Ab-initio calculations of real-time evolution for lattice gauge theory have very interesting potential applications but present challenging computational aspects. We show that tensor renormalization group methods developed in the context of Euclidean-time lattice field theory can be applied to calculation of Trotterized evolution operators at real-time. We discuss the optimization of truncation procedures for the real-time equivalent of the partition function. We apply the numerical methods to the 1D Quantum Ising Model with an external transverse field in the ordered phase and compare with universal quantum computing for $N_{s}=4$ and 8 sites.Ab-initio calculations of real-time evolution for lattice gauge theory have very interesting potential applications but present challenging computational aspects. We show that tensor renormalization group methods developed in the context of Euclidean-time lattice field theory can be applied to calculation of Trotterized evolution operators at real-time. We discuss the optimization of truncation procedures for the real-time equivalent of the partition function. We apply the numerical methods to the 1D Quantum Ising Model with an external transverse field in the ordered phase and compare with universal quantum computing for $N_{s}=4$ and 8 sites.

International Symposium on Lattice Field Theory (LATTICE)May 3, 20240 citations

Quantum real-time evolution using tensor renormalization group methods

Michael Hite, Yannick Meurice

We introduce an approach for approximate real-time evolution of quantum systems using tensor renormalization group (TRG) methods originally developed for imaginary time. We use higher-order TRG to generate a coarse-grained time evolution operator for a $1+1\mathrm{D}$ transverse Ising model with a longitudinal field. We show that the standard tensor norm used for the singular value decomposition-based truncation is degenerate and propose an alternate method to discriminate. We show that it is effective and efficient in evolving Gaussian wave packets for one and two particles in the disordered phase, while ordered phase behavior is more challenging to capture. We compare our algorithm with local simulators for universal quantum computers and discuss possible benchmarking in the near future.We introduce an approach for approximate real-time evolution of quantum systems using tensor renormalization group (TRG) methods originally developed for imaginary time. We use higher-order TRG to generate a coarse-grained time evolution operator for a $1+1\mathrm{D}$ transverse Ising model with a longitudinal field. We show that the standard tensor norm used for the singular value decomposition-based truncation is degenerate and propose an alternate method to discriminate. We show that it is effective and efficient in evolving Gaussian wave packets for one and two particles in the disordered phase, while ordered phase behavior is more challenging to capture. We compare our algorithm with local simulators for universal quantum computers and discuss possible benchmarking in the near future.

arXiv:2411.05301Physical review. D/Physical review. D.Feb 5, 20251 citation

Tensor field theory with applications to quantum computing

Yannick Meurice, Ryo Sakai, Judah Unmuth-Yockey

We discuss the successes and limitations of statistical sampling for a sequence of models studied in the context of lattice QCD and emphasize the need for new methods to deal with finite-density and real-time evolution. We show that these lattice models can be reformulated using tensorial methods where the field integrations in the path-integral formalism are replaced by discrete sums. These formulations involve various types of duality and provide exact coarse-graining formulas which can be combined with truncations to obtain practical implementations of the Wilson renormalization group program. Tensor reformulations are naturally discrete and provide manageable transfer matrices. Combining truncations with the time continuum limit, we derive Hamiltonians suitable to perform quantum simulation experiments, for instance using cold atoms, or to be programmed on existing quantum computers. We review recent progress concerning the tensor field theory treatment of non-compact scalar models, supersymmetric models, economical four-dimensional algorithms, noise-robust enforcement of Gauss's law, symmetry preserving truncations and topological considerations.We discuss the successes and limitations of statistical sampling for a sequence of models studied in the context of lattice QCD and emphasize the need for new methods to deal with finite-density and real-time evolution. We show that these lattice models can be reformulated using tensorial methods where the field integrations in the path-integral formalism are replaced by discrete sums. These formulations involve various types of duality and provide exact coarse-graining formulas which can be combined with truncations to obtain practical implementations of the Wilson renormalization group program. Tensor reformulations are naturally discrete and provide manageable transfer matrices. Combining truncations with the time continuum limit, we derive Hamiltonians suitable to perform quantum simulation experiments, for instance using cold atoms, or to be programmed on existing quantum computers. We review recent progress concerning the tensor field theory treatment of non-compact scalar models, supersymmetric models, economical four-dimensional algorithms, noise-robust enforcement of Gauss's law, symmetry preserving truncations and topological considerations.

