Algebraic Geometry of Matrix Product States

Abstract

We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions. Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS. We make four conjectures on the identifiability of MPS parameters.

References in corpus (5)

• Matrix product states represent ground states faithfully
• Entropy and Entanglement in Quantum Ground States
• Scalable reconstruction of density matrices
• Wick's theorem for matrix product states
• The Invariant Ring Of m Matrices Under The Adjoint Action By a Product Of General Linear Groups

Cited by in corpus (12)

• Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1
• Tensor Networks in a Nutshell
• Tensor Analysis and Fusion of Multimodal Brain Images
• Tropical Tensor Network for Ground States of Spin Glasses
• Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems
• Tensor Networks contraction and the Belief Propagation algorithm
• Lectures on Quantum Tensor Networks
• Matrix Product States for Quantum Stochastic Modelling
• The Invariant Ring Of m Matrices Under The Adjoint Action By a Product Of General Linear Groups
• The Linear Span of Uniform Matrix Product States
• Quantum Tensor Networks, Stochastic Processes, and Weighted Automata
• Belief propagation in monoidal categories