial Lindbladian, which is meticulously constructed to obey a detailed-balance condition with the Gibbs state of interest, ensuring it is a stationary point of the evolution, while simultaneously having efficiently implementable time steps. The overall complexity then depends primarily on the mixing time of the Lindbladian, which can vary drastically, but which has been previously bounded in the regime of high enough temperatures [Rouzé et al. arXiv:2403.12691 and arXiv:2411.04885]. In this work, we calculate the spectral gap of the Lindbladian for free fermions using third quantisation, and also prove a logarithmic bound on its mixing time by analysing corresponding covariance matrices. Then we prove a constant gap of the perturbed Lindbladian corresponding to interacting fermions up to some maximal coupling strength. This is achieved by using theorems about stability of the gap for lattice fermions. Our methods apply at any constant temperature and independently of the system size. The gap then provides an upper bound on the mixing time, and hence on the overall complexity of the quantum algorithm, proving that the purified Gibbs state of weakly interacting (quasi-)local fermionic systems of any dimension can be prepared in O~(n3polylog⁡(1/ε))\widetilde{\mathcal{O}} (n^3 \operatorname{polylog}(1/ε))O(n3polylog(1/ε)) time on O(n)\mathcal{O}(n)O(n) qubits, where nnn denotes the size of the system and εεε the desired accuracy. As an application, we explain how to calculate partition functions for the considered systems. We provide exact numerical simulations for small system sizes supporting the theory and also identify different suitable jump operators and filter functions for the sought-after regime of intermediate coupling in the Fermi-Hubbard model.