Symmetry-protected sign problem and magic in quantum phases of matter

Tyler D. Ellison1,2, Kohtaro Kato3,4, Zi-Wen Liu2, and Timothy H. Hsieh2

1University of Washington, Seattle, WA 98195, USA
2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
4Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Osaka 560-8531, Japan

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We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic to study the complexity of symmetry-protected topological (SPT) phases of matter. In particular, we say a state has a symmetry-protected sign problem or symmetry-protected magic, if finite-depth quantum circuits composed of symmetric gates are unable to transform the state into a non-negative real wave function or stabilizer state, respectively. We prove that states belonging to certain SPT phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, we find that one-dimensional $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional $\mathbb{Z}_2$ SPT states (e.g. Levin-Gu state) have symmetry-protected magic. Furthermore, we comment on the relation between a symmetry-protected sign problem and the computational wire property of one-dimensional SPT states. In an appendix, we also introduce explicit decorated domain wall models of SPT phases, which may be of independent interest.

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[1] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The resource theory of stabilizer quantum computation. New Journal of Physics, 16 (1): 013009, jan 2014. 10.1088/​1367-2630/​16/​1/​013009. URL https:/​/​​10.1088.

[2] Daniel Gottesman. The Heisenberg Representation of Quantum Computers. Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, eds. S. P. Corney, R. Delbourgo, and P. D. Jarvis, pp. 32-43 (Cambridge, MA, International Press, 1999). URL https:/​/​​abs/​quant-ph/​9807006.

[3] Tarun Grover and Matthew P. A. Fisher. Entanglement and the sign structure of quantum states. Phys. Rev. A, 92: 042308, Oct 2015. 10.1103/​PhysRevA.92.042308. URL https:/​/​​doi/​10.1103/​PhysRevA.92.042308.

[4] M. B. Hastings. How quantum are non-negative wavefunctions? Journal of Mathematical Physics, 57 (1): 015210, 2016. 10.1063/​1.4936216. URL https:/​/​​10.1063/​1.4936216.

[5] Giacomo Torlai, Juan Carrasquilla, Matthew T. Fishman, Roger G. Melko, and Matthew P. A. Fisher. Wave-function positivization via automatic differentiation. Phys. Rev. Research, 2: 032060, Sep 2020. 10.1103/​PhysRevResearch.2.032060. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.032060.

[6] Matthias Troyer and Uwe-Jens Wiese. Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations. Phys. Rev. Lett., 94: 170201, May 2005. 10.1103/​PhysRevLett.94.170201. URL https:/​/​​doi/​10.1103/​PhysRevLett.94.170201.

[7] Milad Marvian, Daniel A. Lidar, and Itay Hen. On the computational complexity of curing non-stoquastic hamiltonians. Nature Communications, 10 (1): 1571, Apr 2019. ISSN 2041-1723. 10.1038/​s41467-019-09501-6. URL https:/​/​​10.1038/​s41467-019-09501-6.

[8] Joel Klassen, Milad Marvian, Stephen Piddock, Marios Ioannou, Itay Hen, and Barbara Terhal. Hardness and ease of curing the sign problem for two-local qubit hamiltonians. arXiv:1906.08800, 2020. URL https:/​/​​abs/​1906.08800.

[9] Joel Klassen and Barbara M. Terhal. Two-local qubit Hamiltonians: when are they stoquastic? Quantum, 3: 139, May 2019. ISSN 2521-327X. 10.22331/​q-2019-05-06-139. URL https:/​/​​10.22331/​q-2019-05-06-139.

[10] Ryan Levy and Bryan K. Clark. Mitigating the sign problem through basis rotations. Phys. Rev. Lett., 126: 216401, May 2021. 10.1103/​PhysRevLett.126.216401. URL https:/​/​​doi/​10.1103/​PhysRevLett.126.216401.

[11] Zhou-Quan Wan, Shi-Xin Zhang, and Hong Yao. Mitigating sign problem by automatic differentiation. arXiv:2010.01141, October 2020. URL https:/​/​​abs/​2010.01141.

[12] Dominik Hangleiter, Ingo Roth, Daniel Nagaj, and Jens Eisert. Easing the monte carlo sign problem. Science Advances, 6 (33), 2020. 10.1126/​sciadv.abb8341. URL https:/​/​​content/​6/​33/​eabb8341.

