Fundamental thresholds of realistic quantum error correction circuits from classical spin models

Davide Vodola; Manuel Rispler; Seyong Kim; Markus Müller

doi:10.22331/q-2022-01-05-618

Fundamental thresholds of realistic quantum error correction circuits from classical spin models

Davide Vodola^1,2, Manuel Rispler³, Seyong Kim⁴, and Markus Müller^5,6

¹Dipartimento di Fisica e Astronomia ``Augusto Righi'' dell'Università di Bologna, I-40127 Bologna, Italy
²INFN, Sezione di Bologna, I-40127 Bologna, Italy
³QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
⁴Department of Physics, Sejong University, 05006 Seoul, Republic of Korea
⁵Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, 52428 Jülich, Germany
⁶Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany

Published:	2022-01-05, volume 6, page 618
Eprint:	arXiv:2104.04847v2
Doi:	https://doi.org/10.22331/q-2022-01-05-618
Citation:	Quantum 6, 618 (2022).

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Abstract

Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological noise models. Here, we extend this mapping to admit realistic, multi-parameter noise models of faulty QEC circuits, derive the associated strongly correlated classical spin models, and illustrate this approach for a quantum repetition code with faulty stabilizer readout circuits. We use Monte-Carlo simulations to study the resulting phase diagram and benchmark our results against a minimum-weight perfect matching decoder. The presented method provides an avenue to assess fundamental thresholds of QEC circuits, independent of specific decoding strategies, and can thereby help guiding the development of near-term QEC hardware.

Popular summary

One of the main obstacles for the realization of a fully functioning quantum computer is the noise affecting the qubits. To protect quantum information against such disturbances, scientists have developed techniques for quantum error correction (QEC). Here, one uses quantum codes that redundantly encode the fragile quantum information in a collection of data qubits. Then quantum circuits can be periodically applied, to detect where in the qubit register errors have occurred. This is done via indirect measurements of correlations among the data qubits via coupling and measurement of auxiliary qubits. The information accumulated from these measurements can then be used to find a suitable correction, which removes the detected errors with a high success probability.

It has been proven that the process of finding the most suitable correction for the quantum codes is equivalent to studying the phase diagram of a classical disordered statistical mechanics model. This connection between quantum and classical physics allows one to harness the knowledge of phase diagram of the classical models to gain insight into the fundamental robustness limits of the quantum codes against noise. For example, locating the phase transitions between ordered and disordered phases in the classical models reveals the parameter regimes for which QEC succeeds or fails, respectively.

So far, however, these mappings have been largely limited to QEC codes with simplified noise models where data qubits can suffer errors and the measurement of the auxiliary qubits is faulty with a certain probability. These simplified descriptions, however, ignore the possibility that errors can occur while executing the quantum circuits used for the QEC procedures, and that such errors can propagate and proliferate through the quantum hardware.

In this work, we extend the approach of mapping QEC codes to classical statistical mechanics models to the more realistic scenario ,in which errors in the quantum circuits and propagation of errors is taken into account explicitly. We illustrate this mapping for a one-dimensional quantum repetition code. In realistic quantum hardware, different circuit elements such as single or two-qubit gates typically have different failure probabilities. In our study, we model this by a multi-parameter microscopic noise model for the QEC circuits, and derive the associated classical statistical physics model. A numerical study of its phase diagram finally allows us to determine the largest possible parameter region, in which successful QEC is feasible.

