Recovery With Incomplete Knowledge: Fundamental Bounds on Real-Time Quantum Memories

Arshag Danageozian

Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA

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Abstract

The recovery of fragile quantum states from decoherence is the basis of building a quantum memory, with applications ranging from quantum communications to quantum computing. Many recovery techniques, such as quantum error correction, rely on the $apriori$ knowledge of the environment noise parameters to achieve their best performance. However, such parameters are likely to drift in time in the context of implementing long-time quantum memories. This necessitates using a "spectator" system, which estimates the noise parameter in real-time, then feed-forwards the outcome to the recovery protocol as a classical side-information. The memory qubits and the spectator system hence comprise the building blocks for a real-time (i.e. drift-adapting) quantum memory. In this article, I consider spectator-based (incomplete knowledge) recovery protocols as a real-time parameter estimation problem (generally with nuisance parameters present), followed by the application of the "best-guess" recovery map to the memory qubits, as informed by the estimation outcome. I present information-theoretic and metrological bounds on the performance of this protocol, quantified by the diamond distance between the "best-guess" recovery and optimal recovery outcomes, thereby identifying the cost of adaptation in real-time quantum memories. Finally, I provide fundamental bounds for multi-cycle recovery in the form of recurrence inequalities. The latter suggests that incomplete knowledge of the noise could be an advantage, as errors from various cycles can cohere. These results are illustrated for the approximate [4,1] code of the amplitude-damping channel and relations to various fields are discussed.

Noise drift in quantum hardware is an important obstacle for scalable quantum computation and communication, e.g. when building a long-time quantum memory that protects encoded information from decoherence. For better protection of quantum information under time-varying noise, this work proposes the inclusion of spectator qubits that are co-located with their memory counterparts. These qubits monitor the change in the noise parameters in real-time by performing quantum multi-parameter estimation, thereby providing the most up-to-date information on the optimal correction parameters for errors (i.e. decoherence) in the quantum memory. The author derives a fundamental bound on the performance of such a noise-adaptive quantum memory as a function of the physical properties of the spectator qubits used, thereby providing a quantitative formula for selecting better versus worse spectators for a given quantum hardware. The article provides a consistent framework for combining quantum information theory, quantum error correction, and quantum parameter estimation, in the pursuit of quantum memories with longer coherence times.

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► References

[1] Mustafa Gündoğan, Jasminder S Sidhu, Victoria Henderson, Luca Mazzarella, Janik Wolters, Daniel KL Oi, and Markus Krutzik. ``Proposal for space-borne quantum memories for global quantum networking''. npj Quantum Information 7, 128 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00460-9

[2] Julius Wallnöfer, Frederik Hahn, Mustafa Gündoğan, Jasminder S Sidhu, Fabian Wiesner, Nathan Walk, Jens Eisert, and Janik Wolters. ``Simulating quantum repeater strategies for multiple satellites''. Communications Physics 5, 169 (2022).
https:/​/​doi.org/​10.1038/​s42005-022-00945-9

[3] Mustafa Gündoğan, Jasminder J Sidhu, Daniel KL Oi, and Markus Krutzik. ``Time-delayed single quantum repeater node for global quantum communications with a single satellite'' (2023).

[4] Jasminder S Sidhu, Yingkai Ouyang, Earl T Campbell, and Pieter Kok. ``Tight bounds on the simultaneous estimation of incompatible parameters''. Physical Review X 11, 011028 (2021).
https:/​/​doi.org/​10.1103/​PhysRevX.11.011028

[5] Ye Wang, Mark Um, Junhua Zhang, Shuoming An, Ming Lyu, Jing-Ning Zhang, L-M Duan, Dahyun Yum, and Kihwan Kim. ``Single-qubit quantum memory exceeding ten-minute coherence time''. Nature Photonics 11, 646–650 (2017).
https:/​/​doi.org/​10.1038/​s41566-017-0007-1

[6] P-J Stas, Yan Qi Huan, Bartholomeus Machielse, Erik N Knall, Aziza Suleymanzade, Benjamin Pingault, Madison Sutula, Sophie W Ding, Can M Knaut, Daniel R Assumpcao, et al. ``Robust multi-qubit quantum network node with integrated error detection''. Science 378, 557–560 (2022).
https:/​/​doi.org/​10.1126/​science.add9771

[7] Benjamin Schumacher and Michael A Nielsen. ``Quantum data processing and error correction''. Physical Review A 54, 2629 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.2629

