The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek-Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of $N$ coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.
Topology played an important role in physics research during the last few decades. In particular, the quantum geometric tensor that provides local information about topological properties has attracted much attention. It will reveal interesting topological properties in non-Abelian systems, which have not been realized in practice. Here, we use a four-qubit quantum system in superconducting circuits to construct a degenerate Hamiltonian with parametric modulation. By manipulating the Hamiltonian with periodic drivings, we simulate the Bernevig-Hughes-Zhang model and obtain the quantum geometri…
We show that standard identities and theorems for lattice models with $U(1)$ symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogy between the continuous lattice equations of motion and the discrete selection rules of the tensors. We construct a gauge-invariant transfer matrix in arbitrary dimensions. We show the equivalence with its gauge-fixed version in a maximal temporal gauge and explain how a discrete Gauss's law is always enforced. We propose a noise-robust way to implement Gauss's law in arbitrary dimensions. We reformulat…
In this thesis we extend the formalism of tensor network algorithms to incorporate global internal symmetries. We describe how to both numerically protect the symmetry and exploit it for computational gain in tensor network simulations. Our formalism is independent of the details of a specific tensor network decomposition since the symmetry constraints are imposed at the level of individual tensors. Moreover, the formalism can be applied to a wide spectrum of physical symmetries described by any discrete or continuous group that is compact and reducible. We describe in detail the implementatio…
The complete geometry of quantum states in parameter space is characterized by the quantum geometric tensor, which contains the quantum metric and Berry curvature as the real and imaginary parts, respectively. When the quantum states are degenerate, the quantum metric and Berry curvature take non-Abelian forms. The non-Abelian (Abelian) Berry curvature and Abelian quantum metric have been experimentally measured. However, an experimentally feasible scheme to extract all the components of the non-Abelian quantum metric tensor is still lacking. Here we propose a generic protocol to directly extr…
We propose a generalized quantum geometric tenor to understand topological quantum phase transitions, which can be defined on the parameter space with the adiabatic evolution of a quantum many-body system. The generalized quantum geometric tenor contains two different local measurements, the non-Abelian Riemannian metric and the non-Abelian Berry curvature, which are recognized as two natural geometric characterizations for the change of the ground-state properties when the parameter of the Hamiltonian varies. Our results show the symmetry-breaking and topological quantum phase transitions can…
The full exploitation of non-abelian symmetries in tensor network states (TNS) derived from a given lattice Hamiltonian is highly attractive in various aspects. From a theoretical perspective, it can offer deep insights into the entanglement structure and quantum information content of strongly correlated quantum many-body states. From a practical perspective, it allows one to push numerical efficiency by orders of magnitude. Physical expectation values based on TNS require the full contraction of a given tensor network, with the elementary ingredient being a pairwise contraction. While well-e…
We generalize the $K$ matrix formulation to non-trivial non-Abelian families of 2+1D topological orders. Given a topological order $\mathcal C$, any topological order in the same non-Abelian family as $\mathcal C$ can be efficiently described by $\boldsymbol{a}=(a_I)$ where $a_I$ are Abelian anyons in $\mathcal C$, together with a symmetric invertible matrix $K$, $K_{IJ}=k_{IJ}-t_{a_I,a_J}$ where $k_{IJ}$ are integers, $k_{II}$ are even and $t_{a_I,a_J}$ are the mutual statistics between $a_I,a_J$. In particular, when $\mathcal C$ is a root whose rank is the smallest in the family, $K$ becomes…
We discuss recent progress in Tensor Lattice Field Theory and economical, symmetry preserving, truncations suitable for quantum computations or 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.
Skeleton of isometric Tensor Network States for Abelian String-Net Models
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Center for Theoretical Physics - a Leinweber Institute, Massachusetts Institute of Technology（麻省理工学院莱因韦伯理论物理中心）
A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of orthonormal local as well as effective basis sets. The two crucial ingredients, the Clebsch-Gordan algebra for multiplet spaces as well as the Wigner-Eckart theorem for operators, are accounted for in a natural, well-organized, and computationally straightforward way. The unifying tensor-representation for quantum symmetry spaces, dubbed QSpace, is particularly suitable to deal with standard renormalization group algorithms such as the numerical renormalization group (NRG…
In this thesis I develop a formalism whereby a tensor network may be understood in terms of a unitary braided tensor category, and represented in a particularly efficient manner corresponding to the exploitation of this mathematical structure. This approach permits the exploitation of the global Abelian and non-Abelian internal group symmetries of a lattice model, both to facilitate the study of particular symmetry sectors and to provide a considerable reduction in computational cost. It also enables the study of models with non-trivial exchange statistics (fermions, Abelian and non-Abelian an…
Topological phases of matter became a new standard to classify quantum systems in many cases, yet key quantities like the quantum geometric tensor providing local information about topological properties are still experimentally hard to access. In non-Abelian systems this accessibility to geometric properties can be even more restrictive due to the degeneracy of the states. We propose universal protocols to determine quantum geometric properties in non-Abelian systems. First, we show that for a weak resonant driving of the local parameters the coherent Rabi oscillations are related to the quan…
Hamiltonian formulation of lattice gauge theories (LGTs) is the most natural framework for the purpose of quantum simulation, an area of research that is growing with advances in quantum-computing algorithms and hardware. It, therefore, remains an important task to identify the most accurate, while computationally economic, Hamiltonian formulation(s) in such theories, considering the necessary truncation imposed on the Hilbert space of gauge bosons with any finite computing resources. This paper is a first step toward addressing this question in the case of non-Abelian LGTs, which further requ…
TensorKit.jl: A Julia package for large-scale tensor computations, with a hint of category theory
Center for Computational Quantum Physics, Flatiron Institute（计算量子物理中心，_flatiron_研究所）；Department of Physics and Astronomy, Ghent University（根特大学物理与天文学系）
