scholarly journals Variational methods for characterizing matrix product operator symmetries

2021 ◽  
Vol 104 (19) ◽  
Author(s):  
Anna Francuz ◽  
Laurens Lootens ◽  
Frank Verstraete ◽  
Jacek Dziarmaga
Keyword(s):  
Variational Methods ◽  
Matrix Product ◽  
Product Operator

scholarly journals Anyons and matrix product operator algebras

2017 ◽  
Vol 378 ◽  
pp. 183-233 ◽  
Author(s):  
N. Bultinck ◽  
M. Mariën ◽  
D.J. Williamson ◽  
M.B. Şahinoğlu ◽  
J. Haegeman ◽  
...  
Keyword(s):  
Operator Algebras ◽  
Matrix Product ◽  
Product Operator

Scaling and efficient classical simulation of the quantum Fourier transform

2017 ◽  
Vol 17 (1&2) ◽  
pp. 1-14
Author(s):  
Kieran J. Woolfe ◽  
Charles D. Hill ◽  
Lloyd C. L. Hollenberg
Keyword(s):  
Fourier Transform ◽  
Matrix Product ◽  
Product State ◽  
Quantum Fourier Transform ◽  
Numerical Evidence ◽  
Matrix Product State ◽  
Product Operator ◽  
Classical Simulation ◽  
Schmidt Rank

We provide numerical evidence that the quantum Fourier transform can be efficiently represented in a matrix product operator with a size growing relatively slowly with the number of qubits. Additionally, we numerically show that the tensors in the operator converge to a common tensor as the number of qubits in the transform increases. Together these results imply that the application of the quantum Fourier transform to a matrix product state with n qubits of maximum Schmidt rank χ can be simulated in O(n (log(n))2 χ 2 ) time. We perform such simulations and quantify the error involved in representing the transform as a matrix product operator and simulating the quantum Fourier transform of periodic states.

scholarly journals Matrix product operator representations

2010 ◽  
Vol 12 (2) ◽  
pp. 025012 ◽  
Author(s):  
B Pirvu ◽  
V Murg ◽  
J I Cirac ◽  
F Verstraete
Keyword(s):  
Matrix Product ◽  
Product Operator ◽  
Operator Representations

scholarly journals An efficient matrix product operator representation of the quantum chemical Hamiltonian

2015 ◽  
Vol 143 (24) ◽  
pp. 244118 ◽  
Author(s):  
Sebastian Keller ◽  
Michele Dolfi ◽  
Matthias Troyer ◽  
Markus Reiher
Keyword(s):  
Quantum Chemical ◽  
Matrix Product ◽  
Operator Representation ◽  
Product Operator

scholarly journals Tangent-space methods for truncating uniform MPS

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Bram Vanhecke ◽  
Maarten Van Damme ◽  
Jutho Haegeman ◽  
Laurens Vanderstraeten ◽  
Frank Verstraete
Keyword(s):  
Tangent Space ◽  
Computational Cost ◽  
Infinite Matrix ◽  
Matrix Product ◽  
Product State ◽  
Matrix Product State ◽  
Product Operator ◽  
Tensor Network ◽  
Network Simulations ◽  
Central Bottleneck

An essential primitive in quantum tensor network simulations is the problem of approximating a matrix product state with one of a smaller bond dimension. This problem forms the central bottleneck in algorithms for time evolution and for contracting projected entangled pair states. We formulate a tangent-space based variational algorithm to achieve this goal for uniform (infinite) matrix product states. The algorithm exhibits a favourable scaling of the computational cost, and we demonstrate its usefulness by several examples involving the multiplication of a matrix product state with a matrix product operator.

scholarly journals Matrix product operator symmetries and intertwiners in string-nets with domain walls

2021 ◽  
Vol 10 (3) ◽  
Author(s):  
Laurens Lootens ◽  
Jürgen Fuchs ◽  
Jutho Haegeman ◽  
Christoph Schweigert ◽  
Frank Verstraete
Keyword(s):  
Domain Walls ◽  
Error Correcting Codes ◽  
Matrix Product ◽  
Topological Phases ◽  
Coupled Equations ◽  
Product Operator ◽  
Tensor Network ◽  
Topological Field ◽  
Non Local ◽  
Quantum Error

We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.

scholarly journals Automated construction of $U(1)$-invariant matrix-product operators from graph representations

2017 ◽  
Vol 3 (5) ◽  
Author(s):  
Sebastian Paeckel ◽  
Thomas Köhler ◽  
Salvatore R. Manmana
Keyword(s):  
Building Blocks ◽  
Matrix Product ◽  
Heisenberg Chain ◽  
Graph Representations ◽  
Quantum Numbers ◽  
Construction Scheme ◽  
Product Operator ◽  
Local Quantum ◽  
Local Operators ◽  
Algorithmic Construction

We present an algorithmic construction scheme for matrix-product-operator (MPO) representations of arbitrary U(1)U(1)-invariant operators whenever there is an expression of the local structure in terms of a finite-states machine (FSM). Given a set of local operators as building blocks, the method automatizes two major steps when constructing a U(1)U(1)-invariant MPO representation: (i) the bookkeeping of auxiliary bond-index shifts arising from the application of operators changing the local quantum numbers and (ii) the appearance of phase factors due to particular commutation rules. The automatization is achieved by post-processing the operator strings generated by the FSM. Consequently, MPO representations of various types of U(1)U(1)-invariant operators can be constructed generically in MPS algorithms reducing the necessity of expensive MPO arithmetics. This is demonstrated by generating arbitrary products of operators in terms of FSM, from which we obtain exact MPO representations for the variance of the Hamiltonian of a S=1S=1 Heisenberg chain.

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