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

[9] Emanuel Knill and Raymond Laflamme. ``Theory of quantum error-correcting codes''. Physical Review A 55, 900 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.55.900

[10] Lorenza Viola and Seth Lloyd. ``Dynamical suppression of decoherence in two-state quantum systems''. Physical Review A 58, 2733 (1998).
https:/​/​doi.org/​10.1103/​PhysRevA.58.2733

[11] Lorenza Viola, Emanuel Knill, and Seth Lloyd. ``Dynamical decoupling of open quantum systems''. Physical Review Letters 82, 2417 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.82.2417

[12] Andrew S Fletcher, Peter W Shor, and Moe Z Win. ``Channel-adapted quantum error correction for the amplitude damping channel''. IEEE Transactions on Information Theory 54, 5705–5718 (2008).
https:/​/​doi.org/​10.1109/​TIT.2008.2006458

[13] Debbie W Leung, Michael A Nielsen, Isaac L Chuang, and Yoshihisa Yamamoto. ``Approximate quantum error correction can lead to better codes''. Physical Review A 56, 2567 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.56.2567

[14] Andrew S Fletcher, Peter W Shor, and Moe Z Win. ``Optimum quantum error recovery using semidefinite programming''. Physical Review A 75, 012338 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.012338

[15] Heinz-Peter Breuer, Francesco Petruccione, et al. ``The theory of open quantum systems''. Oxford University Press on Demand. (2002).
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[16] Clemens Müller, Jürgen Lisenfeld, Alexander Shnirman, and Stefano Poletto. ``Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits''. Physical Review B 92, 035442 (2015).
https:/​/​doi.org/​10.1103/​PhysRevB.92.035442

[17] PV Klimov, Julian Kelly, Z Chen, Matthew Neeley, Anthony Megrant, Brian Burkett, Rami Barends, Kunal Arya, Ben Chiaro, Yu Chen, et al. ``Fluctuations of energy-relaxation times in superconducting qubits''. Physical review letters 121, 090502 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.090502

[18] Josu Etxezarreta Martinez, Patricio Fuentes, Pedro Crespo, and Javier Garcia-Frias. ``Time-varying quantum channel models for superconducting qubits''. npj Quantum Information 7, 1–10 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00448-5

[19] Samudra Dasgupta and Travis S Humble. ``Characterizing the stability of nisq devices''. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE). Pages 419–429. IEEE (2020).
https:/​/​doi.org/​10.1109/​QCE49297.2020.00059

[20] Samudra Dasgupta and Travis S Humble. ``Stability of noisy quantum computing devices'' (2021).

[21] Cristian Bonato and Dominic W Berry. ``Adaptive tracking of a time-varying field with a quantum sensor''. Physical Review A 95, 052348 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.052348

[22] Luis Cortez, Areeya Chantasri, Luis Pedro García-Pintos, Justin Dressel, and Andrew N Jordan. ``Rapid estimation of drifting parameters in continuously measured quantum systems''. Physical Review A 95, 012314 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.012314

[23] Timothy Proctor, Melissa Revelle, Erik Nielsen, Kenneth Rudinger, Daniel Lobser, Peter Maunz, Robin Blume-Kohout, and Kevin Young. ``Detecting and tracking drift in quantum information processors''. Nature communications 11, 1–9 (2020).
https:/​/​doi.org/​10.1038/​s41467-020-19074-4

[24] Swarnadeep Majumder, Leonardo Andreta de Castro, and Kenneth R Brown. ``Real-time calibration with spectator qubits''. npj Quantum Information 6, 1–9 (2020).
https:/​/​doi.org/​10.1038/​s41534-020-0251-y

[25] Riddhi Swaroop Gupta, Luke CG Govia, and Michael J Biercuk. ``Integration of spectator qubits into quantum computer architectures for hardware tune-up and calibration''. Physical Review A 102, 042611 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.042611

[26] Akram Youssry, Gerardo A Paz-Silva, and Christopher Ferrie. ``Noise detection with spectator qubits and quantum feature engineering''. New Journal of Physics 25, 073004 (2023).
https:/​/​doi.org/​10.1088/​1367-2630/​ace2e4

[27] Arshag Danageozian, Ashe Miller, Pratik J Barge, Narayan Bhusal, and Jonathan P Dowling. ``Noisy coherent population trapping: applications to noise estimation and qubit state preparation''. Journal of Physics B: Atomic, Molecular and Optical Physics 55, 155503 (2022).
https:/​/​doi.org/​10.1088/​1361-6455/​ac7760

