Fractalizing quantum codes

Trithep Devakul1,2 and Dominic J. Williamson3

1Department of Physics, Princeton University, Princeton, NJ 08540, USA
2Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We introduce "fractalization", a procedure by which spin models are extended to higher-dimensional "fractal" spin models. This allows us to interpret type-II fracton phases, fractal symmetry-protected topological phases, and more, in terms of well understood lower-dimensional spin models. Fractalization is also useful for deriving new spin models and quantum codes from known ones. We construct higher dimensional generalizations of fracton models that host extended fractal excitations. Finally, by applying fractalization to a 2D subsystem code, we produce a family of locally generated 3D subsystem codes that are conjectured to saturate a quantum information storage tradeoff bound.

► BibTeX data

► References

[1] Jeongwan Haah. An Invariant of Topologically Ordered States Under Local Unitary Transformations. Commun. Math. Phys., 342 (3): 771–801, 2016a. ISSN 14320916. 10.1007/​s00220-016-2594-y.

[2] Sagar Vijay, Jeongwan Haah, and Liang Fu. Fracton topological order, generalized lattice gauge theory, and duality. Phys. Rev. B, 94: 235157, Dec 2016. 10.1103/​PhysRevB.94.235157. URL https:/​/​​doi/​10.1103/​PhysRevB.94.235157.

[3] Rahul M. Nandkishore and Michael Hermele. Fractons. Annual Review of Condensed Matter Physics, 10 (1): 295–313, 2019. 10.1146/​annurev-conmatphys-031218-013604. URL https:/​/​​10.1146/​annurev-conmatphys-031218-013604.

[4] Michael Pretko, Xie Chen, and Yizhi You. Fracton phases of matter. International Journal of Modern Physics A, 35 (06): 2030003, 2020. 10.1142/​S0217751X20300033. URL https:/​/​​10.1142/​S0217751X20300033.

[5] Claudio Castelnovo and Claudio Chamon. Topological quantum glassiness. Philosophical Magazine, 92 (1-3): 304–323, 2012. 10.1080/​14786435.2011.609152. URL https:/​/​​10.1080/​14786435.2011.609152.

[6] Sergey Bravyi, Bernhard Leemhuis, and Barbara M. Terhal. Topological order in an exactly solvable 3d spin model. Annals of Physics, 326 (4): 839 – 866, 2011. ISSN 0003-4916. https:/​/​​10.1016/​j.aop.2010.11.002. URL http:/​/​​science/​article/​pii/​S0003491610001910.

[7] Jeongwan Haah. Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A, 83: 042330, Apr 2011. 10.1103/​PhysRevA.83.042330. URL https:/​/​​doi/​10.1103/​PhysRevA.83.042330.

[8] Isaac H. Kim. 3d local qupit quantum code without string logical operator. 2012. URL https:/​/​​abs/​1202.0052.

[9] Beni Yoshida. Exotic topological order in fractal spin liquids. Phys. Rev. B, 88: 125122, Sep 2013. 10.1103/​PhysRevB.88.125122. URL https:/​/​​doi/​10.1103/​PhysRevB.88.125122.

[10] Sagar Vijay, Jeongwan Haah, and Liang Fu. A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations. Phys. Rev. B, 92: 235136, Dec 2015. 10.1103/​PhysRevB.92.235136. URL https:/​/​​doi/​10.1103/​PhysRevB.92.235136.

[11] Dominic J. Williamson. Fractal symmetries: Ungauging the cubic code. Phys. Rev. B, 94 (15): 155128, oct 2016. ISSN 24699969. 10.1103/​PhysRevB.94.155128. URL https:/​/​​doi/​10.1103/​PhysRevB.94.155128.

[12] Arpit Dua, Isaac H. Kim, Meng Cheng, and Dominic J. Williamson. Sorting topological stabilizer models in three dimensions. Phys. Rev. B, 100: 155137, Oct 2019a. 10.1103/​PhysRevB.100.155137. URL https:/​/​​doi/​10.1103/​PhysRevB.100.155137.

[13] Sergey Bravyi and Jeongwan Haah. Energy landscape of 3D spin hamiltonians with topological order. Phys. Rev. Lett., 107 (15): 150504, oct 2011. ISSN 00319007. 10.1103/​PhysRevLett.107.150504. URL https:/​/​​doi/​10.1103/​PhysRevLett.107.150504.

