Efficient Machine Learning Representations of Surface Code with Boundaries, Defects, Domain Walls and Twists

Abstract

Machine learning representations of many-body quantum states have recently been introduced as an ansatz to describe the ground states and unitary evolutions of many-body quantum systems. We explore one of the most important representations, restricted Boltzmann machine (RBM) representation, in stabilizer formalism. We give the general method of constructing RBM representation for stabilizer code states and find the exact RBM representation for several types of stabilizer groups with the number of hidden neurons equal or less than the number of visible neurons, which indicates that the representation is extremely efficient. Then we analyze the surface code with boundaries, defects, domain walls and twists in full detail and find that all the models can be efficiently represented via RBM ansatz states. Besides, the case for Kitaev's $D(\Zb_d)$ model, which is a generalized model of surface code, is also investigated.

References in corpus (32)

• Fault-tolerant quantum computation by anyons
• Quantum Machine Learning
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Quantum support vector machine for big data classification
• Machine learning phases of matter
• A Theory of Fault-Tolerant Quantum Computation
• Many-body quantum state tomography with neural networks
• Stabilizer Codes and Quantum Error Correction
• Quantum-enhanced machine learning
• Models for gapped boundaries and domain walls
• Quantum Entanglement in Neural Network States
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Quantum codes on a lattice with boundary
• Efficient Representation of Quantum Many-body States with Deep Neural Networks
• The computational complexity of PEPS
• Machine Learning Topological States
• Quantum Loop Topography for Machine Learning
• Equivalence of restricted Boltzmann machines and tensor network states
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Approximating quantum many-body wave-functions using artificial neural networks
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• A Neural Decoder for Topological Codes
• Solving the Bose-Hubbard model with machine learning
• Neural network representation of tensor network and chiral states
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• Using Recurrent Neural Networks to Optimize Dynamical Decoupling for Quantum Memory
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Inductive supervised quantum learning
• How local is the information in MPS/PEPS tensor networks?
• An Efficient Algorithmic Way to Construct Boltzmann Machine Representations for Arbitrary Stabilizer Code

Cited by in corpus (22)

• Quantum Neural Network States: A Brief Review of Methods and Applications
• Advantages of versatile neural-network decoding for topological codes
• Efficient Representation of Topologically Ordered States with Restricted Boltzmann Machines
• Quantum Codes from Neural Networks
• Expressive power of complex-valued restricted Boltzmann machines for solving non-stoquastic Hamiltonians
• Approximating power of machine-learning ansatz for quantum many-body states
• Restricted Boltzmann Machines and Matrix Product States of 1D Translational Invariant Stabilizer Codes
• Determination of the Semion Code Threshold using Neural Decoders
• Transforming Generalized Ising Model into Boltzmann Machine
• Entanglement Area Law for Shallow and Deep Quantum Neural Network States
• Compact Neural-network Quantum State representations of Jastrow and Stabilizer states
• On maximum-likelihood decoding with circuit-level errors
• Neural-network Quantum States for Spin-1 systems: spin-basis and parameterization effects on compactness of representations
• Restricted Boltzmann machine representation for the groundstate and excited states of Kitaev Honeycomb model
• Non-stabilizerness of Neural Quantum States
• Specialising Neural-network Quantum States for the Bose Hubbard Model
• Finding Quantum Critical Points with Neural-Network Quantum States
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• Neural Quantum State Study of Fracton Models
• Time-dependent variational principle for open quantum systems with artificial neural networks
• Representing arbitrary ground states of toric code by a restricted Boltzmann machine
• Variational Transformer Ansatz for the Density Operator of Steady States in Dissipative Quantum Many-Body Systems