Lectures on Quantum Tensor Networks

Abstract

Situated as a language between computer science, quantum physics and mathematics, tensor network theory has steadily grown in popularity and can now be found in applications ranging across the entire field of quantum information processing. This book aims to present the best contemporary practices in the use of tensor networks as a reasoning tool, placing quantum states, operators and processes on the same compositional footing. The book has 7 parts and over 40 subsections which took shape in over a decade of teaching. In addition to covering the foundations, the book covers important applications such as matrix product states, open quantum systems and entanglement $-$ all cast into the diagrammatic tensor network language. The intended audience includes those in quantum information science wishing to learn about tensor networks. It includes scientists who have employed tensor networks in their modeling codes who have interest in the tools graphical reasoning capacity. The audience further includes the graduate student researcher, whom with some effort, should find this book accessible. I would appreciate it if you emailed me about any mistakes or typos you find.

178 page book in LaTeX. Please email me about any mistakes or typos. Current version always available on overleaf, see http://www.overleaf.com/read/jkccbhcdqwnh

References in corpus (60)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Elementary gates for quantum computation
• Distributed Entanglement
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement renormalization
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• A simple formula for the average gate fidelity of a quantum dynamical operation
• Classical simulation of quantum many-body systems with a tree tensor network
• Scalable Noise Estimation with Random Unitary Operators
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations
• Ancilla-assisted quantum process tomography
• High fidelity spin entanglement using optimal control
• Tomography of Quantum Operations
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Novel schemes for measurement-based quantum computation
• Multi-particle entanglement and its applications to cryptography
• Measurement-based quantum computation beyond the one-way model
• Quantum Process Tomography of the Quantum Fourier Transform
• Computing Local Invariants of Qubit Systems
• Lecture Notes of Tensor Network Contractions
• Closed timelike curves via post-selection: theory and experimental demonstration
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Strings, Projected Entangled Pair States, and variational Monte Carlo methods
• A Simple Proof that Toffoli and Hadamard are Quantum Universal
• The maximally entangled symmetric state in terms of the geometric measure
• Physical accessibility of non-completely positive maps
• Tensor Networks in a Nutshell
• Approximating strongly correlated spin and fermion wavefunctions with correlator product states
• Non-perturbative k-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins
• Optimum Quantum Error Recovery using Semidefinite Programming
• Machine learning non-Markovian quantum dynamics
• Imprinting a complete information about a quantum channel on its output state
• Density Matrix Renormalization Group in the Heisenberg Picture
• Phase groups and the origin of non-locality for qubits
• Renyi Entropy and Free Energy
• TensorNetwork: A Library for Physics and Machine Learning
• Gate fidelity fluctuations and quantum process invariants
• Quantum Criticality and the Kondo Lattice
• Quantum boolean functions
• Entanglement and tensor network states
• Dynamical simulations of classical stochastic systems using matrix product states
• Tensor Network Contractions for #SAT
• A Prehistory of n-Categorical Physics
• A 2 rebit gate universal for quantum computing
• Planar Para Algebras, Reflection Positivity
• Atemporal diagrams for quantum circuits
• Categorical Quantum Circuits
• Algebraically contractible topological tensor network states
• Bipartite entanglement-annihilating maps: necessary and sufficient conditions
• Quons: A 3D Language for Quantum Information
• Algebraic Geometry of Matrix Product States
• Quantum Gate Fidelity in Terms of Choi Matrices
• Solving search problems by strongly simulating quantum circuits
• Undecidability in Tensor Network States
• Geometric local invariants and pure three-qubit states
• Virtual Parallel Computing and a Search Algorithm using Matrix Product States
• A Mathematical Picture Language Program
• Charged String Tensor Networks
• Pushing Tensor Networks to the Limit

Cited by in corpus (14)

• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Complex Quantum Networks: a Topical Review
• Mutual Information Scaling for Tensor Network Machine Learning
• Quantum Annealing Algorithms for Boolean Tensor Networks
• Qibolab: an open-source hybrid quantum operating system
• Construction of genuinely multipartite entangled subspaces and the associated bounds on entanglement measures for mixed states
• Heralded Optical Entanglement Generation via the Graph Picture of Linear Quantum Networks
• Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties
• Language Modeling with Reduced Densities
• State Diagrams to determine Tree Tensor Network Operators
• Interplay between local and non-local frustration in the 1D ANNNI chain I -- The even case
• Approximate Equivalence Checking of Noisy Quantum Circuits
• At the Interface of Algebra and Statistics
• State of practice: evaluating GPU performance of state vector and tensor network methods