[14] Sergey Bravyi and Jeongwan Haah. Quantum self-correction in the 3D cubic code model. Phys. Rev. Lett., 111 (20): 200501, nov 2013. ISSN 00319007. 10.1103/​PhysRevLett.111.200501. URL https:/​/​​doi/​10.1103/​PhysRevLett.111.200501.

[15] Benjamin J. Brown, Daniel Loss, Jiannis K. Pachos, Chris N. Self, and James R. Wootton. Quantum memories at finite temperature. Rev. Mod. Phys., 88 (4): 45005, nov 2016. ISSN 15390756. 10.1103/​RevModPhys.88.045005. URL https:/​/​​doi/​10.1103/​RevModPhys.88.045005.

[16] Abhinav Prem, Jeongwan Haah, and Rahul Nandkishore. Glassy quantum dynamics in translation invariant fracton models. Phys. Rev. B, 95 (15): 155133, apr 2017. ISSN 24699969. 10.1103/​PhysRevB.95.155133. URL https:/​/​​doi/​10.1103/​PhysRevB.95.155133.

[17] Shriya Pai, Michael Pretko, and Rahul M. Nandkishore. Localization in fractonic random circuits. Phys. Rev. X, 9: 021003, Apr 2019. 10.1103/​PhysRevX.9.021003. URL https:/​/​​doi/​10.1103/​PhysRevX.9.021003.

[18] Andrey Gromov, Andrew Lucas, and Rahul M. Nandkishore. Fracton hydrodynamics. Phys. Rev. Research, 2: 033124, Jul 2020. 10.1103/​PhysRevResearch.2.033124. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.033124.

[19] Shriya Pai and Michael Pretko. Dynamical scar states in driven fracton systems. Phys. Rev. Lett., 123: 136401, Sep 2019. 10.1103/​PhysRevLett.123.136401. URL https:/​/​​doi/​10.1103/​PhysRevLett.123.136401.

[20] Huan He, Yizhi You, and Abhinav Prem. Lieb-schultz-mattis–type constraints on fractonic matter. Phys. Rev. B, 101: 165145, Apr 2020. 10.1103/​PhysRevB.101.165145. URL https:/​/​​doi/​10.1103/​PhysRevB.101.165145.

[21] Han Yan. Hyperbolic fracton model, subsystem symmetry, and holography. Phys. Rev. B, 99: 155126, Apr 2019. 10.1103/​PhysRevB.99.155126. URL https:/​/​​doi/​10.1103/​PhysRevB.99.155126.

[22] Michael Pretko and Leo Radzihovsky. Fracton-Elasticity Duality. Phys. Rev. Lett., 120 (19): 195301, may 2018. ISSN 10797114. 10.1103/​PhysRevLett.120.195301. URL https:/​/​​doi/​10.1103/​PhysRevLett.120.195301.

[23] Andrey Gromov. Chiral Topological Elasticity and Fracton Order. Phys. Rev. Lett., 122 (7), dec 2019a. ISSN 10797114. 10.1103/​PhysRevLett.122.076403. URL http:/​/​​abs/​1712.06600.

[24] Michael Pretko, Zhengzheng Zhai, and Leo Radzihovsky. Crystal-to-fracton tensor gauge theory dualities. Phys. Rev. B, 100: 134113, Oct 2019. 10.1103/​PhysRevB.100.134113. URL https:/​/​​doi/​10.1103/​PhysRevB.100.134113.

[25] Andrey Gromov and Piotr Surówka. On duality between Cosserat elasticity and fractons. SciPost Phys., 8 (4), aug 2019. 10.21468/​SciPostPhys.8.4.065. URL http:/​/​​10.21468/​SciPostPhys.8.4.065.

[26] Gábor B. Halász, Timothy H. Hsieh, and Leon Balents. Fracton Topological Phases from Strongly Coupled Spin Chains. Phys. Rev. Lett., 119 (25): 257202, dec 2017. ISSN 10797114. 10.1103/​PhysRevLett.119.257202. URL https:/​/​​doi/​10.1103/​PhysRevLett.119.257202.

[27] Abhinav Prem, Sagar Vijay, Yang-Zhi Chou, Michael Pretko, and Rahul M. Nandkishore. Pinch point singularities of tensor spin liquids. Phys. Rev. B, 98: 165140, Oct 2018. 10.1103/​PhysRevB.98.165140. URL https:/​/​​doi/​10.1103/​PhysRevB.98.165140.

