arXiv:hep-lat/9610012

Hubbard Model and Luscher fermions

P. Sawicki

Published Oct 9, 1996• 0 citations

Abstract

We discuss numerical complexity of the Lüscher algorithm applied to the Hubbard Model. In particular we present comparison to a certain algorithm based on direct computation of the fermionic determinant.

DOI: 10.48550/arxiv.hep-lat/9610012

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arXiv:hep-lat/9610012

Hubbard Model and Luscher fermions

P. Sawicki

Published Oct 9, 1996• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.hep-lat/9610012

Related papers

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Simulating interactions between fermions and bosons is central to understanding correlated phenomena, yet these systems are inherently difficult to treat classically. Previous quantum algorithms for fermion-boson models exhibit computation costs that scale polynomially with the bosonic truncation parameter, $Λ$ . In this work we identify the efficient unitary transformation enabling fast-forwarded evolution of the fermion-boson interaction term, yielding an interaction-picture based simulation algorithm with complexity polylogarithmic in $Λ$ . We apply this transformation to explicitly construct an efficient quantum algorithm for the Hubbard-Holstein model and discuss its generalisation to other fermion-boson interacting models. This approach yields an important asymptotic improvement in the dependence on the bosonic cutoff and establishes that, for certain models, fermion-boson interactions can be simulated with resources comparable to those required for purely fermionic systems.Simulating interactions between fermions and bosons is central to understanding correlated phenomena, yet these systems are inherently difficult to treat classically. Previous quantum algorithms for fermion-boson models exhibit computation costs that scale polynomially with the bosonic truncation parameter, $Λ$. In this work we identify the efficient unitary transformation enabling fast-forwarded evolution of the fermion-boson interaction term, yielding an interaction-picture based simulation algorithm with complexity polylogarithmic in $Λ$. We apply this transformation to explicitly construct an efficient quantum algorithm for the Hubbard-Holstein model and discuss its generalisation to other fermion-boson interacting models. This approach yields an important asymptotic improvement in the dependence on the bosonic cutoff and establishes that, for certain models, fermion-boson interactions can be simulated with resources comparable to those required for purely fermionic systems.

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We study the basic features of the two-dimensional quantum Hubbard Model at half-filling by means of the Lüscher algorithm and the algorithm based on direct update of the determinant of the fermionic matrix. We implement the Lüscher idea employing the transfer matrix formalism which allows to formulate the problem on the lattice in (2+1) dimensions. We discuss the numerical complexity of the Lüscher technique, systematic errors introduced by polynomial approximation and introduce some improvements which reduce long autocorrelations. In particular we show that preconditioning of the fermionic matrix speeds up the algorithm and extends the available range of parameters. We investigate the magnetic and the one-particle properties of the Hubbard Model at half-filling and show that they are in qualitativeWe study the basic features of the two-dimensional quantum Hubbard Model at half-filling by means of the Lüscher algorithm and the algorithm based on direct update of the determinant of the fermionic matrix. We implement the Lüscher idea employing the transfer matrix formalism which allows to formulate the problem on the lattice in (2+1) dimensions. We discuss the numerical complexity of the Lüscher technique, systematic errors introduced by polynomial approximation and introduce some improvements which reduce long autocorrelations. In particular we show that preconditioning of the fermionic matrix speeds up the algorithm and extends the available range of parameters. We investigate the magnetic and the one-particle properties of the Hubbard Model at half-filling and show that they are in qualitative

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Λ

. In this work we identify the efficient unitary transformation enabling fast-forwarded evolution of the fermion-boson interaction term, yielding an interaction-picture based simulation algorithm with complexity polylogarithmic in

Λ

. We apply this transformation to explicitly construct an efficient quantum algorithm for the Hubbard-Holstein model and discuss its generalisation to other fermion-boson interacting models. This approach yields an important asymptotic improvement in the dependence on the bosonic cutoff and establishes that, for certain models, fermion-boson interactions can be simulated with resources comparable to those required for purely fermionic systems.

6 \times 6

and

8 \times 8

at temperature

T/t = 1/6.4 \simeq 0.156

and interaction strength

U/t = 8.0

, using Trotter number

N_t = 20

(Trotter step

ε= 0.32

). Our results demonstrate that WV-HMC remains effective even in parameter regimes where standard (non-thimble) determinant quantum Monte Carlo methods fail. In this work, fermion matrix inversions are performed using direct solvers, leading to a computational cost of

O(N^3)

, where

N

denotes the number of degrees of freedom and is proportional to the spacetime lattice volume. An alternative algorithm employing pseudofermions and iterative solvers, which reduces the cost to

O(N^2)

at the expense of careful parameter tuning, will be discussed in a separate publication.