[13] Christopher David White, ChunJun Cao, and Brian Swingle. Conformal field theories are magical. Phys. Rev. B, 103: 075145, Feb 2021. 10.1103/​PhysRevB.103.075145. URL https:/​/​​doi/​10.1103/​PhysRevB.103.075145.

[14] S Sarkar, C Mukhopadhyay, and A Bayat. Characterization of an operational quantum resource in a critical many-body system. New Journal of Physics, 22 (8): 083077, aug 2020. 10.1088/​1367-2630/​aba919. URL https:/​/​​10.1088.

[15] Zi-Wen Liu and Andreas Winter. Many-body quantum magic. arXiv:2010.13817, October 2020. URL https:/​/​​abs/​2010.13817.

[16] Zohar Ringel and Dmitry L. Kovrizhin. Quantized gravitational responses, the sign problem, and quantum complexity. Science Advances, 3 (9), 2017. 10.1126/​sciadv.1701758. URL https:/​/​​content/​3/​9/​e1701758.

[17] Omri Golan, Adam Smith, and Zohar Ringel. Intrinsic sign problem in fermionic and bosonic chiral topological matter. Phys. Rev. Research, 2: 043032, Oct 2020. 10.1103/​PhysRevResearch.2.043032. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.043032.

[18] Adam Smith, Omri Golan, and Zohar Ringel. Intrinsic sign problems in topological quantum field theories. Phys. Rev. Research, 2: 033515, Sep 2020. 10.1103/​PhysRevResearch.2.033515. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.033515.

[19] Maxime Dupont, Snir Gazit, and Thomas Scaffidi. Evidence for deconfined $u(1)$ gauge theory at the transition between toric code and double semion. Phys. Rev. B, 103: L140412, Apr 2021a. 10.1103/​PhysRevB.103.L140412. URL https:/​/​​doi/​10.1103/​PhysRevB.103.L140412.

[20] Maxime Dupont, Snir Gazit, and Thomas Scaffidi. From trivial to topological paramagnets: The case of ${\mathbb{z}}_{2}$ and ${\mathbb{z}}_{2}^{3}$ symmetries in two dimensions. Phys. Rev. B, 103: 144437, Apr 2021b. 10.1103/​PhysRevB.103.144437. URL https:/​/​​doi/​10.1103/​PhysRevB.103.144437.

[21] C. L. Kane and E. J. Mele. ${Z}_{2}$ topological order and the quantum spin hall effect. Phys. Rev. Lett., 95: 146802, Sep 2005a. 10.1103/​PhysRevLett.95.146802. URL https:/​/​​doi/​10.1103/​PhysRevLett.95.146802.

[22] C. L. Kane and E. J. Mele. Quantum spin hall effect in graphene. Phys. Rev. Lett., 95: 226801, Nov 2005b. 10.1103/​PhysRevLett.95.226801. URL https:/​/​​doi/​10.1103/​PhysRevLett.95.226801.

[23] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett., 86: 5188–5191, May 2001. 10.1103/​PhysRevLett.86.5188. URL https:/​/​​doi/​10.1103/​PhysRevLett.86.5188.

[24] Robert Raussendorf, Dong-Sheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, and David T. Stephen. Symmetry-protected topological phases with uniform computational power in one dimension. Phys. Rev. A, 96: 012302, Jul 2017. 10.1103/​PhysRevA.96.012302. URL https:/​/​​doi/​10.1103/​PhysRevA.96.012302.

[25] 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.

[26] 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.

[27] 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.

[28] Trithep Devakul and Dominic J. Williamson. Universal quantum computation using fractal symmetry-protected cluster phases. Phys. Rev. A, 98: 022332, Aug 2018. 10.1103/​PhysRevA.98.022332. URL https:/​/​​doi/​10.1103/​PhysRevA.98.022332.

[29] Akimasa Miyake. Quantum computation on the edge of a symmetry-protected topological order. Phys. Rev. Lett., 105: 040501, Jul 2010. 10.1103/​PhysRevLett.105.040501. URL https:/​/​​doi/​10.1103/​PhysRevLett.105.040501.

[30] Jacob Miller and Akimasa Miyake. Hierarchy of universal entanglement in 2D measurement-based quantum computation. npj Quantum Information, 2 (9): 16036, 2016. 10.1038/​npjqi.2016.36.

