Universality Classes of Stabilizer Code Hamiltonians

Weinstein, Zack; Ortiz, Gerardo; Nussinov, Zohar

doi:10.1103/PhysRevLett.123.230503

Title: Universality Classes of Stabilizer Code Hamiltonians

Abstract

Stabilizer code quantum Hamiltonians have been introduced with the intention of physically realizing a quantum memory because of their resilience to decoherence. In order to analyze their finite temperature thermodynamics, we show how to generically solve their partition function using duality techniques. By unveiling each model’s universality class and effective dimension, insights may be gained on their finite temperature dynamics and robustness. In this work, our technique is demonstrated in particular on the 4D toric code and Haah’s code; we find that the former falls into the 4D Ising universality class, whereas Haah’s code exhibits dimensional reduction and falls into the 1D Ising universality class.

Authors:

^[1];

^[2];

^[1]

Washington Univ., St. Louis, MO (United States). Dept. of Physics
Indiana Univ., Bloomington IN (United States). Dept. of Physics

Publication Date:: 2019-12-03

Research Org.:: Indiana Univ., Bloomington IN (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF)

OSTI Identifier:: 1630819

Grant/Contract Number:: SC0020343; NSF 1411229

Resource Type:: Accepted Manuscript

Journal Name:: Physical Review Letters

Additional Journal Information:: Journal Volume: 123; Journal Issue: 23; Journal ID: ISSN 0031-9007

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats


                    Weinstein, Zack, Ortiz, Gerardo, and Nussinov, Zohar. Universality Classes of Stabilizer Code Hamiltonians.  United States: N. p., 2019. 
Web.  doi:10.1103/PhysRevLett.123.230503.

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                    Weinstein, Zack, Ortiz, Gerardo, & Nussinov, Zohar. Universality Classes of Stabilizer Code Hamiltonians.  United States.  https://doi.org/10.1103/PhysRevLett.123.230503

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                    Weinstein, Zack, Ortiz, Gerardo, and Nussinov, Zohar. Tue .  
"Universality Classes of Stabilizer Code Hamiltonians".  United States.  https://doi.org/10.1103/PhysRevLett.123.230503.  https://www.osti.gov/servlets/purl/1630819.

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@article{osti_1630819,

  title        = {Universality Classes of Stabilizer Code Hamiltonians},

  author       = {Weinstein, Zack and Ortiz, Gerardo and Nussinov, Zohar},

  abstractNote = {Stabilizer code quantum Hamiltonians have been introduced with the intention of physically realizing a quantum memory because of their resilience to decoherence. In order to analyze their finite temperature thermodynamics, we show how to generically solve their partition function using duality techniques. By unveiling each model’s universality class and effective dimension, insights may be gained on their finite temperature dynamics and robustness. In this work, our technique is demonstrated in particular on the 4D toric code and Haah’s code; we find that the former falls into the 4D Ising universality class, whereas Haah’s code exhibits dimensional reduction and falls into the 1D Ising universality class.},

  doi          = {10.1103/PhysRevLett.123.230503},

  journal      = {Physical Review Letters},

  number       = 23,

  volume       = 123,

  place        = {United States},

  year         = {2019},

  month        = {12}

}

Copy to clipboard

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Works referenced in this record:

On thermalization in Kitaev's 2D model
journal, January 2009

Alicki, R.; Fannes, M.; Horodecki, M.
Journal of Physics A: Mathematical and Theoretical, Vol. 42, Issue 6
DOI: 10.1088/1751-8113/42/6/065303

On Thermal Stability of Topological Qubit in Kitaev's 4D Model
journal, March 2010

Alicki, R.; Horodecki, M.; Horodecki, P.
Open Systems & Information Dynamics, Vol. 17, Issue 01
DOI: 10.1142/S1230161210000023

Approximate quantum error correction for generalized amplitude-damping errors
journal, February 2014

Cafaro, Carlo; van Loock, Peter
Physical Review A, Vol. 89, Issue 2
DOI: 10.1103/PhysRevA.89.022316

Fracton topological order, generalized lattice gauge theory, and duality
journal, December 2016

Vijay, Sagar; Haah, Jeongwan; Fu, Liang
Physical Review B, Vol. 94, Issue 23
DOI: 10.1103/PhysRevB.94.235157

Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters
journal, October 1971

Wegner, Franz J.
Journal of Mathematical Physics, Vol. 12, Issue 10
DOI: 10.1063/1.1665530

Effective and exact holographies from symmetries and dualities
journal, October 2012

Nussinov, Zohar; Ortiz, Gerardo; Cobanera, Emilio
Annals of Physics, Vol. 327, Issue 10
DOI: 10.1016/j.aop.2012.07.001

Compass models: Theory and physical motivations
journal, January 2015

Nussinov, Zohar; van den Brink, Jeroen
Reviews of Modern Physics, Vol. 87, Issue 1
DOI: 10.1103/RevModPhys.87.1

Absence of finite temperature phase transitions in the X-Cube model and its $Z_{p}$ generalization
journal, January 2020

Weinstein, Zack; Cobanera, Emilio; Ortiz, Gerardo
Annals of Physics, Vol. 412
DOI: 10.1016/j.aop.2019.168018

Good quantum error-correcting codes exist
journal, August 1996

Calderbank, A. R.; Shor, Peter W.
Physical Review A, Vol. 54, Issue 2
DOI: 10.1103/PhysRevA.54.1098

Topological quantum memory
journal, September 2002

Dennis, Eric; Kitaev, Alexei; Landahl, Andrew
Journal of Mathematical Physics, Vol. 43, Issue 9
DOI: 10.1063/1.1499754

Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems
journal, February 2008

Nussinov, Zohar; Ortiz, Gerardo
Physical Review B, Vol. 77, Issue 6
DOI: 10.1103/PhysRevB.77.064302

Classification of subsystem symmetry-protected topological phases
journal, December 2018

Devakul, Trithep; Williamson, Dominic J.; You, Yizhi
Physical Review B, Vol. 98, Issue 23
DOI: 10.1103/PhysRevB.98.235121

Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection
journal, January 2005

Chamon, Claudio
Physical Review Letters, Vol. 94, Issue 4
DOI: 10.1103/PhysRevLett.94.040402

Glassy quantum dynamics in translation invariant fracton models
journal, April 2017

Prem, Abhinav; Haah, Jeongwan; Nandkishore, Rahul
Physical Review B, Vol. 95, Issue 15
DOI: 10.1103/PhysRevB.95.155133

Topological order in a three-dimensional toric code at finite temperature
journal, October 2008

Castelnovo, Claudio; Chamon, Claudio
Physical Review B, Vol. 78, Issue 15
DOI: 10.1103/PhysRevB.78.155120

Spurious Topological Entanglement Entropy from Subsystem Symmetries
journal, April 2019

Williamson, Dominic J.; Dua, Arpit; Cheng, Meng
Physical Review Letters, Vol. 122, Issue 14
DOI: 10.1103/PhysRevLett.122.140506

Operator quantum error-correcting subsystems for self-correcting quantum memories
journal, January 2006

Bacon, Dave
Physical Review A, Vol. 73, Issue 1
DOI: 10.1103/PhysRevA.73.012340

An Anyon Model in a Toric Honeycomb Lattice
journal, January 2011

Liang, Long; Yu, Yue
Communications in Theoretical Physics, Vol. 55, Issue 1
DOI: 10.1088/0253-6102/55/1/16

Fast decoders for qudit topological codes
journal, June 2014

Anwar, Hussain; Brown, Benjamin J.; Campbell, Earl T.
New Journal of Physics, Vol. 16, Issue 6
DOI: 10.1088/1367-2630/16/6/063038

Scheme for reducing decoherence in quantum computer memory
journal, October 1995

Shor, Peter W.
Physical Review A, Vol. 52, Issue 4
DOI: 10.1103/PhysRevA.52.R2493

Generalized global symmetries
journal, February 2015

Gaiotto, Davide; Kapustin, Anton; Seiberg, Nathan
Journal of High Energy Physics, Vol. 2015, Issue 2
DOI: 10.1007/JHEP02(2015)172

Topological Quantum Distillation
journal, October 2006

Bombin, H.; Martin-Delgado, M. A.
Physical Review Letters, Vol. 97, Issue 18
DOI: 10.1103/PhysRevLett.97.180501

Theory of fault-tolerant quantum computation
journal, January 1998

Gottesman, Daniel
Physical Review A, Vol. 57, Issue 1
DOI: 10.1103/PhysRevA.57.127