[28] Darshil Doshi and Andrey Gromov. Vortices and Fractons. may 2020. URL http:/​/​​abs/​2005.03015.

[29] Michael Pretko. Emergent gravity of fractons: Mach's principle revisited. Phys. Rev. D, 96 (2): 24051, jul 2017. ISSN 24700029. 10.1103/​PhysRevD.96.024051. URL https:/​/​​doi/​10.1103/​PhysRevD.96.024051.

[30] Benjamin J. Brown and Dominic J. Williamson. Parallelized quantum error correction with fracton topological codes. Phys. Rev. Research, 2: 013303, Mar 2020. 10.1103/​PhysRevResearch.2.013303. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.013303.

[31] Sagar Vijay and Liang Fu. A Generalization of Non-Abelian Anyons in Three Dimensions. 2017. URL http:/​/​​abs/​1706.07070.

[32] Dominic J. Williamson and Meng Cheng. Designer non-Abelian fractons from topological layers. apr 2020. URL http:/​/​​abs/​2004.07251.

[33] Daniel Bulmash and Maissam Barkeshli. Gauging fractons: Immobile non-abelian quasiparticles, fractals, and position-dependent degeneracies. Phys. Rev. B, 100: 155146, Oct 2019. 10.1103/​PhysRevB.100.155146. URL https:/​/​​doi/​10.1103/​PhysRevB.100.155146.

[34] Abhinav Prem and Dominic Williamson. Gauging permutation symmetries as a route to non-Abelian fractons. SciPost Phys., 7 (5): 068, nov 2019. ISSN 2542-4653. 10.21468/​scipostphys.7.5.068. URL https:/​/​​10.21468/​SciPostPhys.7.5.068.

[35] David T. Stephen, José Garre-Rubio, Arpit Dua, and Dominic J. Williamson. Subsystem symmetry enriched topological order in three dimensions. Phys. Rev. Research, 2: 033331, Aug 2020. 10.1103/​PhysRevResearch.2.033331. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.033331.

[36] Shriya Pai and Michael Hermele. Fracton fusion and statistics. Phys. Rev. B, 100: 195136, Nov 2019. 10.1103/​PhysRevB.100.195136. URL https:/​/​​doi/​10.1103/​PhysRevB.100.195136.

[37] Jeongwan Haah. Bifurcation in entanglement renormalization group flow of a gapped spin model. Phys. Rev. B, 89: 075119, Feb 2014. 10.1103/​PhysRevB.89.075119. URL https:/​/​​doi/​10.1103/​PhysRevB.89.075119.

[38] Wilbur Shirley, Kevin Slagle, Zhenghan Wang, and Xie Chen. Fracton models on general three-dimensional manifolds. Phys. Rev. X, 8: 031051, Aug 2018. 10.1103/​PhysRevX.8.031051. URL https:/​/​​doi/​10.1103/​PhysRevX.8.031051.

[39] Arpit Dua, Pratyush Sarkar, Dominic J. Williamson, and Meng Cheng. Bifurcating entanglement-renormalization group flows of fracton stabilizer models. Phys. Rev. Research, 2: 033021, Jul 2020. 10.1103/​PhysRevResearch.2.033021. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.033021.

[40] Kevin Slagle, Abhinav Prem, and Michael Pretko. Symmetric tensor gauge theories on curved spaces. Annals of Physics, 410: 167910, 2019a. ISSN 0003-4916. https:/​/​​10.1016/​j.aop.2019.167910. URL http:/​/​​science/​article/​pii/​S0003491619301654.

[41] Kevin Slagle, David Aasen, and Dominic Williamson. Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order. SciPost Phys., 6: 43, 2019b. 10.21468/​SciPostPhys.6.4.043. URL https:/​/​​10.21468/​SciPostPhys.6.4.043.

[42] Xiao-Gang Wen. Systematic construction of gapped nonliquid states. Phys. Rev. Research, 2: 033300, Aug 2020. 10.1103/​PhysRevResearch.2.033300. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.033300.

[43] David Aasen, Daniel Bulmash, Abhinav Prem, Kevin Slagle, and Dominic J. Williamson. Topological defect networks for fractons of all types. Phys. Rev. Research, 2: 043165, Oct 2020. 10.1103/​PhysRevResearch.2.043165. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.043165.

[44] Juven Wang. Non-liquid cellular states. 2020. URL https:/​/​​abs/​2002.12932v2.