[31] Jacob Miller and Akimasa Miyake. Latent computational complexity of symmetry-protected topological order with fractional symmetry. Phys. Rev. Lett., 120: 170503, Apr 2018. 10.1103/​PhysRevLett.120.170503. URL https:/​/​​doi/​10.1103/​PhysRevLett.120.170503.

[32] Beni Yoshida. Gapped boundaries, group cohomology and fault-tolerant logical gates. Annals of Physics, 377: 387 – 413, 2017. ISSN 0003-4916. https:/​/​​10.1016/​j.aop.2016.12.014. URL http:/​/​​science/​article/​pii/​S0003491616302858.

[33] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B, 82 (15): 155138, October 2010. 10.1103/​PhysRevB.82.155138.

[34] Dominic V. Else and Chetan Nayak. Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge. Phys. Rev. B, 90: 235137, Dec 2014. 10.1103/​PhysRevB.90.235137. URL https:/​/​​doi/​10.1103/​PhysRevB.90.235137.

[35] Sergey Bravyi, Matthew B Hastings, and Spyridon Michalakis. Topological quantum order: stability under local perturbations. Journal of mathematical physics, 51 (9): 093512, 2010. URL https:/​/​​doi/​10.1063/​1.3490195. URL https:/​/​​10.1063/​1.3490195.

[36] Sergey Bravyi and Matthew B Hastings. A short proof of stability of topological order under local perturbations. Communications in mathematical physics, 307 (3): 609, 2011. URL https:/​/​​article/​10.1007/​s00220-011-1346-2. URL https:/​/​​10.1007/​s00220-011-1346-2.

[37] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. Symmetry protected topological orders and the group cohomology of their symmetry group. Phys. Rev. B, 87: 155114, Apr 2013. 10.1103/​PhysRevB.87.155114. URL https:/​/​​doi/​10.1103/​PhysRevB.87.155114.

[38] Davide Gaiotto and Theo Johnson-Freyd. Symmetry protected topological phases and generalized cohomology. Journal of High Energy Physics, 2019 (5): 7, 2019. URL https:/​/​​article/​10.1007/​JHEP05(2019)007. URL https:/​/​​10.1007/​JHEP05(2019)007.

[39] Anton Kapustin. Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology. arXiv:1403.1467, March 2014. URL https:/​/​​abs/​1403.1467.

[40] Kazuya Yonekura. On the cobordism classification of symmetry protected topological phases. Communications in Mathematical Physics, 368: 1121–1173, Jun 2019. 10.1007/​s00220-019-03439-y. URL https:/​/​​10.1007/​s00220-019-03439-y.

[41] Ling-Yan Hung and Xiao-Gang Wen. Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases. Phys. Rev. B, 87: 165107, Apr 2013. 10.1103/​PhysRevB.87.165107. URL https:/​/​​doi/​10.1103/​PhysRevB.87.165107.

[42] Xie Chen, Yuan-Ming Lu, and Ashvin Vishwanath. Symmetry-protected topological phases from decorated domain walls. Nature communications, 5 (1): 1–11, 2014. URL https:/​/​​10.1038/​ncomms4507.

[43] Xie Chen, Zheng-Xin Liu, and Xiao-Gang Wen. Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations. Phys. Rev. B, 84: 235141, Dec 2011a. 10.1103/​PhysRevB.84.235141. URL https:/​/​​doi/​10.1103/​PhysRevB.84.235141.

[44] Dominic J. Williamson, Nick Bultinck, Michael Mariën, Mehmet B. Şahinoğlu, Jutho Haegeman, and Frank Verstraete. Matrix product operators for symmetry-protected topological phases: Gauging and edge theories. Phys. Rev. B, 94: 205150, Nov 2016. 10.1103/​PhysRevB.94.205150. URL https:/​/​​doi/​10.1103/​PhysRevB.94.205150.

[45] R. Shankar and Ashvin Vishwanath. Equality of bulk wave functions and edge correlations in some topological superconductors: A spacetime derivation. Phys. Rev. Lett., 107: 106803, Sep 2011. 10.1103/​PhysRevLett.107.106803. URL https:/​/​​doi/​10.1103/​PhysRevLett.107.106803.