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
journal, April 2009

Bravyi, Sergey; Terhal, Barbara
New Journal of Physics, Vol. 11, Issue 4
DOI: 10.1088/1367-2630/11/4/043029

Local stabilizer codes in three dimensions without string logical operators
journal, April 2011

Haah, Jeongwan
Physical Review A, Vol. 83, Issue 4
DOI: 10.1103/PhysRevA.83.042330

Arbitrary dimensional Majorana dualities and architectures for topological matter
journal, August 2012

Nussinov, Zohar; Ortiz, Gerardo; Cobanera, Emilio
Physical Review B, Vol. 86, Issue 8
DOI: 10.1103/PhysRevB.86.085415

Fault-tolerant quantum computation by anyons
journal, January 2003

Kitaev, A. Yu.
Annals of Physics, Vol. 303, Issue 1
DOI: 10.1016/S0003-4916(02)00018-0

Fractons
journal, March 2019

Nandkishore, Rahul M.; Hermele, Michael
Annual Review of Condensed Matter Physics, Vol. 10, Issue 1
DOI: 10.1146/annurev-conmatphys-031218-013604

Generalized Elitzur’s theorem and dimensional reductions
journal, July 2005

Batista, C. D.; Nussinov, Zohar
Physical Review B, Vol. 72, Issue 4
DOI: 10.1103/PhysRevB.72.045137

Renormalization Group Decoder for a Four-Dimensional Toric Code
journal, April 2019

Duivenvoorden, K.; Breuckmann, N. P.; Terhal, B. M.
IEEE Transactions on Information Theory, Vol. 65, Issue 4
DOI: 10.1109/TIT.2018.2879937

Foliated fracton order from gauging subsystem symmetries
journal, January 2019

Shirley, Wilbur; Slagle, Kevin; Chen, Xie
SciPost Physics, Vol. 6, Issue 4
DOI: 10.21468/SciPostPhys.6.4.041

Quantum Self-Correction in the 3D Cubic Code Model
journal, November 2013

Bravyi, Sergey; Haah, Jeongwan
Physical Review Letters, Vol. 111, Issue 20
DOI: 10.1103/PhysRevLett.111.200501

Quantum memories at finite temperature
journal, November 2016

Brown, Benjamin J.; Loss, Daniel; Pachos, Jiannis K.
Reviews of Modern Physics, Vol. 88, Issue 4
DOI: 10.1103/RevModPhys.88.045005

Self-correction in Wegner's three-dimensional Ising lattice gauge theory
journal, March 2019

Poulin, David; Melko, Roger G.; Hastings, Matthew B.
Physical Review B, Vol. 99, Issue 9
DOI: 10.1103/PhysRevB.99.094103

Statistics of the Two-Dimensional Ferromagnet. Part I
journal, August 1941

Kramers, H. A.; Wannier, G. H.
Physical Review, Vol. 60, Issue 3
DOI: 10.1103/PhysRev.60.252

The bond-algebraic approach to dualities
journal, October 2011

Cobanera, Emilio; Ortiz, Gerardo; Nussinov, Zohar
Advances in Physics, Vol. 60, Issue 5
DOI: 10.1080/00018732.2011.619814

A symmetry principle for topological quantum order
journal, May 2009

Nussinov, Zohar; Ortiz, Gerardo
Annals of Physics, Vol. 324, Issue 5
DOI: 10.1016/j.aop.2008.11.002

Sufficient symmetry conditions for Topological Quantum Order
journal, September 2009

Nussinov, Z.; Ortiz, G.
Proceedings of the National Academy of Sciences, Vol. 106, Issue 40
DOI: 10.1073/pnas.0803726105

$Z_{2}$ lattice Gerbe theory
journal, November 2014

Johnston, Desmond A.
Physical Review D, Vol. 90, Issue 10
DOI: 10.1103/PhysRevD.90.107701

A statistical mechanics view on Kitaev's proposal for quantum memories
journal, May 2007

Alicki, R.; Fannes, M.; Horodecki, M.
Journal of Physics A: Mathematical and Theoretical, Vol. 40, Issue 24
DOI: 10.1088/1751-8113/40/24/012

