Determining quantum phase diagrams of topological Kitaev-inspired models on NISQ quantum hardware

Xiao Xiao; J. K. Freericks; A. F. Kemper

doi:10.22331/q-2021-09-28-553

Determining quantum phase diagrams of topological Kitaev-inspired models on NISQ quantum hardware

Xiao Xiao¹, J. K. Freericks², and A. F. Kemper¹

¹Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
²Department of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057 USA

Published:	2021-09-28, volume 5, page 553
Eprint:	arXiv:2006.05524v3
Doi:	https://doi.org/10.22331/q-2021-09-28-553
Citation:	Quantum 5, 553 (2021).

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

Abstract

Topological protection is employed in fault-tolerant error correction and in developing quantum algorithms with topological qubits. But, topological protection $\textit{intrinsic to models being simulated}$, also robustly protects calculations, even on NISQ hardware. We leverage it by simulating Kitaev-inspired models on IBM quantum computers and accurately determining their phase diagrams. This requires constructing conventional quantum circuits for Majorana braiding to prepare the ground states of Kitaev-inspired models. The entanglement entropy is then measured to calculate the quantum phase boundaries. We show how maintaining particle-hole symmetry when sampling through the Brillouin zone is critical to obtaining high accuracy. This work illustrates how topological protection intrinsic to a quantum model can be employed to perform robust calculations on NISQ hardware, when one measures the appropriate protected quantum properties. It opens the door for further simulation of topological quantum models on quantum hardware available today.

Featured image: Left panel: the quantum circuits to perform braiding operations on a conventional quantum hardware; right panels: the phase diagram of the Kitaev honeycomb model determined by NISQ quantum hardware; the details of entanglement entropy along the dashed line in the lower right panel is shown in the upper right one.

Popular summary

It is challenging to achieve high accuracy for programs run on current noisy intermediate-scale quantum (NISQ) hardware. Topological quantum computing has long been thought to be a remedy for handling noise and decoherence, because topological qubits are intrinsically protected from environment. However, the creation of topological qubits is so difficult that we are yet to see topological quantum computers available for use. Here we demonstrate that performing calculations of the topological properties of quantum systems is also intrinsically robust, even if we do so on a conventional quantum computer. We do this in two steps. First, we construct nontrivial topological states by braiding non-Abelian quasi-particles with conventional qubits. Second, we use symmetry-enforced quantum circuits to perform calculations with small numbers of qubits. This study provides a paradigm for applying NISQ hardware to study nontrivial topological quantum states.

► BibTeX data

@article{Xiao2021determiningquantum,
  doi = {10.22331/q-2021-09-28-553},
  url = {https://doi.org/10.22331/q-2021-09-28-553},
  title = {Determining quantum phase diagrams of topological {K}itaev-inspired models on {NISQ} quantum hardware},
  author = {Xiao, Xiao and Freericks, J. K. and Kemper, A. F.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {553},
  month = sep,
  year = {2021}
}

► References

[1] M. Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1267 (2005).
https://doi.org/10.1103/RevModPhys.76.1267

[2] A. M. Steane, Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793 (1996).
https://doi.org/10.1103/PhysRevLett.77.793

[3] E. Knill, R. Laflamme, and W. Zurek, Threshold accuracy for quantum computation. arXiv:9610011 [quant-ph] (1996).
arXiv:quant-ph/9610011

[4] D. Aharonov, and M. Ben-Or, Fault-tolerant quantum computation with constant error. SIAM J. Comput., 38, 1207 (2008).
https://doi.org/10.1137/S0097539799359385

[5] B. M. Terhal, Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307 (2015).
https://doi.org/10.1103/RevModPhys.87.307

[6] A. Aliferis, D. Gottesman, and, J. Preskill, Quantum accuracy threshold for concatenated distance-$3$ codes Quantum Inf. Comput. 6, 97 (2005).
https://dl.acm.org/doi/10.5555/2011665.2011666

[7] D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A 73, 012340 (2006).
https://doi.org/10.1103/PhysRevA.73.012340

[8] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).
https://doi.org/10.1103/PhysRevA.86.032324

[9] D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultrahigh error threshold for surface codes with biased noise. Phys. Rev. Lett. 120, 050505 (2018).
https://doi.org/10.1103/PhysRevLett.120.050505

[10] R. Chao, and B. W. Reichardt, Quantum error correction with only two extra qubits. Phys. Rev. Lett. 121, 050502 (2018).
https://doi.org/10.1103/PhysRevLett.121.050502