[45] Daniel Bulmash and Maissam Barkeshli. Generalized $U(1)$ Gauge Field Theories and Fractal Dynamics. arXiv, (1): 1–7, 2018. URL http:/​/​​abs/​1806.01855.

[46] Andrey Gromov. Towards Classification of Fracton Phases: The Multipole Algebra. Phys. Rev. X, 9 (3), dec 2019b. ISSN 21603308. 10.1103/​PhysRevX.9.031035. URL https:/​/​​pdf/​1812.05104.pdf.

[47] Michael Pretko. The fracton gauge principle. Phys. Rev. B, 98: 115134, Sep 2018. 10.1103/​PhysRevB.98.115134. URL https:/​/​​doi/​10.1103/​PhysRevB.98.115134.

[48] Dominic J. Williamson, Zhen Bi, and Meng Cheng. Fractonic matter in symmetry-enriched $u(1)$ gauge theory. Phys. Rev. B, 100: 125150, Sep 2019a. 10.1103/​PhysRevB.100.125150. URL https:/​/​​doi/​10.1103/​PhysRevB.100.125150.

[49] Nathan Seiberg. Field Theories With a Vector Global Symmetry. SciPost Phys., 8: 50, 2020. 10.21468/​SciPostPhys.8.4.050. URL https:/​/​​10.21468/​SciPostPhys.8.4.050.

[50] Nathanan Tantivasadakarn and Sagar Vijay. Searching for fracton orders via symmetry defect condensation. Phys. Rev. B, 101: 165143, Apr 2020. 10.1103/​PhysRevB.101.165143. URL https:/​/​​doi/​10.1103/​PhysRevB.101.165143.

[51] Wilbur Shirley. Fractonic order and emergent fermionic gauge theory. feb 2020. URL http:/​/​​abs/​2002.12026.

[52] Nathanan Tantivasadakarn. Jordan-wigner dualities for translation-invariant hamiltonians in any dimension: Emergent fermions in fracton topological order. Phys. Rev. Research, 2: 023353, Jun 2020. 10.1103/​PhysRevResearch.2.023353. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.023353.

[53] Trithep Devakul, Wilbur Shirley, and Juven Wang. Strong planar subsystem symmetry-protected topological phases and their dual fracton orders. Phys. Rev. Research, 2: 012059, Mar 2020. 10.1103/​PhysRevResearch.2.012059. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.012059.

[54] Trithep Devakul, S. A. Parameswaran, and S. L. Sondhi. Correlation function diagnostics for type-i fracton phases. Phys. Rev. B, 97: 041110(R), Jan 2018. 10.1103/​PhysRevB.97.041110. URL https:/​/​​doi/​10.1103/​PhysRevB.97.041110.

[55] Trithep Devakul, Yizhi You, F. J. Burnell, and S. L. Sondhi. Fractal Symmetric Phases of Matter. SciPost Phys., 6: 7, 2019. 10.21468/​SciPostPhys.6.1.007. URL https:/​/​​10.21468/​SciPostPhys.6.1.007.

[56] D. R. Chowdhury, S. Basu, I. S. Gupta, and P. P. Chaudhuri. Design of caecc - cellular automata based error correcting code. IEEE Transactions on Computers, 43 (6): 759–764, 1994. 10.1109/​12.286310.

[57] Beni Yoshida. Information storage capacity of discrete spin systems. Ann. Phys. (N. Y)., 338: 134–166, nov 2011. 10.1016/​j.aop.2013.07.009. URL http:/​/​​abs/​1111.3275 http:/​/​​10.1016/​j.aop.2013.07.009.

[58] G. M. Nixon and B. J. Brown. Correcting spanning errors with a fractal code. IEEE Transactions on Information Theory, pages 1–1, 2021. 10.1109/​TIT.2021.3068359.

[59] M. E. J. Newman and Cristopher Moore. Glassy dynamics and aging in an exactly solvable spin model. Phys. Rev. E, 60: 5068–5072, Nov 1999. 10.1103/​PhysRevE.60.5068. URL https:/​/​​doi/​10.1103/​PhysRevE.60.5068.

[60] Aleksander Kubica and Beni Yoshida. Ungauging quantum error-correcting codes. 2018. URL http:/​/​​abs/​1805.01836.

[61] Trithep Devakul. Classifying local fractal subsystem symmetry-protected topological phases. Phys. Rev. B, 99: 235131, Jun 2019. 10.1103/​PhysRevB.99.235131. URL https:/​/​​doi/​10.1103/​PhysRevB.99.235131.