[46] Yi-Zhuang You, Zhen Bi, Alex Rasmussen, Kevin Slagle, and Cenke Xu. Wave function and strange correlator of short-range entangled states. Phys. Rev. Lett., 112: 247202, Jun 2014. 10.1103/​PhysRevLett.112.247202. URL https:/​/​​doi/​10.1103/​PhysRevLett.112.247202.

[47] Robijn Vanhove, Matthias Bal, Dominic J. Williamson, Nick Bultinck, Jutho Haegeman, and Frank Verstraete. Mapping topological to conformal field theories through strange correlators. Phys. Rev. Lett., 121: 177203, Oct 2018. 10.1103/​PhysRevLett.121.177203. URL https:/​/​​doi/​10.1103/​PhysRevLett.121.177203.

[48] Nick Bultinck, Robijn Vanhove, Jutho Haegeman, and Frank Verstraete. Global anomaly detection in two-dimensional symmetry-protected topological phases. Phys. Rev. Lett., 120: 156601, Apr 2018. 10.1103/​PhysRevLett.120.156601. URL https:/​/​​doi/​10.1103/​PhysRevLett.120.156601.

[49] Frank Pollmann and Ari M. Turner. Detection of symmetry-protected topological phases in one dimension. Phys. Rev. B, 86: 125441, Sep 2012. 10.1103/​PhysRevB.86.125441. URL https:/​/​​doi/​10.1103/​PhysRevB.86.125441.

[50] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B, 83: 035107, Jan 2011b. 10.1103/​PhysRevB.83.035107. URL https:/​/​​doi/​10.1103/​PhysRevB.83.035107.

[51] Lukasz Fidkowski and Alexei Kitaev. Topological phases of fermions in one dimension. Phys. Rev. B, 83: 075103, Feb 2011. 10.1103/​PhysRevB.83.075103. URL https:/​/​​doi/​10.1103/​PhysRevB.83.075103.

[52] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. 10.1017/​CBO9780511976667.

[53] Daniel Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, January 1997.

[54] Héctor J. García, Igor L. Markov, and Andrew W. Cross. On the geometry of stabilizer states. Quantum Info. Comput., 14: 683–720, May 2014. ISSN 1533-7146. URL https:/​/​​doi/​10.5555/​2638682.2638691.

[55] Victor Veitch, Christopher Ferrie, David Gross, and Joseph Emerson. Negative quasi-probability as a resource for quantum computation. New Journal of Physics, 14 (11): 113011, nov 2012. 10.1088/​1367-2630/​14/​11/​113011. URL https:/​/​​10.1088.

[56] Vlad Gheorghiu. Standard form of qudit stabilizer groups. Physics Letters A, 378 (5): 505–509, 2014. ISSN 0375-9601. https:/​/​​10.1016/​j.physleta.2013.12.009. URL https:/​/​​science/​article/​pii/​S0375960113011080.

[57] Mark Howard and Earl Campbell. Application of a resource theory for magic states to fault-tolerant quantum computing. Phys. Rev. Lett., 118: 090501, Mar 2017. 10.1103/​PhysRevLett.118.090501. URL https:/​/​​doi/​10.1103/​PhysRevLett.118.090501.

[58] Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard. Simulation of quantum circuits by low-rank stabilizer decompositions. Quantum, 3: 181, September 2019. ISSN 2521-327X. 10.22331/​q-2019-09-02-181. URL https:/​/​​10.22331/​q-2019-09-02-181.

[59] Zi-Wen Liu, Kaifeng Bu, and Ryuji Takagi. One-shot operational quantum resource theory. Phys. Rev. Lett., 123: 020401, Jul 2019. 10.1103/​PhysRevLett.123.020401. URL https:/​/​​doi/​10.1103/​PhysRevLett.123.020401.

[60] Xin Wang, Mark M. Wilde, and Yuan Su. Efficiently computable bounds for magic state distillation. Phys. Rev. Lett., 124: 090505, Mar 2020. 10.1103/​PhysRevLett.124.090505. URL https:/​/​​doi/​10.1103/​PhysRevLett.124.090505.

[61] Michael Levin and Zheng-Cheng Gu. Braiding statistics approach to symmetry-protected topological phases. Phys. Rev. B, 86: 115109, Sep 2012. 10.1103/​PhysRevB.86.115109. URL https:/​/​​doi/​10.1103/​PhysRevB.86.115109.

[62] Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson. Pauli stabilizer models of twisted quantum doubles. In preparation.