[11] M. Li, D. Miller, M. Newman, Y. Wu, and K. R. Brown, $2D$ Compass Codes. Phys. Rev. X 9, 021041 (2019).
https://doi.org/10.1103/PhysRevX.9.021041

[12] A. Kitaev, Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2 (2003).
https://doi.org/10.1016/S0003-4916(02)00018-0

[13] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083 (2008).
https://doi.org/10.1103/RevModPhys.80.1083

[14] S. D. Sarma, M. Freedman, and C. Nayak, Majorana zero modes and topological quantum computation. NPJ Quantum Information 1, 15001 (2015).
https://doi.org/10.1038/npjqi.2015.1

[15] A. Kitaev, Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2 (2006).
https://doi.org/10.1016/j.aop.2005.10.005

[16] H. Yao, and S. A. Kivelson, Exact chiral spin liquid with Non-Abelian anyons. Phys. Rev. Lett. 99, 247203 (2007).
https://doi.org/10.1103/PhysRevLett.99.247203

[17] S. Yang, D. L. Zhou, and C. P. Sun, Mosaic spin models with topological order. Phys. Rev. B 76, 180404 (2007).
https://doi.org/10.1103/PhysRevB.76.180404

[18] X.-Y. Feng, G. M. Zhang, and T. Xiang, Topological characterization of quantum phase Transitions in a Spin-$1/2$ Model. Phys. Rev. Lett. 98, 087204 (2007).
https://doi.org/10.1103/PhysRevLett.98.087204

[19] D. H. Lee, G. M. Zhang, and T. Xiang, Edge solitons of topological insulators and fractionalized quasiparticles in Two Dimensions. Phys. Rev. Lett. 99, 196805 (2007).
https://doi.org/10.1103/PhysRevLett.99.196805

[20] T. Si, and Y. Yu, Exactly soluble spin-$1/2$ models on three-dimensional lattices and non-abelian statistics of closed string excitations. arXiv:0709.1302 [cond-mat] (2007).
arXiv:0709.1302

[21] Y. Yu, & Z. Q. Wang, An exactly soluble model with tunable p-wave paired fermion ground states. Europhys. Lett. 84, 57002 (2008).
https://doi.org/10.1209/0295-5075/84/57002

[22] G. Baskaran, G. Santhosh, and R. Shankar, Exact quantum spin liquids with Fermi surfaces in spin-half models. arXiv:0908.1614 [cond-mat] (2009).
arXiv:0908.1614

[23] S. Mandal, and N. Surendran, Exactly solvable Kitaev model in three dimensions. Phys. Rev. B 79, 024426 (2009).
https://doi.org/10.1103/PhysRevB.79.024426

[24] S. Ryu, Three-dimensional topological phase on the diamond lattice. Phys. Rev. B 79, 075124 (2009).
https://doi.org/10.1103/PhysRevB.79.075124

[25] H. Yao, S.-C. Zhang, and S. A. Kivelson, Algebraic spin liquid in an exactly solvable spin model. Phys. Rev. Lett. 102, 217202 (2009).
https://doi.org/10.1103/PhysRevLett.102.217202

[26] C. Wu, D. Arovas, and H. H. Hung, $\Gamma$-matrix generalization of the Kitaev model. Phys. Rev. B 79, 134427 (2009).
https://doi.org/10.1103/PhysRevB.79.134427

[27] K. S. Tikhonov, and M. V. Feigelman, Quantum spin metal state on a decorated honeycomb lattice. Phys. Rev. Lett. 105, 067207 (2010).
https://doi.org/10.1103/PhysRevLett.105.067207

[28] G. W. Chern, Three-dimensional topological phases in a layered honeycomb spin-orbital model. Phys. Rev. B 81, 125134 (2010).
https://doi.org/10.1103/PhysRevB.81.125134

[29] F. Wang, Realization of the exactly solvable Kitaev honeycomb lattice model in a spin-rotation-invariant system. Phys. Rev. B 81, 184416 (2010).
https://doi.org/10.1103/PhysRevB.81.184416

[30] V. Lahtinen, and J. K. Pachos, Topological phase transitions driven by gauge fields in an exactly solvable model. Phys. Rev. B 81, 245132 (2010).
https://doi.org/10.1103/PhysRevB.81.245132

[31] G. Kells, J. Kailasvuori, J. K. Slingerland, and J. Vala, Kaleidoscope of topological phases with multiple Majorana species. New J. Phys. 13, 095014 (2011).
https://doi.org/10.1088/1367-2630/13/9/095014

