scholarly journals Contracting Arbitrary Tensor Networks: General Approximate Algorithm and Applications in Graphical Models and Quantum Circuit Simulations

2020 ◽  
Vol 125 (6) ◽  
Author(s):  
Feng Pan ◽  
Pengfei Zhou ◽  
Sujie Li ◽  
Pan Zhang
Keyword(s):  
Graphical Models ◽  
Quantum Circuit ◽  
Approximate Algorithm ◽  
Tensor Networks ◽  
Circuit Simulations ◽  
Arbitrary Tensor

scholarly journals Hyper-optimized tensor network contraction

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 410
Author(s):  
Johnnie Gray ◽  
Stefanos Kourtis
Keyword(s):  
Computation Time ◽  
Quantum Circuit ◽  
Optimization Approach ◽  
Many Body ◽  
Tensor Networks ◽  
Randomized Protocols ◽  
Classical Simulation ◽  
Tensor Network ◽  
Speed Up ◽  
Many Body Systems

Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to tensor networks with irregular geometries. Finding the best possible contraction path for such networks is a central problem, with an exponential effect on computation time and memory footprint. In this work, we implement new randomized protocols that find very high quality contraction paths for arbitrary and large tensor networks. We test our methods on a variety of benchmarks, including the random quantum circuit instances recently implemented on Google quantum chips. We find that the paths obtained can be very close to optimal, and often many orders or magnitude better than the most established approaches. As different underlying geometries suit different methods, we also introduce a hyper-optimization approach, where both the method applied and its algorithmic parameters are tuned during the path finding. The increase in quality of contraction schemes found has significant practical implications for the simulation of quantum many-body systems and particularly for the benchmarking of new quantum chips. Concretely, we estimate a speed-up of over 10,000× compared to the original expectation for the classical simulation of the Sycamore `supremacy' circuits.

scholarly journals On the geometry of tensor network states

2012 ◽  
Vol 12 (3&4) ◽  
pp. 346-354
Author(s):  
Joseph M. Landsburg ◽  
Yang Qi ◽  
Ke Ye
Keyword(s):  
Graphical Models ◽  
Matrix Multiplication ◽  
Algebraic Statistics ◽  
Geometric Complexity ◽  
Network State ◽  
Geometric Complexity Theory ◽  
Tensor Networks ◽  
Tensor Network ◽  
Network States ◽  
Tensor Network States

We answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops. Grasedyck's question has a surprising connection to the area of Geometric Complexity Theory, in that the result is equivalent to the statement that the boundary of the Mulmuley-Sohoni type variety associated to matrix multiplication is strictly larger than the projections of matrix multiplication (and re-expressions of matrix multiplication and its projections after changes of bases). Tensor Network States are also related to graphical models in algebraic statistics.

scholarly journals Duality of graphical models and tensor networks

2018 ◽  
Vol 8 (2) ◽  
pp. 273-288 ◽  
Author(s):  
Elina Robeva ◽  
Anna Seigal
Keyword(s):  
Graphical Models ◽  
Graphical Model ◽  
Belief Propagation ◽  
Discrete Variables ◽  
Network Algorithms ◽  
Research Areas ◽  
Tensor Networks ◽  
Tensor Network

Abstract In this article we show the duality between tensor networks and undirected graphical models with discrete variables. We study tensor networks on hypergraphs, which we call tensor hypernetworks. We show that the tensor hypernetwork on a hypergraph exactly corresponds to the graphical model given by the dual hypergraph. We translate various notions under duality. For example, marginalization in a graphical model is dual to contraction in the tensor network. Algorithms also translate under duality. We show that belief propagation corresponds to a known algorithm for tensor network contraction. This article is a reminder that the research areas of graphical models and tensor networks can benefit from interaction.

scholarly journals Parallel quantum simulation of large systems on small NISQ computers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
F. Barratt ◽  
James Dborin ◽  
Matthias Bal ◽  
Vid Stojevic ◽  
Frank Pollmann ◽  
...  
Keyword(s):  
Finite Depth ◽  
Quantum Circuit ◽  
Quantum Simulation ◽  
Quantum Spin ◽  
Matrix Product ◽  
One Dimensional ◽  
Matrix Product State ◽  
Tensor Networks ◽  
Large Systems ◽  
Invariant Quantum

AbstractTensor networks permit computational and entanglement resources to be concentrated in interesting regions of Hilbert space. Implemented on NISQ machines they allow simulation of quantum systems that are much larger than the computational machine itself. This is achieved by parallelising the quantum simulation. Here, we demonstrate this in the simplest case; an infinite, translationally invariant quantum spin chain. We provide Cirq and Qiskit code that translates infinite, translationally invariant matrix product state (iMPS) algorithms to finite-depth quantum circuit machines, allowing the representation, optimisation and evolution of arbitrary one-dimensional systems. The illustrative simulated output of these codes for achievable circuit sizes is given.

scholarly journals From Probabilistic Graphical Models to Generalized Tensor Networks for Supervised Learning

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 68169-68182 ◽  
Author(s):  
Ivan Glasser ◽  
Nicola Pancotti ◽  
J. Ignacio Cirac
Keyword(s):  
Supervised Learning ◽  
Graphical Models ◽  
Probabilistic Graphical Models ◽  
Tensor Networks

Bereavement or Breakup: A Network Approach to Modeling Differential Pathways into Depression

2019 ◽  
Author(s):  
Julian Burger ◽  
Margaret S. Stroebe ◽  
Pasqualina Perrig-Chiello ◽  
Henk A.W. Schut ◽  
Stefanie Spahni ◽  
...  
Keyword(s):  
Depressive Symptoms ◽  
Graphical Models ◽  
Contextual Factors ◽  
Marital Disruption ◽  
Depression Symptoms ◽  
Depression Symptom ◽  
Multiple Time ◽  
Cross Sectional ◽  
Network Analyses ◽  
Within Subjects

Background: Prior network analyses demonstrated that the death of a loved one potentially precedes specific depression symptoms, primarily loneliness, which in turn links to other depressive symptoms. In this study, we extend prior research by comparing depression symptom network structures following two types of marital disruption: bereavement versus separation. Methods: We fitted two Gaussian Graphical Models to cross-sectional data from a Swiss survey of older persons (145 bereaved, 217 separated, and 362 married controls), and compared symptom levels across bereaved and separated individuals. Results: Separated compared to widowed individuals were more likely to perceive an unfriendly environment and oneself as a failure. Both types of marital disruption were linked primarily to loneliness, from where different relations emerged to other depressive symptoms. Amongst others, loneliness had a stronger connection to perceiving oneself as a failure in separated compared to widowed individuals. Conversely, loneliness had a stronger connection to getting going in widowed individuals. Limitations: Analyses are based on cross-sectional between-subjects data, and conclusions regarding dynamic processes on the within-subjects level remain putative. Further, some of the estimated parameters in the network exhibited overlapping confidence intervals and their order needs to be interpreted with care. Replications should thus aim for studies with multiple time points and larger samples. Conclusions: The findings of this study add to a growing body of literature indicating that depressive symptom patterns depend on contextual factors. If replicated on the within-subjects level, such findings have implications for setting up patient-tailored treatment approaches in dependence of contextual factors.

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