New perspectives on covariant quantum error correction

Sisi Zhou1,2, Zi-Wen Liu3, and Liang Jiang2

1Department of Physics, Yale University, New Haven, Connecticut 06511, USA
2Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA
3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

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Abstract

Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin–Knill theorem). The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds.

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► References

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).

[2] D. Gottesman, in Quantum information science and its contributions to mathematics, Proceedings of Symposia in Applied Mathematics, Vol. 68 (2010) pp. 13–58.

[3] D. A. Lidar and T. A. Brun, Quantum error correction (Cambridge university press, 2013).

[4] B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Physical Review Letters 102, 110502 (2009).
https:/​/​doi.org/​10.1103/​physrevlett.102.110502

[5] S. Bravyi and R. König, Classification of topologically protected gates for local stabilizer codes, Physical Review Letters 110, 170503 (2013).
https:/​/​doi.org/​10.1103/​physrevlett.110.170503

[6] F. Pastawski and B. Yoshida, Fault-tolerant logical gates in quantum error-correcting codes, Physical Review A 91, 012305 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.012305

[7] T. Jochym-O'Connor, A. Kubica, and T. J. Yoder, Disjointness of stabilizer codes and limitations on fault-tolerant logical gates, Physical Review X 8, 021047 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.021047

[8] D.-S. Wang, G. Zhu, C. Okay, and R. Laflamme, Quasi-exact quantum computation, Physical Review Research 2 (2020).
https:/​/​doi.org/​10.1103/​physrevresearch.2.033116

[9] P. Hayden, S. Nezami, S. Popescu, and G. Salton, Error correction of quantum reference frame information, PRX Quantum 2 (2021).
https:/​/​doi.org/​10.1103/​prxquantum.2.010326

[10] P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, Continuous symmetries and approximate quantum error correction, Physical Review X 10 (2020).
https:/​/​doi.org/​10.1103/​physrevx.10.041018

[11] J. Preskill, Quantum clock synchronization and quantum error correction, (2000), arXiv:quant-ph/​0010098 [quant-ph].
arXiv:quant-ph/0010098

[12] M. P. Woods and Á. M. Alhambra, Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames, Quantum 4, 245 (2020).
https:/​/​doi.org/​10.22331/​q-2020-03-23-245

[13] A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in ads/​cft, Journal of High Energy Physics 2015, 163 (2015).
https:/​/​doi.org/​10.1007/​JHEP04(2015)163

[14] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/​boundary correspondence, Journal of High Energy Physics 2015, 149 (2015).
https:/​/​doi.org/​10.1007/​JHEP06(2015)149

[15] D. Harlow and H. Ooguri, Constraints on symmetries from holography, Physical Review Letters 122, 191601 (2019).
https:/​/​doi.org/​10.1103/​physrevlett.122.191601

[16] D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, (2018), arXiv:1810.05338 [hep-th].
arXiv:1810.05338

[17] T. Kohler and T. Cubitt, Toy models of holographic duality between local hamiltonians, Journal of High Energy Physics 2019, 17 (2019).
https:/​/​doi.org/​10.1007/​JHEP08(2019)017

[18] M. Gschwendtner, R. König, B. Şahinoğlu, and E. Tang, Quantum error-detection at low energies, Journal of High Energy Physics 2019, 21 (2019).
https:/​/​doi.org/​10.1007/​JHEP09(2019)021

[19] F. G. S. L. Brandão, E. Crosson, M. B. Şahinoğlu, and J. Bowen, Quantum error correcting codes in eigenstates of translation-invariant spin chains, Physical Review Letters 123, 110502 (2019).
https:/​/​doi.org/​10.1103/​physrevlett.123.110502

[20] C. Bény and O. Oreshkov, General conditions for approximate quantum error correction and near-optimal recovery channels, Physical Review Letters 104, 120501 (2010).
https:/​/​doi.org/​10.1103/​physrevlett.104.120501

[21] P. Hayden, M. Horodecki, A. Winter, and J. Yard, A decoupling approach to the quantum capacity, Open Systems & Information Dynamics 15, 7 (2008).
https:/​/​doi.org/​10.1142/​S1230161208000043

[22] C. Bény, Z. Zimborás, and F. Pastawski, Approximate recovery with locality and symmetry constraints, (2018), arXiv:1806.10324 [quant-ph].
arXiv:1806.10324

[23] V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nature Photonics 5, 222 (2011).
https:/​/​doi.org/​10.1038/​nphoton.2011.35

[24] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Reviews of Modern Physics 89, 035002 (2017).
https:/​/​doi.org/​10.1103/​RevModPhys.89.035002

[25] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, Quantum-enhanced measurements without entanglement, Reviews of Modern Physics 90, 035006 (2018).
https:/​/​doi.org/​10.1103/​RevModPhys.90.035006