[32] H. Yao, and D. H. Lee, Fermionic magnons, Non-Abelian spinons, and the spin quantum hall effect from an exactly solvable spin-$1/2$ Kitaev model with $SU(2)$ symmetry. Phys. Rev. Lett. 107, 087205 (2011).
https://doi.org/10.1103/PhysRevLett.107.087205

[33] H. H. Lai, and O. I. Motrunich, Power-law behavior of bond energy correlators in a Kitaev-type model with a stable parton Fermi surface. Phys. Rev. B 83, 155104 (2011).
https://doi.org/10.1103/PhysRevB.83.155104

[34] V. Chua, H. Yao, & G. A. Fiete, Exact chiral spin liquid with stable spin Fermi surface on the kagome lattice. Phys. Rev. B 83, 180412 (2011).
https://doi.org/10.1103/PhysRevB.83.180412

[35] R. Nakai, S. Ryu, and A. Furusaki, Time-reversal symmetric Kitaev model and topological superconductor in two dimensions. Phys. Rev. B 85, 155119 (2012).
https://doi.org/10.1103/PhysRevB.85.155119

[36] Z. Nussinov, and J. van den Brink, Compass and Kitaev models: theory and physical motivations. Rev. Mod. Phys. 87, 1 (2015).
https://doi.org/10.1103/RevModPhys.87.1

[37] M. Hermanns, K. O'Brien, and S. Trebst, Weyl spin liquids. Phys. Rev. Lett. 114, 157202 (2015).
https://doi.org/10.1103/PhysRevLett.114.157202

[38] K. O'Brien, M. Hermanns, & S. Trebst, Classification of gapless $Z_2$ spin liquids in three-dimensional Kitaev models. Phys. Rev. B 93, 085101 (2016).
https://doi.org/10.1103/PhysRevB.93.085101

[39] Z. Chen, X. Li, and T. K. Ng, Exactly Solvable BCS-Hubbard Model in Arbitrary Dimensions. Phys. Rev. Lett. 120, 046401 (2018).
https://doi.org/10.1103/PhysRevLett.120.046401

[40] J.-J. Miao, H.-K. Jin, F.-C. Zhang, and Y. Zhou, Exact solution to a class of generalized Kitaev spin-1/2 models in arbitrary dimensions. Sci. China Phys. Mech. Astron. 63, 247011 (2020).
https://doi.org/10.1007/s11433-019-1442-2

[41] J.-J. Miao, H.-K. Jin, F. Wang, F.-C. Zhang, and Y. Zhou, Pristine Mott insulator from an exactly solvable spin-$1/2$ Kitaev model. Phys. Rev. B 99, 155105 (2019).
https://doi.org/10.1103/PhysRevB.99.155105

[42] A. Stern, and N. H. Lindner, Topological quantum computation—from basic concepts to first experiments. Science 339, 1179-1184 (2013).
https://doi.org/10.1126/science.1231473

[43] H. Yao, and X.-L. Qi, Entanglement entropy and entanglement spectrum of the Kitaev model. Phys. Rev. Lett. 105, 080501 (2010).
https://doi.org/10.1103/PhysRevLett.105.080501

[44] K. Meichanetzidis, M. Cirio, J. K. Pachos and V. Lahtinen, Anatomy of fermionic entanglement and criticality in Kitaev spin liquids. Phys. Rev. B 94, 115158 (2016).
https://doi.org/10.1103/PhysRevB.94.115158

[45] P. Schmoll, and R. Orus, Kitaev honeycomb tensor networks: Exact unitary circuits and applications. Phys. Rev. B 95, 045112 (2017).
https://doi.org/10.1103/PhysRevB.95.045112

[46] F. Verstraete, J. I. Cirac,, and J. I. Latorre, Quantum circuits for strongly correlated quantum systems. Phys. Rev. A 79, 032316 (2009).
https://doi.org/10.1103/PhysRevA.79.032316

[47] A. Cervera-Lierta, Exact Ising model simulation on a quantum computer. Quantum 2, 114 (2018).
https://doi.org/10.22331/q-2018-12-21-114

[48] A. J. Ferris, Fourier transform for fermionic systems and the spectral tensor network. Phys. Rev. Lett. 113, 010401 (2014).
https://doi.org/10.1103/PhysRevLett.113.010401

[49] D. A. Ivanov, Non-Abelian Statistics of Half-Quantum Vortices in $p$-Wave Superconductors. Phys. Rev. Lett. 86, 268 (2001).
https://doi.org/10.1103/PhysRevLett.86.268

[50] G. Vidal, and C. M. Dawson, Universal quantum circuit for two-qubit transformations with three controlled-NOT gates. Phys. Rev. A 69, 010301 (2004).
https://doi.org/10.1103/PhysRevA.69.010301