► BibTeX data

@article{Vodola2022fundamental,
  doi = {10.22331/q-2022-01-05-618},
  url = {https://doi.org/10.22331/q-2022-01-05-618},
  title = {Fundamental thresholds of realistic quantum error correction circuits from classical spin models},
  author = {Vodola, Davide and Rispler, Manuel and Kim, Seyong and M{\"{u}}ller, Markus},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {618},
  month = jan,
  year = {2022}
}

► References

[1] D. A. Lidar and O. Biham, ``Simulating Ising spin glasses on a quantum computer,'' Phys. Rev. E 56, 3661 (1997).
https://doi.org/10.1103/PhysRevE.56.3661

[2] R. D. Somma, C. D. Batista, and G. Ortiz, ``Quantum approach to classical statistical mechanics,'' Phys. Rev. Lett. 99, 030603 (2007).
https://doi.org/10.1103/PhysRevLett.99.030603

[3] M. Van den Nest, W. Dür, and H. J. Briegel, ``Classical spin models and the quantum-stabilizer formalism,'' Phys. Rev. Lett. 98, 117207 (2007).
https://doi.org/10.1103/PhysRevLett.98.117207

[4] J. Geraci and D. A. Lidar, ``On the exact evaluation of certain instances of the Potts partition function by quantum computers,'' Commun. Math. Phys. 279, 735 (2008).
https://doi.org/10.1007/s00220-008-0438-0

[5] G. De las Cuevas, W. Dür, H. J. Briegel, and M. A. Martin-Delgado, ``Unifying all classical spin models in a lattice gauge theory,'' Phys. Rev. Lett. 102, 230502 (2009).
https://doi.org/10.1103/PhysRevLett.102.230502

[6] G. De las Cuevas, W. Dür, M. V. den Nest, and M. A. Martin-Delgado, ``Quantum algorithms for classical lattice models,'' New J. Phys. 13, 093021 (2011).
https://doi.org/10.1088/1367-2630/13/9/093021

[7] Y. Xu, G. De las Cuevas, W. Dür, H. J. Briegel, and M. A. Martin-Delgado, ``The U(1) lattice gauge theory universally connects all classical models with continuous variables, including background gravity,'' J. Stat. Mech-Theory E 2011, P02013 (2011).
https://doi.org/10.1088/1742-5468/2011/02/p02013

[8] G. De las Cuevas and T. S. Cubitt, ``Simple universal models capture all classical spin physics,'' Science 351, 1180 (2016).
https://doi.org/10.1126/science.aab3326

[9] M. H. Zarei and A. Montakhab, ``Dual correspondence between classical spin models and quantum Calderbank-Shor-Steane states,'' Phys. Rev. A 98, 012337 (2018).
https://doi.org/10.1103/PhysRevA.98.012337

[10] Y. Li and M. P. A. Fisher, ``Statistical mechanics of quantum error correcting codes,'' Phys. Rev. B 103, 104306 (2021).
https://doi.org/10.1103/PhysRevB.103.104306

[11] A. Nahum, S. Vijay, and J. Haah, ``Operator spreading in random unitary circuits,'' Phys. Rev. X 8, 021014 (2018).
https://doi.org/10.1103/PhysRevX.8.021014

[12] T. Zhou and A. Nahum, ``Emergent statistical mechanics of entanglement in random unitary circuits,'' Phys. Rev. B 99, 174205 (2019).
https://doi.org/10.1103/PhysRevB.99.174205

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

[14] D. A. Lidar and T. A. Brun, eds., Quantum Error Correction (Cambridge University Press, Cambridge, England, 2013).
https://doi.org/10.1017/CBO9781139034807

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

[16] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, ``Topological quantum memory,'' J. Math. Phys. 43, 4452 (2002).
https://doi.org/10.1063/1.1499754

[17] H. Bombin and M. A. Martin-Delgado, ``Topological quantum distillation,'' Phys. Rev. Lett. 97, 180501 (2006).
https://doi.org/10.1103/PhysRevLett.97.180501

[18] H. Bombin and M. A. Martin-Delgado, ``Topological computation without braiding,'' Phys. Rev. Lett. 98, 160502 (2007).
https://doi.org/10.1103/PhysRevLett.98.160502

[19] H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, ``Error threshold for color codes and random three-body Ising models,'' Phys. Rev. Lett. 103, 090501 (2009).
https://doi.org/10.1103/PhysRevLett.103.090501