Oct 13, 202010 citations

Tensor lattice field theory for renormalization and quantum computing

Yannick Meurice, Ryo Sakai, Judah Unmuth-Yockey

One goal in understanding quantum chromodynamics (QCD) includes solving how quarks and gluons combine to form the hadrons and nuclei seen in nature. With lattice QCD, progress has been made regarding the calculation of masses and couplings. However, the real-time evolution and the critical behavior at finite density of strong particles in colliders, stars, or after the big bang remain a challenging problem despite their potential to detect the existence of new physics. The tensor methods for lattice field theories provide a route to handle strongly correlated systems across different subfields using renormalization group methods or quantum computing.One goal in understanding quantum chromodynamics (QCD) includes solving how quarks and gluons combine to form the hadrons and nuclei seen in nature. With lattice QCD, progress has been made regarding the calculation of masses and couplings. However, the real-time evolution and the critical behavior at finite density of strong particles in colliders, stars, or after the big bang remain a challenging problem despite their potential to detect the existence of new physics. The tensor methods for lattice field theories provide a route to handle strongly correlated systems across different subfields using renormalization group methods or quantum computing.

arXiv:2010.06539Reviews of Modern PhysicsMay 26, 202274 citations

Quantum Field Theories with Tensor Renormalization Group

Shinichiro Akiyama, Y. Kuramashi, Yusuke Yoshimura

We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1) $d$ ) complex $\phi^4$ theory at finite density, (1+1) $d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1) $d$ Nambu--Jona-Lasinio model at finite density and (1+1) $d$ and (2+1) $d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)$d$) complex $\phi^4$ theory at finite density, (1+1)$d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1)$d$ Nambu--Jona-Lasinio model at finite density and (1+1)$d$ and (2+1)$d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.

arXiv:2111.04240International Symposium on Lattice Field Theory (LATTICE)May 16, 20224 citations

Lattice renormalization of quantum simulations

Marcela S. Carena, Henry Lamm, Ying-Ying Li +1 more

With advances in quantum computing, new opportunities arise to tackle challenging calculations in quantum field theory. We show that trotterized time-evolution operators can be related by analytic continuation to the Euclidean transfer matrix on an anisotropic lattice. In turn, trotterization entails renormalization of the temporal and spatial lattice spacings. Based on the tools of Euclidean lattice field theory, we propose two schemes to determine Minkowski lattice spacings, using Euclidean data and thereby overcoming the demands on quantum resources for scale setting. In addition, we advocate using a fixed-anisotropy approach to the continuum to reduce both circuit depth and number of independent simulations. We demonstrate these methods with qiskit noiseless simulators for a $2+1\mathrm{D}$ discrete non-Abelian ${D}_{4}$ gauge theory with two spatial plaquettes.With advances in quantum computing, new opportunities arise to tackle challenging calculations in quantum field theory. We show that trotterized time-evolution operators can be related by analytic continuation to the Euclidean transfer matrix on an anisotropic lattice. In turn, trotterization entails renormalization of the temporal and spatial lattice spacings. Based on the tools of Euclidean lattice field theory, we propose two schemes to determine Minkowski lattice spacings, using Euclidean data and thereby overcoming the demands on quantum resources for scale setting. In addition, we advocate using a fixed-anisotropy approach to the continuum to reduce both circuit depth and number of independent simulations. We demonstrate these methods with qiskit noiseless simulators for a $2+1\mathrm{D}$ discrete non-Abelian ${D}_{4}$ gauge theory with two spatial plaquettes.

arXiv:2107.01166Physical review. D/Physical review. D.Nov 29, 202139 citations

N_{s}=4

and 8 sites.

N_{s}=4

and 8 sites.

1+1\mathrm{D}

transverse Ising model with a longitudinal field. We show that the standard tensor norm used for the singular value decomposition-based truncation is degenerate and propose an alternate method to discriminate. We show that it is effective and efficient in evolving Gaussian wave packets for one and two particles in the disordered phase, while ordered phase behavior is more challenging to capture. We compare our algorithm with local simulators for universal quantum computers and discuss possible benchmarking in the near future.

d

) complex

\phi^4

theory at finite density, (1+1)

d

pure U(1) lattice gauge theory with a

\theta

term, (3+1)

d

Nambu--Jona-Lasinio model at finite density and (1+1)

d

and (2+1)

d

Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.

2+1\mathrm{D}

discrete non-Abelian

{D}_{4}

gauge theory with two spatial plaquettes.