[51] J. A. Smolin, J. M. Gambetta, and G. Smith, Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise. Phys. Rev. Lett. 108, 070502 (2012).
https://doi.org/10.1103/PhysRevLett.108.070502

[52] G. Aleksandrowicz, et al. Qiskit: An open-source framework for quantum computing. https://doi.org/10.5281/ZENODO.2562111 (2019).
https://doi.org/10.5281/ZENODO.2562111

[53] I. Peschel, Calculation of reduced density matrices from correlation functions. J. Phys. A: Math.Gen. 36, L205 (2003).
https://doi.org/10.1088/0305-4470/36/14/101

[54] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in Quantum Critical Phenomena. Phys. Rev. Lett. 90, 227902 (2003).
https://doi.org/10.1103/PhysRevLett.90.227902

[55] H. Jiang, C.-Y. Wang, B. Huang, and Y.-M. Lu, Field induced quantum spin liquid with spinon Fermi surfaces in the Kitaev model. arXiv:1809.08247 [cond-mat] (2018).
arXiv:1809.08247

[56] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
https://doi.org/10.1103/PhysRevB.78.195125

Cited by

[1] Junmo Jeon and SungBin Lee, "Pattern-dependent proximity effect and Majorana edge mode in one-dimensional quasicrystals", Physical Review B 105 6, 064502 (2022).

[2] Bilal Khalid, Shree Hari Sureshbabu, Arnab Banerjee, and Sabre Kais, "Finite-Size Scaling on a Digital Quantum Simulator Using Quantum Restricted Boltzmann Machine", Frontiers in Physics 10, 915863 (2022).

[3] Ammar Jahin, Andy C. Y. Li, Thomas Iadecola, Peter P. Orth, Gabriel N. Perdue, Alexandru Macridin, M. Sohaib Alam, and Norm M. Tubman, "Fermionic approach to variational quantum simulation of Kitaev spin models", Physical Review A 106 2, 022434 (2022).

[4] Xiao Xiao, J. K. Freericks, and A. F. Kemper, "Robust measurement of wave function topology on NISQ quantum computers", Quantum 7, 987 (2023).

[5] Nikolaos Petropoulos, Robert Bogdan Staszewski, Dirk Leipold, and Elena Blokhina, "Topological order detection and qubit encoding in Su–Schrieffer–Heeger type quantum dot arrays", Journal of Applied Physics 131 7, 074401 (2022).

[6] Marko J. Rančić, "Exactly solving the Kitaev chain and generating Majorana-zero-modes out of noisy qubits", Scientific Reports 12 1, 19882 (2022).

[7] Philippe Suchsland, Panagiotis Kl. Barkoutsos, Ivano Tavernelli, Mark H. Fischer, and Titus Neupert, "Simulating a ring-like Hubbard system with a quantum computer", Physical Review Research 4 1, 013165 (2022).

[8] Bruno Murta, Pedro M. Q. Cruz, and J. Fernández-Rossier, "Preparing valence-bond-solid states on noisy intermediate-scale quantum computers", Physical Review Research 5 1, 013190 (2023).

[9] John P. T. Stenger, Gilad Ben-Shach, David Pekker, and Nicholas T. Bronn, "Simulating spectroscopy experiments with a superconducting quantum computer", Physical Review Research 4 4, 043106 (2022).

[10] Lindsay Bassman, Miroslav Urbanek, Mekena Metcalf, Jonathan Carter, Alexander F. Kemper, and Wibe A. de Jong, "Simulating quantum materials with digital quantum computers", Quantum Science and Technology 6 4, 043002 (2021).

[11] John P. T. Stenger, Nicholas T. Bronn, Daniel J. Egger, and David Pekker, "Simulating the dynamics of braiding of Majorana zero modes using an IBM quantum computer", Physical Review Research 3 3, 033171 (2021).

[12] Tatiana A. Bespalova and Oleksandr Kyriienko, "Quantum simulation and ground state preparation for the honeycomb Kitaev model", arXiv:2109.13883, (2021).

[13] Marcel Cech, Igor Lesanovsky, and Federico Carollo, "Thermodynamics of Quantum Trajectories on a Quantum Computer", Physical Review Letters 131 12, 120401 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-08-11 15:23:36) and SAO/NASA ADS (last updated successfully 2023-12-07 09:39:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2023-12-07 09:39:44: Encountered the unhandled forward link type postedcontent_cite while looking for citations to DOI 10.22331/q-2021-09-28-553.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.