[20] H. G. Katzgraber, H. Bombin, R. S. Andrist, and M. A. Martin-Delgado, ``Topological color codes on union jack lattices: a stable implementation of the whole clifford group,'' Phys. Rev. A 81, 012319 (2010).
https://doi.org/10.1103/PhysRevA.81.012319

[21] H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber, and M. A. Martin-Delgado, ``Strong resilience of topological codes to depolarization,'' Phys. Rev. X 2, 021004 (2012).
https://doi.org/10.1103/PhysRevX.2.021004

[22] R. S. Andrist, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, ``Error tolerance of topological codes with independent bit-flip and measurement errors,'' Phys. Rev. A 94, 012318 (2016).
https://doi.org/10.1103/PhysRevA.94.012318

[23] C. T. Chubb and S. T. Flammia, ``Statistical mechanical models for quantum codes with correlated noise,'' Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8, 269 (2021).
https://doi.org/10.4171/AIHPD/105

[24] T. Ohno, G. Arakawa, I. Ichinose, and T. Matsui, ``Phase structure of the random-plaquette $Z_2$ gauge model: accuracy threshold for a toric quantum memory,'' Nucl. Phys. B 697, 462 (2004).
https://doi.org/10.1016/j.nuclphysb.2004.07.003

[25] R. S. Andrist, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, ``Tricolored lattice gauge theory with randomness: fault tolerance in topological color codes,'' New J. Phys. 13, 83006 (2011).
https://doi.org/10.1088/1367-2630/13/8/083006
http://stacks.iop.org/1367-2630/13/i=8/a=083006

[26] T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, ``Loss-tolerant optical qubits,'' Phys. Rev. Lett. 95, 100501 (2005).
https://doi.org/10.1103/PhysRevLett.95.100501

[27] T. M. Stace, S. D. Barrett, and A. C. Doherty, ``Thresholds for Topological Codes in the Presence of Loss,'' Phys. Rev. Lett. 102, 200501 (2009).
https://doi.org/10.1103/PhysRevLett.102.200501

[28] T. M. Stace and S. D. Barrett, ``Error correction and degeneracy in surface codes suffering loss,'' Phys. Rev. A 81, 022317 (2010).
https://doi.org/10.1103/PhysRevA.81.022317

[29] D. Vodola, D. Amaro, M. A. Martin-Delgado, and M. Müller, ``Twins percolation for qubit losses in topological color codes,'' Phys. Rev. Lett. 121, 060501 (2018).
https://doi.org/10.1103/PhysRevLett.121.060501

[30] D. Amaro, J. Bennett, D. Vodola, and M. Müller, ``Analytical percolation theory for topological color codes under qubit loss,'' Phys. Rev. A 101, 032317 (2020).
https://doi.org/10.1103/PhysRevA.101.032317

[31] C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, ``Quantum error correction with the toric Gottesman-Kitaev-Preskill code,'' Phys. Rev. A 99, 032344 (2019).
https://doi.org/10.1103/PhysRevA.99.032344

[32] A. D. Córcoles, E. Magesan, S. J. Srinivasan, A. W. Cross, M. Steffen, J. M. Gambetta, and J. M. Chow, ``Demonstration of a quantum error detection code using a square lattice of four superconducting qubits,'' Nat. Commun. 6, 6979 (2015).
https://doi.org/10.1038/ncomms7979

[33] M. Saffman, ``Quantum computing with atomic qubits and Rydberg interactions: progress and challenges,'' J. Phys. B: At., Mol. Opt. Phys. 49, 202001 (2016).
https://doi.org/10.1088/0953-4075/49/20/202001

[34] V. Negnevitsky, M. Marinelli, K. K. Mehta, H. Y. Lo, C. Flühmann, and J. P. Home, ``Repeated multi-qubit readout and feedback with a mixed-species trapped-ion register,'' Nature 563, 527 (2018).
https://doi.org/10.1038/s41586-018-0668-z