[63] Trithep Devakul, Dominic J. Williamson, and Yizhi You. Classification of subsystem symmetry-protected topological phases. Phys. Rev. B, 98: 235121, Dec 2018. 10.1103/​PhysRevB.98.235121. URL https:/​/​​doi/​10.1103/​PhysRevB.98.235121.

[64] Yizhi You, Trithep Devakul, F. J. Burnell, and S. L. Sondhi. Subsystem symmetry protected topological order. Phys. Rev. B, 98: 035112, Jul 2018. 10.1103/​PhysRevB.98.035112. URL https:/​/​​doi/​10.1103/​PhysRevB.98.035112.

[65] Beni Yoshida. Topological phases with generalized global symmetries. Phys. Rev. B, 93: 155131, Apr 2016. 10.1103/​PhysRevB.93.155131. URL https:/​/​​doi/​10.1103/​PhysRevB.93.155131.

[66] Robert Raussendorf, Sergey Bravyi, and Jim Harrington. Long-range quantum entanglement in noisy cluster states. Phys. Rev. A, 71: 062313, Jun 2005. 10.1103/​PhysRevA.71.062313. URL https:/​/​​doi/​10.1103/​PhysRevA.71.062313.

[67] Yu-An Chen, Tyler D. Ellison, and Nathanan Tantivasadakarn. Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions. Phys. Rev. Research, 3: 013056, Jan 2021. 10.1103/​PhysRevResearch.3.013056. URL https:/​/​​doi/​10.1103/​PhysRevResearch.3.013056.

[68] Sam Roberts, Beni Yoshida, Aleksander Kubica, and Stephen D. Bartlett. Symmetry-protected topological order at nonzero temperature. Phys. Rev. A, 96: 022306, Aug 2017. 10.1103/​PhysRevA.96.022306. URL https:/​/​​doi/​10.1103/​PhysRevA.96.022306.

[69] Sam Roberts and Stephen D. Bartlett. Symmetry-protected self-correcting quantum memories. Phys. Rev. X, 10: 031041, Aug 2020. 10.1103/​PhysRevX.10.031041. URL https:/​/​​doi/​10.1103/​PhysRevX.10.031041.

[70] Lokman Tsui and Xiao-Gang Wen. Lattice models that realize $\mathbb{Z}_{n}$-1 symmetry-protected topological states for even $n$. Phys. Rev. B, 101: 035101, Jan 2020. 10.1103/​PhysRevB.101.035101. URL https:/​/​​doi/​10.1103/​PhysRevB.101.035101.

[71] Juven C. Wang, Zheng-Cheng Gu, and Xiao-Gang Wen. Field-theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology, and beyond. Phys. Rev. Lett., 114: 031601, Jan 2015. 10.1103/​PhysRevLett.114.031601. URL https:/​/​​doi/​10.1103/​PhysRevLett.114.031601.

[72] F. J. Burnell, Xie Chen, Lukasz Fidkowski, and Ashvin Vishwanath. Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order. Phys. Rev. B, 90: 245122, Dec 2014. 10.1103/​PhysRevB.90.245122. URL https:/​/​​doi/​10.1103/​PhysRevB.90.245122.

[73] Lukasz Fidkowski, Jeongwan Haah, and Matthew B. Hastings. Exactly solvable model for a $4+1\mathrm{D}$ beyond-cohomology symmetry-protected topological phase. Phys. Rev. B, 101: 155124, Apr 2020. 10.1103/​PhysRevB.101.155124. URL https:/​/​​doi/​10.1103/​PhysRevB.101.155124.

[74] Jeongwan Haah, Lukasz Fidkowski, and Matthew B. Hastings. Nontrivial Quantum Cellular Automata in Higher Dimensions. arXiv:1812.01625, December 2018. URL https:/​/​​abs/​1812.01625.

[75] Sergey Bravyi, David P. Divincenzo, Roberto Oliveira, and Barbara M. Terhal. The complexity of stoquastic local hamiltonian problems. Quantum Info. Comput., 8 (5): 361–385, May 2008. ISSN 1533-7146. URL https:/​/​​doi/​10.5555/​2011772.2011773.

[76] Lalit Gupta and Itay Hen. Elucidating the interplay between non-stoquasticity and the sign problem. Advanced Quantum Technologies, 3 (1): 1900108, 2020. URL https:/​/​​doi/​abs/​10.1002/​qute.201900108. URL https:/​/​​10.1002/​qute.201900108.

