[Paper Review] Extracting Wilson loop operators and fractional statistics from a single bulk ground state

Ze-Pei Cian, Mohammad Hafezi+1 more|arXiv (Cornell University)|Sep 28, 2022

Quantum many-body systems4 citations

TL;DR

This paper presents an unbiased numerical optimization method to extract Wilson loop operators (WLOs) and fractional statistics from a single ground state wave function of a gapped Hamiltonian on a disk, without prior knowledge of the Hamiltonian. By formulating WLOs as matrix product operators (MPOs) and optimizing their variational parameters, the method identifies operators that preserve the ground state subspace, enabling the extraction of topological order—specifically the modular S and T matrices—through braiding and anyon creation/annihilation operations.

ABSTRACT

An essential aspect of topological phases of matter is the existence of Wilson loop operators that keep the ground state subspace invariant. Here we present and implement an \it unbiased m numerical optimization scheme to systematically find the Wilson loop operators given a single ground state wave function of a gapped Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We subsequently use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully extract topological order from a single wave function. We apply our method to the ground state of the perturbed toric code and doubled semion models with a magnetic field that is up to a half of the critical value. From a contemporary perspective, this can be thought of as a machine learning approach to discover emergent 1-form symmetries of a ground state wave function. From an application perspective, our approach can be relevant to find Wilson loop operators in current quantum simulators.

Motivation & Objective

• To develop a systematic, Hamiltonian-free method to diagnose topological order from a single ground state wave function.
• To identify Wilson loop operators (WLOs) that keep the ground state invariant, even when the underlying Hamiltonian is unknown or perturbed.
• To extract fractional statistics and topological invariants such as the modular S and T matrices from the same wave function.
• To enable characterization of topological order in quantum simulators where the exact Hamiltonian may not be known.
• To establish a framework for discovering emergent 1-form symmetries and anyonic excitations from wave function data alone.

Proposed method

• Formulate Wilson loop operators as matrix product operators (MPOs) with a variational ansatz, parameterized by tensors along a loop path.
• Define a cost function based on the expectation value of the MPO in the given ground state, aiming to maximize it toward unity for valid WLOs.
• Perform iterative numerical optimization (e.g., gradient-based) of the MPO tensors to minimize the cost function and identify WLOs that preserve the ground state subspace.
• Use the optimized WLOs as building blocks to construct operators that create, move, and annihilate anyons via further optimization.
• Compute braiding statistics and topological twists by measuring expectation values of composite anyon operators and extracting modular S and T matrices.
• Validate the method on perturbed toric code and doubled semion models with magnetic field perturbations up to half the critical value.

Experimental results

Research questions

• RQ1 Can Wilson loop operators be systematically extracted from a single ground state wave function without knowledge of the Hamiltonian?
• RQ2 How can one identify emergent 1-form symmetries from a ground state wave function using variational optimization?
• RQ3 Can the full topological order—specifically the modular S and T matrices—be reconstructed from a single wave function using only numerical optimization?
• RQ4 What is the role of matrix product operator (MPO) parametrization in enabling the discovery of anyonic excitations and their statistics?
• RQ5 To what extent can this method be applied to real-world quantum simulators with unknown or perturbed Hamiltonians?

Key findings

• The optimization procedure successfully identifies Wilson loop operators whose expectation values approach unity, indicating they preserve the ground state subspace.
• For the perturbed toric code with $ h_x = 0.15 $, $ h_z = 0.05 $, the method converges in approximately 400 iterations, with the WLO expectation value stabilizing near 1.
• The extracted WLOs allow construction of anyon creation and braiding operators, enabling the computation of topological S and T matrices.
• The method correctly recovers the fractional statistics of anyons, including anyon braiding phases and topological twists.
• The approach is robust under perturbations up to half the critical magnetic field strength in both the toric code and doubled semion models.
• The framework can be extended to extract F and R symbols and potentially full G-crossed braided tensor categories for symmetry-enriched topological phases.