[62] Hans J. Briegel and Robert Raussendorf. Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett., 86 (5): 910–913, jan 2001. ISSN 00319007. 10.1103/​PhysRevLett.86.910. URL https:/​/​​doi/​10.1103/​PhysRevLett.86.910.

[63] Robert Raussendorf, Cihan Okay, Dong-Sheng Wang, David T. Stephen, and Hendrik Poulsen Nautrup. Computationally universal phase of quantum matter. Phys. Rev. Lett., 122: 090501, Mar 2019. 10.1103/​PhysRevLett.122.090501. URL https:/​/​​doi/​10.1103/​PhysRevLett.122.090501.

[64] Trithep Devakul and Dominic J. Williamson. Universal quantum computation using fractal symmetry-protected cluster phases. Phys. Rev. A, 98 (2): 022332, aug 2018. ISSN 24699934. 10.1103/​PhysRevA.98.022332. URL https:/​/​​doi/​10.1103/​PhysRevA.98.022332 http:/​/​​abs/​1806.04663.

[65] David T. Stephen, Hendrik Poulsen Nautrup, Juani Bermejo-Vega, Jens Eisert, and Robert Raussendorf. Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter. Quantum, 3: 142, May 2019. ISSN 2521-327X. 10.22331/​q-2019-05-20-142. URL https:/​/​​10.22331/​q-2019-05-20-142.

[66] Austin K. Daniel, Rafael N. Alexander, and Akimasa Miyake. Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices. Quantum, 4: 228, February 2020. ISSN 2521-327X. 10.22331/​q-2020-02-10-228. URL https:/​/​​10.22331/​q-2020-02-10-228.

[67] A. Yu Kitaev. Fault-tolerant quantum computation by anyons. Ann. Phys. (N. Y)., 303 (1): 2–30, jan 2003. ISSN 00034916. 10.1016/​S0003-4916(02)00018-0. URL http:/​/​​science/​article/​pii/​S0003491602000180.

[68] Dave Bacon. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A, 73: 012340, Jan 2006. 10.1103/​PhysRevA.73.012340. URL https:/​/​​doi/​10.1103/​PhysRevA.73.012340.

[69] David Poulin. Stabilizer formalism for operator quantum error correction. Phys. Rev. Lett., 95: 230504, Dec 2005. 10.1103/​PhysRevLett.95.230504. URL https:/​/​​doi/​10.1103/​PhysRevLett.95.230504.

[70] Sergey Bravyi. Subsystem codes with spatially local generators. Phys. Rev. A, 83: 012320, Jan 2011. 10.1103/​PhysRevA.83.012320. URL https:/​/​​doi/​10.1103/​PhysRevA.83.012320.

[71] Steven T. Flammia, Jeongwan Haah, Michael J. Kastoryano, and Isaac H. Kim. Limits on the storage of quantum information in a volume of space. Quantum, 1: 4, oct 2016. 10.22331/​q-2017-04-25-4. URL http:/​/​​10.22331/​q-2017-04-25-4.

[72] F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. North-Holland Pub. Co. ; sole distributors for the U.S.A. and Canada, Elsevier/​North-Holland Amsterdam ; New York : New York, 1977. ISBN 0444850090 0444850104.

[73] Jeongwan Haah. Commuting Pauli Hamiltonians as Maps between Free Modules. Commun. Math. Phys., 324 (2): 351–399, 2013. ISSN 00103616. 10.1007/​s00220-013-1810-2. URL http:/​/​​10.1007/​s00220-013-1810-2.

[74] Sergey Bravyi, David Poulin, and Barbara Terhal. Tradeoffs for reliable quantum information storage in 2d systems. Phys. Rev. Lett., 104: 050503, Feb 2010. 10.1103/​PhysRevLett.104.050503. URL https:/​/​​doi/​10.1103/​PhysRevLett.104.050503.

[75] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Complete classification of one-dimensional gapped quantum phases in interacting spin systems. Phys. Rev. B, 84: 235128, Dec 2011. 10.1103/​PhysRevB.84.235128. URL https:/​/​​doi/​10.1103/​PhysRevB.84.235128.

[76] Wilbur Shirley, Kevin Slagle, and Xie Chen. Foliated fracton order from gauging subsystem symmetries. SciPost Phys., 6: 41, 2019. 10.21468/​SciPostPhys.6.4.041. URL https:/​/​​10.21468/​SciPostPhys.6.4.041.