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
text, January 2008

Bravyi, Sergey; Terhal, Barbara
arXiv
DOI: 10.48550/arxiv.0810.1983

On thermalization in Kitaev's 2D model
text, January 2008

Alicki, R.; Fannes, M.; Horodecki, M.
arXiv
DOI: 10.48550/arxiv.0810.4584

An anyon model in a toric honeycomb lattice
text, January 2010

Liang, Long; Yu, Yue
arXiv
DOI: 10.48550/arxiv.1006.0804

Effective and exact holographies from symmetries and dualities
text, January 2011

Nussinov, Zohar; Ortiz, Gerardo; Cobanera, Emilio
arXiv
DOI: 10.48550/arxiv.1110.2179

Renormalization group decoder for a four-dimensional toric code
text, January 2017

Duivenvoorden, Kasper; Breuckmann, Nikolas P.; Terhal, Barbara M.
arXiv
DOI: 10.48550/arxiv.1708.09286

Foliated fracton order from gauging subsystem symmetries
text, January 2018

Shirley, Wilbur; Slagle, Kevin; Chen, Xie
arXiv
DOI: 10.48550/arxiv.1806.08679

Spurious topological entanglement entropy from subsystem symmetries
text, January 2018

Williamson, Dominic J.; Dua, Arpit; Cheng, Meng
arXiv
DOI: 10.48550/arxiv.1808.05221

Generalized Elitzur's Theorem and Dimensional Reduction
text, January 2004

Batista, Cristian D.; Nussinov, Zohar
arXiv
DOI: 10.48550/arxiv.cond-mat/0410599

A symmetry principle for Topological Quantum Order
text, January 2007

Nussinov, Zohar; Ortiz, Gerardo
arXiv
DOI: 10.48550/arxiv.cond-mat/0702377

A statistical mechanics view on Kitaev's proposal for quantum memories
text, January 2007

Alicki, R.; Fannes, M.; Horodecki, M.
arXiv
DOI: 10.48550/arxiv.quant-ph/0702102

Fault-tolerant quantum computation by anyons
text, January 1997

Kitaev, A. Yu.
arXiv
DOI: 10.48550/arxiv.quant-ph/9707021

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Title: Universality Classes of Stabilizer Code Hamiltonians

Abstract

Citation Formats

On thermalization in Kitaev's 2D model journal, January 2009

On Thermal Stability of Topological Qubit in Kitaev's 4D Model journal, March 2010

Approximate quantum error correction for generalized amplitude-damping errors journal, February 2014

Fracton topological order, generalized lattice gauge theory, and duality journal, December 2016

Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters journal, October 1971

Effective and exact holographies from symmetries and dualities journal, October 2012

Compass models: Theory and physical motivations journal, January 2015

Absence of finite temperature phase transitions in the X-Cube model and its Z p generalization journal, January 2020

Good quantum error-correcting codes exist journal, August 1996

Topological quantum memory journal, September 2002

Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems journal, February 2008

Classification of subsystem symmetry-protected topological phases journal, December 2018

Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection journal, January 2005

Glassy quantum dynamics in translation invariant fracton models journal, April 2017

Topological order in a three-dimensional toric code at finite temperature journal, October 2008

Spurious Topological Entanglement Entropy from Subsystem Symmetries journal, April 2019

Operator quantum error-correcting subsystems for self-correcting quantum memories journal, January 2006

An Anyon Model in a Toric Honeycomb Lattice journal, January 2011

Fast decoders for qudit topological codes journal, June 2014

Scheme for reducing decoherence in quantum computer memory journal, October 1995

Generalized global symmetries journal, February 2015

Topological Quantum Distillation journal, October 2006

Theory of fault-tolerant quantum computation journal, January 1998

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes journal, April 2009

Local stabilizer codes in three dimensions without string logical operators journal, April 2011

Arbitrary dimensional Majorana dualities and architectures for topological matter journal, August 2012

Fault-tolerant quantum computation by anyons journal, January 2003

Fractons journal, March 2019

Generalized Elitzur’s theorem and dimensional reductions journal, July 2005

Renormalization Group Decoder for a Four-Dimensional Toric Code journal, April 2019

Foliated fracton order from gauging subsystem symmetries journal, January 2019

Quantum Self-Correction in the 3D Cubic Code Model journal, November 2013

Quantum memories at finite temperature journal, November 2016

Self-correction in Wegner's three-dimensional Ising lattice gauge theory journal, March 2019

Statistics of the Two-Dimensional Ferromagnet. Part I journal, August 1941

The bond-algebraic approach to dualities journal, October 2011

A symmetry principle for topological quantum order journal, May 2009

Sufficient symmetry conditions for Topological Quantum Order journal, September 2009

