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Tensor field theory with applications to quantum computing

Yannick Meurice, Ryo Sakai, Judah Unmuth-Yockey

Published Oct 13, 2020• 10 citations
OpenAlex

Abstract

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.

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•Jul 7, 2020•0 citations
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.
In most lattice simulations, the variables of integration are compact and character expansion (for instance Fourier analysis for U(1) models) can be used to rewrite the partition function and average observables as discrete sums of contracted tensors. These reformulations have been used for RG blocking but they also naturally fit the needs of quantum computing. We discuss FAQ about tensorial reformulations: effects of truncations on symmetries, boundary conditions, Grassmann variables, and other recent aspects of quantum computing in the same context.
We discuss recent progress in Tensor Lattice Field Theory and economical, symmetry preserving, truncations suitable for quantum computations/simulations. We focus on spin and gauge models with continuous Abelian symmetries such as the Abelian Higgs model and emphasize noise-robust implementations of Gauss’s law. We discuss recent progress concerning the comparison between field digitizations and character expansions, symmetry breaking in tensor language, wave-packet preparation and possible new implementations of Abelian models using Rydberg atoms.
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.
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 various observables. 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 Ns=4N_{s}=4Ns​=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 Ns=4N_{s}=4Ns​=4 and 8 sites.
The study of the ground states of local Hamiltonians in quantum many-body and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum many-body theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on non-interacting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a real-space renormalization group flow via continuous coarse-graining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.