Tree tensor network approach to simulating Shor's algorithm
Abstract
Constructively simulating quantum systems furthers our understanding of qualitative and quantitative features which may be analytically intractable. In this paper, we directly simulate and explore the entanglement structure present in the paradigmatic example for exponential quantum speedups: Shor's algorithm. To perform our simulation, we construct a dynamic tree tensor network which manifestly captures two salient circuit features for modular exponentiation. These are the natural tworegister bipartition and the invariance of entanglement with respect to permutations of the topregister qubits. Our construction help identify the entanglement entropy properties, which we summarize by a scaling relation. Further, the tree network is efficiently projected onto a matrix product state from which we efficiently execute the quantum Fourier transform. In conclusion, future simulation of quantum information states with tensor networks exploiting circuit symmetries is discussed.
 Authors:

 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Univ. of Tennessee, Knoxville, TN (United States)
 Publication Date:
 Research Org.:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1414718
 DOE Contract Number:
 AC0500OR22725
 Resource Type:
 Journal Article
 Journal Name:
 Physical Review A
 Additional Journal Information:
 Journal Volume: 96; Journal Issue: 6; Journal ID: ISSN 24699926
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
Citation Formats
Dumitrescu, Eugene F. Tree tensor network approach to simulating Shor's algorithm. United States: N. p., 2017.
Web. doi:10.1103/PhysRevA.96.062322.
Dumitrescu, Eugene F. Tree tensor network approach to simulating Shor's algorithm. United States. doi:10.1103/PhysRevA.96.062322.
Dumitrescu, Eugene F. Wed .
"Tree tensor network approach to simulating Shor's algorithm". United States. doi:10.1103/PhysRevA.96.062322.
@article{osti_1414718,
title = {Tree tensor network approach to simulating Shor's algorithm},
author = {Dumitrescu, Eugene F.},
abstractNote = {Constructively simulating quantum systems furthers our understanding of qualitative and quantitative features which may be analytically intractable. In this paper, we directly simulate and explore the entanglement structure present in the paradigmatic example for exponential quantum speedups: Shor's algorithm. To perform our simulation, we construct a dynamic tree tensor network which manifestly captures two salient circuit features for modular exponentiation. These are the natural tworegister bipartition and the invariance of entanglement with respect to permutations of the topregister qubits. Our construction help identify the entanglement entropy properties, which we summarize by a scaling relation. Further, the tree network is efficiently projected onto a matrix product state from which we efficiently execute the quantum Fourier transform. In conclusion, future simulation of quantum information states with tensor networks exploiting circuit symmetries is discussed.},
doi = {10.1103/PhysRevA.96.062322},
journal = {Physical Review A},
issn = {24699926},
number = 6,
volume = 96,
place = {United States},
year = {2017},
month = {12}
}
Works referenced in this record:
The densitymatrix renormalization group in the age of matrix product states
journal, January 2011
 Schollwöck, Ulrich
 Annals of Physics, Vol. 326, Issue 1
Quantum Entanglement in Neural Network States
journal, May 2017
 Deng, DongLing; Li, Xiaopeng; Das Sarma, S.
 Physical Review X, Vol. 7, Issue 2
A practical introduction to tensor networks: Matrix product states and projected entangled pair states
journal, October 2014
 Orús, Román
 Annals of Physics, Vol. 349
Classical simulation of quantum manybody systems with a tree tensor network
journal, August 2006
 Shi, Y. Y.; Duan, L. M.; Vidal, G.
 Physical Review A, Vol. 74, Issue 2
Solving search problems by strongly simulating quantum circuits
journal, February 2013
 Johnson, T. H.; Biamonte, J. D.; Clark, S. R.
 Scientific Reports, Vol. 3, Issue 1
Efficient classical simulation of the approximate quantum Fourier transform
journal, October 2007
 Yoran, Nadav; Short, Anthony J.
 Physical Review A, Vol. 76, Issue 4
Fast quantum modular exponentiation
journal, May 2005
 Van Meter, Rodney; Itoh, Kohei M.
 Physical Review A, Vol. 71, Issue 5
Density matrix formulation for quantum renormalization groups
journal, November 1992
 White, Steven R.
 Physical Review Letters, Vol. 69, Issue 19
TensorNetwork Simulations of the Surface Code under Realistic Noise
journal, July 2017
 Darmawan, Andrew S.; Poulin, David
 Physical Review Letters, Vol. 119, Issue 4
PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
journal, October 1997
 Shor, Peter W.
 SIAM Journal on Computing, Vol. 26, Issue 5
Simulating Quantum Computation by Contracting Tensor Networks
journal, January 2008
 Markov, Igor L.; Shi, Yaoyun
 SIAM Journal on Computing, Vol. 38, Issue 3
Matrix product states for critical spin chains: Finitesize versus finiteentanglement scaling
journal, August 2012
 Pirvu, B.; Vidal, G.; Verstraete, F.
 Physical Review B, Vol. 86, Issue 7
Hardwareefficient variational quantum eigensolver for small molecules and quantum magnets
journal, September 2017
 Kandala, Abhinav; Mezzacapo, Antonio; Temme, Kristan
 Nature, Vol. 549, Issue 7671
Advances on tensor network theory: symmetries, fermions, entanglement, and holography
journal, November 2014
 Orús, Román
 The European Physical Journal B, Vol. 87, Issue 11
Universality of entanglement and quantumcomputation complexity
journal, May 2004
 Orús, Román; Latorre, José I.
 Physical Review A, Vol. 69, Issue 5