Z 2 lattice Gerbe theory journal, November 2014

A statistical mechanics view on Kitaev's proposal for quantum memories journal, May 2007

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes text, January 2008

On thermalization in Kitaev's 2D model text, January 2008

An anyon model in a toric honeycomb lattice text, January 2010

Effective and exact holographies from symmetries and dualities text, January 2011

Renormalization group decoder for a four-dimensional toric code text, January 2017

Foliated fracton order from gauging subsystem symmetries text, January 2018

Spurious topological entanglement entropy from subsystem symmetries text, January 2018

Generalized Elitzur's Theorem and Dimensional Reduction text, January 2004

A symmetry principle for Topological Quantum Order text, January 2007

A statistical mechanics view on Kitaev's proposal for quantum memories text, January 2007

Fault-tolerant quantum computation by anyons text, January 1997

On thermalization in Kitaev's 2D model
journal, January 2009

On Thermal Stability of Topological Qubit in Kitaev's 4D Model
journal, March 2010

Approximate quantum error correction for generalized amplitude-damping errors
journal, February 2014

Fracton topological order, generalized lattice gauge theory, and duality
journal, December 2016

Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters
journal, October 1971

Effective and exact holographies from symmetries and dualities
journal, October 2012

Compass models: Theory and physical motivations
journal, January 2015

Absence of finite temperature phase transitions in the X-Cube model and its $Z_{p}$ generalization
journal, January 2020

Good quantum error-correcting codes exist
journal, August 1996

Topological quantum memory
journal, September 2002

Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems
journal, February 2008

Classification of subsystem symmetry-protected topological phases
journal, December 2018

Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection
journal, January 2005

Glassy quantum dynamics in translation invariant fracton models
journal, April 2017

Topological order in a three-dimensional toric code at finite temperature
journal, October 2008

Spurious Topological Entanglement Entropy from Subsystem Symmetries
journal, April 2019

Operator quantum error-correcting subsystems for self-correcting quantum memories
journal, January 2006

An Anyon Model in a Toric Honeycomb Lattice
journal, January 2011

Fast decoders for qudit topological codes
journal, June 2014

Scheme for reducing decoherence in quantum computer memory
journal, October 1995

Generalized global symmetries
journal, February 2015

Topological Quantum Distillation
journal, October 2006

Theory of fault-tolerant quantum computation
journal, January 1998

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
journal, April 2009

Local stabilizer codes in three dimensions without string logical operators
journal, April 2011

Arbitrary dimensional Majorana dualities and architectures for topological matter
journal, August 2012

Fault-tolerant quantum computation by anyons
journal, January 2003

Fractons
journal, March 2019

Generalized Elitzur’s theorem and dimensional reductions
journal, July 2005

Renormalization Group Decoder for a Four-Dimensional Toric Code
journal, April 2019

Foliated fracton order from gauging subsystem symmetries
journal, January 2019

Quantum Self-Correction in the 3D Cubic Code Model
journal, November 2013

Quantum memories at finite temperature
journal, November 2016

Self-correction in Wegner's three-dimensional Ising lattice gauge theory
journal, March 2019

Statistics of the Two-Dimensional Ferromagnet. Part I
journal, August 1941

The bond-algebraic approach to dualities
journal, October 2011

A symmetry principle for topological quantum order
journal, May 2009

Sufficient symmetry conditions for Topological Quantum Order
journal, September 2009

$Z_{2}$ lattice Gerbe theory
journal, November 2014

A statistical mechanics view on Kitaev's proposal for quantum memories
journal, May 2007

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
text, January 2008

On thermalization in Kitaev's 2D model
text, January 2008

An anyon model in a toric honeycomb lattice
text, January 2010

Effective and exact holographies from symmetries and dualities
text, January 2011

Renormalization group decoder for a four-dimensional toric code
text, January 2017

Foliated fracton order from gauging subsystem symmetries
text, January 2018

Spurious topological entanglement entropy from subsystem symmetries
text, January 2018

Generalized Elitzur's Theorem and Dimensional Reduction
text, January 2004

A symmetry principle for Topological Quantum Order
text, January 2007

A statistical mechanics view on Kitaev's proposal for quantum memories
text, January 2007

Fault-tolerant quantum computation by anyons
text, January 1997