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arXiv:2409.04550 · Physical Review Research

Solving free fermion problems on a quantum computer

Maarten Stroeks, Daan Lenterman, Barbara M. Terhal, Yaroslav Herasymenko

Published Nov 18, 2025• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/zmwm-gdmw

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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.
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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.
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