[26] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Reviews of Modern Physics 90, 035005 (2018).
https:/​/​doi.org/​10.1103/​RevModPhys.90.035005

[27] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nature Photonics 12, 724 (2018).
https:/​/​doi.org/​10.1038/​s41566-018-0301-6

[28] B. Escher, R. de Matos Filho, and L. Davidovich, General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology, Nature Physics 7, 406 (2011).
https:/​/​doi.org/​10.1038/​nphys1958

[29] R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, The elusive heisenberg limit in quantum-enhanced metrology, Nature Communications 3, 1063 (2012).
https:/​/​doi.org/​10.1038/​ncomms2067

[30] R. Demkowicz-Dobrzański and L. Maccone, Using entanglement against noise in quantum metrology, Physical Review Letters 113, 250801 (2014).
https:/​/​doi.org/​10.1103/​physrevlett.113.250801

[31] H. Yuan and C.-H. F. Fung, Quantum parameter estimation with general dynamics, npj Quantum Information 3, 1 (2017a).
https:/​/​doi.org/​10.1038/​s41534-017-0014-6

[32] R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski, Adaptive quantum metrology under general markovian noise, Physical Review X 7, 041009 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.041009

[33] S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Achieving the heisenberg limit in quantum metrology using quantum error correction, Nature Communications 9, 78 (2018).
https:/​/​doi.org/​10.1038/​s41467-017-02510-3

[34] S. Zhou and L. Jiang, Asymptotic theory of quantum channel estimation, PRX Quantum 2 (2021).
https:/​/​doi.org/​10.1103/​prxquantum.2.010343

[35] A. Fujiwara and H. Imai, A fibre bundle over manifolds of quantum channels and its application to quantum statistics, Journal of Physics A: Mathematical and Theoretical 41, 255304 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​25/​255304

[36] M. Hayashi, Comparison between the cramer-rao and the mini-max approaches in quantum channel estimation, Communications in Mathematical Physics 304, 689 (2011).
https:/​/​doi.org/​10.1007/​s00220-011-1239-4

[37] H. Yuan and C.-H. F. Fung, Fidelity and fisher information on quantum channels, New Journal of Physics 19, 113039 (2017b).
https:/​/​doi.org/​10.1088/​1367-2630/​aa874c

[38] V. Katariya and M. M. Wilde, Geometric distinguishability measures limit quantum channel estimation and discrimination, Quantum Information Processing 20 (2021).
https:/​/​doi.org/​10.1007/​s11128-021-02992-7

[39] G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: manipulations and monotones, New Journal of Physics 10, 033023 (2008).
https:/​/​doi.org/​10.1088/​1367-2630/​10/​3/​033023

[40] I. Marvian and R. W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Physical Review A 94, 052324 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.052324

[41] I. Marvian and R. W. Spekkens, Extending noether’s theorem by quantifying the asymmetry of quantum states, Nature communications 5, 1 (2014).
https:/​/​doi.org/​10.1038/​ncomms4821

[42] K. Fang and Z.-W. Liu, No-go theorems for quantum resource purification, Physical Review Letters 125, 060405 (2020).
https:/​/​doi.org/​10.1103/​physrevlett.125.060405

[43] B. Regula, K. Bu, R. Takagi, and Z.-W. Liu, Benchmarking one-shot distillation in general quantum resource theories, Physical Review A 101 (2020).
https:/​/​doi.org/​10.1103/​physreva.101.062315

[44] I. Marvian, Coherence distillation machines are impossible in quantum thermodynamics, Nature Communications 11, 1 (2020).
https:/​/​doi.org/​10.1038/​s41467-019-13846-3

[45] B. Schumacher, Sending entanglement through noisy quantum channels, Physical Review A 54, 2614 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.2614

[46] A. Gilchrist, N. K. Langford, and M. A. Nielsen, Distance measures to compare real and ideal quantum processes, Physical Review A 71, 062310 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.062310

[47] A. Kubica and R. Demkowicz-Dobrzański, Using quantum metrological bounds in quantum error correction: A simple proof of the approximate eastin-knill theorem, Physical Review Letters 126 (2021).
https:/​/​doi.org/​10.1103/​physrevlett.126.150503

[48] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Quantum error correction for metrology, Physical Review Letters 112, 150802 (2014).
https:/​/​doi.org/​10.1103/​physrevlett.112.150802

[49] G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, Increasing sensing resolution with error correction, Physical Review Letters 112, 150801 (2014).
https:/​/​doi.org/​10.1103/​physrevlett.112.150801

[50] W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, Improved quantum metrology using quantum error correction, Physical Review Letters 112, 080801 (2014).
https:/​/​doi.org/​10.1103/​physrevlett.112.080801

[51] X.-M. Lu, S. Yu, and C. Oh, Robust quantum metrological schemes based on protection of quantum fisher information, Nature Communications 6, 7282 (2015).
https:/​/​doi.org/​10.1038/​ncomms8282