[35] C. K. Andersen, A. Remm, S. Lazar, S. Krinner, J. Heinsoo, J.-C. Besse, M. Gabureac, A. Wallraff, and C. Eichler, ``Entanglement stabilization using ancilla-based parity detection and real-time feedback in superconducting circuits,'' npj Quantum Inf. 5, 69 (2019).
https://doi.org/10.1038/s41534-019-0185-4

[36] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, ``Trapped-ion quantum computing: Progress and challenges,'' Applied Physics Reviews 6 (2019).
https://doi.org/10.1063/1.5088164

[37] M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, ``Superconducting qubits: Current state of play,'' Annual Review of Condensed Matter Physics 11, 369 (2020).
https://doi.org/10.1146/annurev-conmatphys-031119-050605

[38] C. K. Andersen, A. Remm, S. Lazar, S. Krinner, N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler, and A. Wallraff, ``Repeated quantum error detection in a surface code,'' Nat. Phys. 16, 875 (2020).
https://doi.org/10.1038/s41567-020-0920-y

[39] S. Pezzagna and J. Meijer, ``Quantum computer based on color centers in diamond,'' Applied Physics Reviews 8, 011308 (2021).
https://doi.org/10.1063/5.0007444

[40] A. Chatterjee, P. Stevenson, S. De Franceschi, A. Morello, N. P. de Leon, and F. Kuemmeth, ``Semiconductor qubits in practice,'' Nat. Rev. Phys. 3, 157 (2021).
https://doi.org/10.1038/s42254-021-00283-9

[41] L. Egan et al., ``Fault-tolerant control of an error-corrected qubit,'' Nature 598, 281 (2021).
https://doi.org/10.1038/s41586-021-03928-y

[42] L. P. Pryadko, ``On maximum-likelihood decoding with circuit-level errors,'' Quantum 4, 304 (2020), 1909.06732v4.
https://doi.org/10.22331/q-2020-08-06-304
arXiv:1909.06732v4

[43] P. Iyer and D. Poulin, ``Hardness of Decoding Quantum Stabilizer Codes,'' IEEE Trans. Inf. Theory 61, 5209 (2015).
https://doi.org/10.1109/TIT.2015.2422294

[44] A. M. Stephens, ``Fault-tolerant thresholds for quantum error correction with the surface code,'' Phys. Rev. A 89, 022321 (2014).
https://doi.org/10.1103/PhysRevA.89.022321

[45] D. Wang, A. Fowler, C. Hill, and L. Hollenberg, ``Graphical algorithms and threshold error rates for the 2d color code,'' Quant. Inf. Comp 10, 780 (2010a).
https://doi.org/10.26421/QIC10.9-10-5

[46] A. J. Landahl, J. T. Anderson, and P. R. Rice, ``Fault-tolerant quantum computing with color codes,'' (2011), arXiv:1108.5738 [quant-ph].
arXiv:1108.5738

[47] M. E. Beverland, A. Kubica, and K. M. Svore, ``Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes,'' PRX Quantum 2, 020341 (2021).
https://doi.org/10.1103/PRXQuantum.2.020341

[48] R. Chao and B. Reichardt, ``Fault-tolerant quantum computation with few qubits,'' npj Quantum Inf. href https://doi.org/10.1038/s41534-018-0085-z 4 (2017).
https://doi.org/10.1038/s41534-018-0085-z
https://www.nature.com/articles/s41534-018-0085-z

[49] C. Chamberland and M. E. Beverland, ``Flag fault-tolerant error correction with arbitrary distance codes,'' Quantum 2, 53 (2018).
https://doi.org/10.22331/q-2018-02-08-53

[50] C. Chamberland, A. Kubica, T. J. Yoder, and G. Zhu, ``Triangular color codes on trivalent graphs with flag qubits,'' New J. Phys. 22, 023019 (2020).
https://doi.org/10.1088/1367-2630/ab68fd

[51] C. Chamberland and K. Noh, ``Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits,'' npj Quantum Inf. 6 (2020).
https://doi.org/10.1038/s41534-020-00319-5