[77] Luiz H. Santos. Rokhsar-kivelson models of bosonic symmetry-protected topological states. Phys. Rev. B, 91: 155150, Apr 2015. 10.1103/​PhysRevB.91.155150. URL https:/​/​​doi/​10.1103/​PhysRevB.91.155150.

[78] Zohar Ringel and Steven H. Simon. Hidden order and flux attachment in symmetry-protected topological phases: A laughlin-like approach. Phys. Rev. B, 91: 195117, May 2015. 10.1103/​PhysRevB.91.195117. URL https:/​/​​doi/​10.1103/​PhysRevB.91.195117.

[79] Thomas Scaffidi and Zohar Ringel. Wave functions of symmetry-protected topological phases from conformal field theories. Phys. Rev. B, 93: 115105, Mar 2016. 10.1103/​PhysRevB.93.115105. URL https:/​/​​doi/​10.1103/​PhysRevB.93.115105.

[80] Dominic V. Else, Ilai Schwarz, Stephen D. Bartlett, and Andrew C. Doherty. Symmetry-protected phases for measurement-based quantum computation. Phys. Rev. Lett., 108: 240505, Jun 2012. 10.1103/​PhysRevLett.108.240505. URL https:/​/​​doi/​10.1103/​PhysRevLett.108.240505.

[81] Iman Marvian. Symmetry-protected topological entanglement. Phys. Rev. B, 95: 045111, Jan 2017. 10.1103/​PhysRevB.95.045111. URL https:/​/​​doi/​10.1103/​PhysRevB.95.045111.

[82] J.I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete. Matrix product density operators: Renormalization fixed points and boundary theories. Annals of Physics, 378: 100–149, Mar 2017. ISSN 0003-4916. 10.1016/​j.aop.2016.12.030. URL http:/​/​​10.1016/​j.aop.2016.12.030.

[83] Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang. Obstacles to variational quantum optimization from symmetry protection. Phys. Rev. Lett., 125: 260505, Dec 2020. 10.1103/​PhysRevLett.125.260505. URL https:/​/​​doi/​10.1103/​PhysRevLett.125.260505.

[84] William K Wootters. A wigner-function formulation of finite-state quantum mechanics. Annals of Physics, 176 (1): 1 – 21, 1987. ISSN 0003-4916. https:/​/​​10.1016/​0003-4916(87)90176-X. URL http:/​/​​science/​article/​pii/​000349168790176X.

[85] D. Gross. Hudson’s theorem for finite-dimensional quantum systems. Journal of Mathematical Physics, 47 (12): 122107, 2006. 10.1063/​1.2393152. URL https:/​/​​10.1063/​1.2393152.

[86] Hakop Pashayan, Joel J. Wallman, and Stephen D. Bartlett. Estimating outcome probabilities of quantum circuits using quasiprobabilities. Phys. Rev. Lett., 115: 070501, Aug 2015. 10.1103/​PhysRevLett.115.070501. URL https:/​/​​doi/​10.1103/​PhysRevLett.115.070501.

[87] Beni Yoshida. Topological color code and symmetry-protected topological phases. Phys. Rev. B, 91: 245131, Jun 2015. 10.1103/​PhysRevB.91.245131. URL https:/​/​​doi/​10.1103/​PhysRevB.91.245131.

[88] Jeongwan Haah. Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices. arXiv:1812.11193, December 2018. URL https:/​/​​abs/​1812.11193. URL https:/​/​​10.1063/​5.0021068.

[89] M. Popp, F. Verstraete, M. A. Martín-Delgado, and J. I. Cirac. Localizable entanglement. Phys. Rev. A, 71: 042306, Apr 2005. 10.1103/​PhysRevA.71.042306. URL https:/​/​​doi/​10.1103/​PhysRevA.71.042306.

[90] Nathanan Tantivasadakarn. Dimensional reduction and topological invariants of symmetry-protected topological phases. Phys. Rev. B, 96: 195101, Nov 2017. 10.1103/​PhysRevB.96.195101. URL https:/​/​​doi/​10.1103/​PhysRevB.96.195101.

[91] Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang. Symmetry fractionalization, defects, and gauging of topological phases. Phys. Rev. B, 100: 115147, Sep 2019. 10.1103/​PhysRevB.100.115147. URL https:/​/​​doi/​10.1103/​PhysRevB.100.115147.

