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Title: 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 two-register bipartition and the invariance of entanglement with respect to permutations of the top-register 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:
ORCiD logo [1]
  1. 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:  
AC05-00OR22725
Resource Type:
Journal Article
Journal Name:
Physical Review A
Additional Journal Information:
Journal Volume: 96; Journal Issue: 6; Journal ID: ISSN 2469-9926
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 two-register bipartition and the invariance of entanglement with respect to permutations of the top-register 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 = {2469-9926},
number = 6,
volume = 96,
place = {United States},
year = {2017},
month = {12}
}

Works referenced in this record:

The density-matrix renormalization group in the age of matrix product states
journal, January 2011


Quantum Entanglement in Neural Network States
journal, May 2017


Classical simulation of quantum many-body systems with a tree tensor network
journal, August 2006


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
  • DOI: 10.1038/srep01235

Efficient classical simulation of the approximate quantum Fourier transform
journal, October 2007


Fast quantum modular exponentiation
journal, May 2005


Density matrix formulation for quantum renormalization groups
journal, November 1992


Tensor-Network Simulations of the Surface Code under Realistic Noise
journal, July 2017


Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
journal, October 1997


Simulating Quantum Computation by Contracting Tensor Networks
journal, January 2008

  • Markov, Igor L.; Shi, Yaoyun
  • SIAM Journal on Computing, Vol. 38, Issue 3
  • DOI: 10.1137/050644756

Matrix product states for critical spin chains: Finite-size versus finite-entanglement scaling
journal, August 2012


Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets
journal, September 2017

  • Kandala, Abhinav; Mezzacapo, Antonio; Temme, Kristan
  • Nature, Vol. 549, Issue 7671
  • DOI: 10.1038/nature23879

Advances on tensor network theory: symmetries, fermions, entanglement, and holography
journal, November 2014


Universality of entanglement and quantum-computation complexity
journal, May 2004