[52] P. Schindler, J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, and R. Blatt, ``Experimental repetitive quantum error correction,'' Science 332, 1059 (2011).
https://doi.org/10.1126/science.1203329

[53] G. Waldherr et al., ``Quantum error correction in a solid-state hybrid spin register,'' Nature 506, 204 (2014).
https://doi.org/10.1038/nature12919

[54] J. Kelly et al., ``State preservation by repetitive error detection in a superconducting quantum circuit,'' Nature 519, 66 (2015).
https://doi.org/10.1038/nature14270

[55] Z. Chen et al., ``Exponential suppression of bit or phase errors with cyclic error correction,'' Nature 595, 383 (2021).
https://doi.org/10.1038/s41586-021-03588-y

[56] S. J. Devitt, W. J. Munro, and K. Nemoto, ``Quantum error correction for beginners,'' Rep. Prog. Phys. 76, 076001 (2013).
https://doi.org/10.1088/0034-4885/76/7/076001

[57] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
https://doi.org/10.1017/CBO9780511976667

[58] D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology (1997).
arXiv:quant-ph/9705052

[59] A. Honecker, M. Picco, and P. Pujol, ``Universality Class of the Nishimori Point in the 2D $\pm{}\mathit{J}$ Random-Bond Ising Model,'' Phys. Rev. Lett. 87, 047201 (2001).
https://doi.org/10.1103/PhysRevLett.87.047201

[60] R. Raussendorf and J. Harrington, ``Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions,'' Phys. Rev. Lett. 98, 190504 (2007).
https://doi.org/10.1103/PhysRevLett.98.190504

[61] M. Rispler, P. Cerfontaine, V. Langrock, and B. M. Terhal, ``Towards a realistic GaAs-spin qubit device for a classical error-corrected quantum memory,'' Phys. Rev. A 102, 022416 (2020).
https://doi.org/10.1103/PhysRevA.102.022416

[62] C. Wang, J. Harrington, and J. Preskill, ``Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory,'' Ann. Phys. 303, 31 (2003).
https://doi.org/10.1016/S0003-4916(02)00019-2

[63] W. Huang et al., ``Fidelity benchmarks for two-qubit gates in silicon,'' Nature 569, 532 (2019).
https://doi.org/10.1038/s41586-019-1197-0

[64] D. Greenbaum, ``Introduction to quantum gate set tomography,'' (2015), arXiv:1509.02921 [quant-ph].
arXiv:1509.02921

[65] M. Palassini and S. Caracciolo, ``Universal finite-size scaling functions in the 3D Ising spin glass,'' Phys. Rev. Lett. 82, 5128 (1999).
https://doi.org/10.1103/PhysRevLett.82.5128

[66] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, ``Equation of state calculations by fast computing machines,'' The Journal of Chemical Physics 21, 1087 (1953).
https://doi.org/10.1063/1.1699114

[67] R. H. Swendsen and J.-S. Wang, ``Replica Monte Carlo simulation of spin-glasses,'' Phys. Rev. Lett. 57, 2607 (1986).
https://doi.org/10.1103/PhysRevLett.57.2607

[68] D. J. Earl and M. W. Deem, ``Parallel tempering: Theory, applications, and new perspectives,'' Phys. Chem. Chem. Phys. 7, 3910 (2005).
https://doi.org/10.1039/B509983H

[69] A. Hagberg, D. A. Schult, and P. J. Swart, ``Exploring network structure, dynamics, and function using networkx,'' in Proceedings of the 7th Python in Science Conference, edited by G. Varoquaux, T. Vaught, and J. Millman (Pasadena, CA USA, 2008) pp. 11 – 15.

[70] D. Wang, A. Fowler, A. Stephens, and L. Hollenberg, ``Threshold error rates for the toric and planar codes,'' Quant. Inf. Comp 10, 456 (2010b).
https://doi.org/10.26421/QIC10.5-6-6

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