[52] F. Reiter, A. S. Sørensen, P. Zoller, and C. Muschik, Dissipative quantum error correction and application to quantum sensing with trapped ions, Nature Communications 8, 1822 (2017).
https:/​/​doi.org/​10.1038/​s41467-017-01895-5

[53] P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, Quantum metrology with full and fast quantum control, Quantum 1, 27 (2017).
https:/​/​doi.org/​10.22331/​q-2017-09-06-27

[54] T. Kapourniotis and A. Datta, Fault-tolerant quantum metrology, Physical Review A 100, 022335 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.022335

[55] D. Layden and P. Cappellaro, Spatial noise filtering through error correction for quantum sensing, npj Quantum Information 4, 30 (2018).
https:/​/​doi.org/​10.1038/​s41534-018-0082-2

[56] D. Layden, S. Zhou, P. Cappellaro, and L. Jiang, Ancilla-free quantum error correction codes for quantum metrology, Physical Review Letters 122, 040502 (2019).
https:/​/​doi.org/​10.1103/​physrevlett.122.040502

[57] S. Zhou and L. Jiang, Optimal approximate quantum error correction for quantum metrology, Physical Review Research 2, 013235 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.013235

[58] C. W. Helstrom, Quantum detection and estimation theory (Academic press, 1976).

[59] A. S. Holevo, Probabilistic and statistical aspects of quantum theory, Vol. 1 (Springer Science & Business Media, 2011).

[60] M. G. Paris, Quantum estimation for quantum technology, International Journal of Quantum Information 7, 125 (2009).
https:/​/​doi.org/​10.1142/​S0219749909004839

[61] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Physical Review Letters 72, 3439 (1994).
https:/​/​doi.org/​10.1103/​physrevlett.72.3439

[62] H. Yuen and M. Lax, Multiple-parameter quantum estimation and measurement of nonselfadjoint observables, IEEE Transactions on Information Theory 19, 740 (1973).
https:/​/​doi.org/​10.1109/​tit.1973.1055103

[63] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Physical Review Letters 96, 010401 (2006).
https:/​/​doi.org/​10.1103/​physrevlett.96.010401

[64] J. Kołodyński and R. Demkowicz-Dobrzański, Efficient tools for quantum metrology with uncorrelated noise, New Journal of Physics 15, 073043 (2013).
https:/​/​doi.org/​10.1088/​1367-2630/​15/​7/​073043

[65] A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Reviews of Modern Physics 89, 041003 (2017).
https:/​/​doi.org/​10.1103/​RevModPhys.89.041003

[66] K. Fang and Z.-W. Liu, No-go theorems for quantum resource purification: new approach and channel theory, (2021), arXiv:2010.11822 [quant-ph].
arXiv:2010.11822

[67] B. Regula and R. Takagi, One-shot manipulation of dynamical quantum resources, (2020), arXiv:2012.02215 [quant-ph].
https:/​/​doi.org/​10.1103/​PhysRevLett.127.060402
arXiv:2012.02215

[68] Y. Ouyang, N. Shettell, and D. Markham, Robust quantum metrology with explicit symmetric states, (2019), arXiv:1908.02378 [quant-ph].
arXiv:1908.02378

[69] D. Gottesman, Quantum fault tolerance in small experiments, (2016), arXiv:1610.03507 [quant-ph].
arXiv:1610.03507

[70] Z.-W. Liu and A. Winter, Resource theories of quantum channels and the universal role of resource erasure, (2019), arXiv:1904.04201 [quant-ph].
arXiv:1904.04201

[71] Y. Liu and X. Yuan, Operational resource theory of quantum channels, Physical Review Research 2, 012035 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.012035

[72] Y. Yang, Y. Mo, J. M. Renes, G. Chiribella, and M. P. Woods, Covariant quantum error correcting codes via reference frames, (2020), arXiv:2007.09154 [quant-ph].
arXiv:2007.09154

[73] H. Komiya, Elementary proof for sion's minimax theorem, Kodai Mathematical Journal 11, 5 (1988).
https:/​/​doi.org/​10.2996/​kmj/​1138038812

[74] M. do Rosário Grossinho and S. A. Tersian, An introduction to minimax theorems and their applications to differential equations, Vol. 52 (Springer Science & Business Media, 2001).

[75] P. Del Moral and A. Niclas, A taylor expansion of the square root matrix function, Journal of Mathematical Analysis and Applications 465, 259 (2018).
https:/​/​doi.org/​10.1016/​j.jmaa.2018.05.005

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[7] Michał Studziński, Marek Mozrzymas, Piotr Kopszak, and Michał Horodecki, "Efficient multi-port teleportation schemes", arXiv:2008.00984.

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[16] D. -S. Wang, "A comparative study of universal quantum computing models: towards a physical unification", arXiv:2108.07909.

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