[77] Arpit Dua, Dominic J. Williamson, Jeongwan Haah, and Meng Cheng. Compactifying fracton stabilizer models. Phys. Rev. B, 99: 245135, Jun 2019b. 10.1103/​PhysRevB.99.245135. URL https:/​/​​doi/​10.1103/​PhysRevB.99.245135.

[78] Dominic J. Williamson, Arpit Dua, and Meng Cheng. Spurious topological entanglement entropy from subsystem symmetries. Phys. Rev. Lett., 122: 140506, Apr 2019b. 10.1103/​PhysRevLett.122.140506. URL https:/​/​​doi/​10.1103/​PhysRevLett.122.140506.

[79] Jeongwan Haah. Algebraic Methods for Quantum Codes on Lattices. Rev. Colomb. Matemáticas, 50 (2): 299–349, 2016b. ISSN 2357-4100. 10.15446/​recolma.v50n2.62214. URL http:/​/​​abs/​1607.01387.

[80] Jeongwan Haah. Classification of translation invariant topological pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices. Journal of Mathematical Physics, 62 (1): 012201, 2021. 10.1063/​5.0021068. URL https:/​/​​10.1063/​5.0021068.

[81] Héctor Bombín. Structure of 2D Topological Stabilizer Codes. Commun. Math. Phys., 327 (2): 387–432, 2014. ISSN 14320916. 10.1007/​s00220-014-1893-4. URL http:/​/​​10.1007/​s00220-014-1893-4.

[82] Dominic J. Williamson and Trithep Devakul. Type-II fractons from coupled spin chains and layers. 2020. URL https:/​/​​abs/​2007.07894.

[83] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. J. Math. Phys., 43 (9): 4452–4505, oct 2001. 10.1063/​1.1499754. URL http:/​/​​abs/​quant-ph/​0110143.

[84] Jeongwan Haah. Two generalizations of the cubic code model. Talk at KITP Conference: Frontiers of Quantum Information Physics, 2017. URL https:/​/​​online/​qinfo_c17/​haah/​.

[85] Julien Dorier, Federico Becca, and Frédéric Mila. Quantum compass model on the square lattice. Phys. Rev. B, 72: 024448, Jul 2005. 10.1103/​PhysRevB.72.024448. URL https:/​/​​doi/​10.1103/​PhysRevB.72.024448.

[86] Dave Bacon, Steven T. Flammia, Aram W. Harrow, and Jonathan Shi. Sparse quantum codes from quantum circuits. In Proc. Annu. ACM Symp. Theory Comput., volume 14-17-June-2015, pages 327–334, New York, New York, USA, jun 2015. Association for Computing Machinery. ISBN 9781450335362. 10.1145/​2746539.2746608. URL http:/​/​​citation.cfm?doid=2746539.2746608.

[87] Kitaev Alexei. Anyons in an exactly solved model and beyond. Ann. Phys. (N. Y)., 321 (1): 2–111, 2006. ISSN 0003-4916. 10.1016/​j.aop.2005.10.005. URL http:/​/​​science/​article/​pii/​S0003491605002381.

Cited by

[1] Joseph Sullivan, Thomas Iadecola, and Dominic J. Williamson, "Planar p-string condensation: Chiral fracton phases from fractional quantum Hall layers and beyond", Physical Review B 103 20, 205301 (2021).

[2] Nathanan Tantivasadakarn, Wenjie Ji, and Sagar Vijay, "Hybrid fracton phases: Parent orders for liquid and nonliquid quantum phases", Physical Review B 103 24, 245136 (2021).

[3] Nikolas P. Breuckmann and Jens Niklas Eberhardt, "Quantum Low-Density Parity-Check Codes", PRX Quantum 2 4, 040101 (2021).

[4] Zheng Zhou, Xue-Feng Zhang, Frank Pollmann, and Yizhi You, "Fractal Quantum Phase Transitions: Critical Phenomena Beyond Renormalization", arXiv:2105.05851.

[5] Jonathan Francisco San Miguel, Arpit Dua, and Dominic J. Williamson, "Bifurcating subsystem symmetric entanglement renormalization in two dimensions", Physical Review B 103 3, 035148 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-22 06:47:39) and SAO/NASA ADS (last updated successfully 2021-10-22 06:47:40). The list may be incomplete as not all publishers provide suitable and complete citation data.