[92] David Fattal, Toby S. Cubitt, Yoshihisa Yamamoto, Sergey Bravyi, and Isaac L. Chuang. Entanglement in the stabilizer formalism. arXiv:quant-ph/​0406168, June 2004. URL https:/​/​​abs/​quant-ph/​0406168.

[93] Noah Linden, Frantisek Matus, Mary Beth Ruskai, and Andreas Winter. The Quantum Entropy Cone of Stabiliser States. In Simone Severini and Fernando Brandao, editors, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013), volume 22 of Leibniz International Proceedings in Informatics (LIPIcs), pages 270–284, Dagstuhl, Germany, 2013. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. ISBN 978-3-939897-55-2. 10.4230/​LIPIcs.TQC.2013.270. URL http:/​/​​opus/​volltexte/​2013/​4327.

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[3] Ning Bao, ChunJun Cao, and Vincent Paul Su, "Magic state distillation from entangled states", Physical Review A 105 2, 022602 (2022).

[4] Cheng-Ju Lin, Weicheng Ye, Yijian Zou, Shengqi Sang, and Timothy H. Hsieh, "Probing sign structure using measurement-induced entanglement", Quantum 7, 910 (2023).

[5] Markus Frembs, Sam Roberts, Earl T Campbell, and Stephen D Bartlett, "Hierarchies of resources for measurement-based quantum computation", New Journal of Physics 25 1, 013002 (2023).

[6] Zi-Wen Liu and Andreas Winter, "Many-Body Quantum Magic", PRX Quantum 3 2, 020333 (2022).

[7] Jong Yeon Lee, Chao-Ming Jian, and Cenke Xu, "Quantum Criticality Under Decoherence or Weak Measurement", PRX Quantum 4 3, 030317 (2023).

[8] Cheng-Ju Lin and Liujun Zou, "Entanglement-enabled symmetry-breaking orders", SciPost Physics Core 7 1, 010 (2024).

[9] Tobias Haug and Lorenzo Piroli, "Quantifying nonstabilizerness of matrix product states", Physical Review B 107 3, 035148 (2023).

[10] Luca Lepori, Michele Burrello, Andrea Trombettoni, and Simone Paganelli, "Strange correlators for topological quantum systems from bulk-boundary correspondence", Physical Review B 108 3, 035110 (2023).

[11] Chae-Yeun Park and Michael J. Kastoryano, "Expressive power of complex-valued restricted Boltzmann machines for solving nonstoquastic Hamiltonians", Physical Review B 106 13, 134437 (2022).

[12] Hrant Topchyan, Vasilii Iugov, Mkhitar Mirumyan, Shahane Khachatryan, Tigran Hakobyan, and Tigran Sedrakyan, "Z3 and (×Z3)3 symmetry protected topological paramagnets", Journal of High Energy Physics 2023 12, 199 (2023).

[13] Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson, "Pauli Stabilizer Models of Twisted Quantum Doubles", PRX Quantum 3 1, 010353 (2022).

[14] Ruochen Ma and Alex Turzillo, "Symmetry Protected Topological Phases of Mixed States in the Doubled Space", arXiv:2403.13280, (2024).

[15] Cheng-Ju Lin and Liujun Zou, "Entanglement-enabled symmetry-breaking orders", arXiv:2207.08828, (2022).

[16] Junjie Chen, Yuxuan Yan, and You Zhou, "Magic of quantum hypergraph states", arXiv:2308.01886, (2023).

[17] Hrant Topchyan, Vasilii Iugov, Mkhitar Mirumyan, Tigran S. Hakobyan, Tigran A. Sedrakyan, and Ara G. Sedrakyan, "Two-dimensional topological paramagnets protected by $\mathbb{Z}_3$ symmetry: Properties of the boundary Hamiltonian", arXiv:2312.15095, (2023).

[18] Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath, and Ruben Verresen, "Building models of topological quantum criticality from pivot Hamiltonians", arXiv:2110.09512, (2021).

[19] Jackson R. Fliss, "Knots, links, and long-range magic", Journal of High Energy Physics 2021 4, 90 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 08:17:50) and SAO/NASA ADS (last updated successfully 2024-05-21 08:17:51). The list may be incomplete as not all publishers provide suitable and complete citation data.