[28] Hongting Song, Areeya Chantasri, Behnam Tonekaboni, and Howard M Wiseman. ``Optimized mitigation of random-telegraph-noise dephasing by spectator-qubit sensing and control''. Physical Review A 107, L030601 (2023).
https:/​/​doi.org/​10.1103/​PhysRevA.107.L030601

[29] Behnam Tonekaboni, Areeya Chantasri, Hongting Song, Yanan Liu, and Howard M Wiseman. ``Greedy versus map-based optimized adaptive algorithms for random-telegraph-noise mitigation by spectator qubits''. Physical Review A 107, 032401 (2023).
https:/​/​doi.org/​10.1103/​PhysRevA.107.032401

[30] K Singh, CE Bradley, S Anand, V Ramesh, R White, and H Bernien. ``Mid-circuit correction of correlated phase errors using an array of spectator qubits''. Science 380, eade5337 (2023).
https:/​/​doi.org/​10.1126/​science.ade5337

[31] Bas Hensen, Hannes Bernien, Anaïs E Dréau, Andreas Reiserer, Norbert Kalb, Machiel S Blok, Just Ruitenberg, Raymond FL Vermeulen, Raymond N Schouten, Carlos Abellán, et al. ``Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres''. Nature 526, 682–686 (2015).
https:/​/​doi.org/​10.1038/​nature15759

[32] Emre Togan, Yiwen Chu, Alexei S Trifonov, Liang Jiang, Jeronimo Maze, Lilian Childress, MV Gurudev Dutt, Anders Søndberg Sørensen, Phillip R Hemmer, Alexander S Zibrov, et al. ``Quantum entanglement between an optical photon and a solid-state spin qubit''. Nature 466, 730–734 (2010).
https:/​/​doi.org/​10.1038/​nature09256

[33] D Andrew Golter and Hailin Wang. ``Optically driven rabi oscillations and adiabatic passage of single electron spins in diamond''. Physical review letters 112, 116403 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.116403

[34] Shu-Hao Wu, Ethan Turner, and Hailin Wang. ``Continuous real-time sensing with a nitrogen-vacancy center via coherent population trapping''. Physical Review A 103, 042607 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.042607

[35] Ethan Turner, Shu-Hao Wu, Xinzhu Li, and Hailin Wang. ``Real-time magnetometry with coherent population trapping in a nitrogen-vacancy center''. Physical Review A 105, L010601 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.105.L010601

[36] Ignas Lekavicius, D Andrew Golter, Thein Oo, and Hailin Wang. ``Transfer of phase information between microwave and optical fields via an electron spin''. Physical Review Letters 119, 063601 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.063601

[37] Michael A Nielsen and Isaac Chuang. ``Quantum computation and quantum information: 10th anniversary edition'' (2002).

[38] Samuel L Braunstein and Carlton M Caves. ``Statistical distance and the geometry of quantum states''. Physical Review Letters 72, 3439 (1994).
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[39] Samuel L Braunstein, Carlton M Caves, and Gerard J Milburn. ``Generalized uncertainty relations: theory, examples, and lorentz invariance''. annals of physics 247, 135–173 (1996).
https:/​/​doi.org/​10.1006/​aphy.1996.0040

[40] Yingkai Ouyang, Kaumudibikash Goswami, Jacquiline Romero, Barry C. Sanders, Min-Hsiu Hsieh, and Marco Tomamichel. ``Approximate reconstructability of quantum states and noisy quantum secret sharing schemes''. Phys. Rev. A 108, 012425 (2023).
https:/​/​doi.org/​10.1103/​PhysRevA.108.012425

[41] Stefano Pirandola, Riccardo Laurenza, Cosmo Lupo, and Jason L Pereira. ``Fundamental limits to quantum channel discrimination''. npj Quantum Information 5, 1–8 (2019).
https:/​/​doi.org/​10.1038/​s41534-019-0162-y

[42] Giuliano Benenti and Giuliano Strini. ``Computing the distance between quantum channels: usefulness of the fano representation''. Journal of Physics B: Atomic, Molecular and Optical Physics 43, 215508 (2010).
https:/​/​doi.org/​10.1088/​0953-4075/​43/​21/​215508

[43] Arnaud Carignan-Dugas, Joel J Wallman, and Joseph Emerson. ``Bounding the average gate fidelity of composite channels using the unitarity''. New Journal of Physics 21, 053016 (2019).
https:/​/​doi.org/​10.1088/​1367-2630/​ab1800

[44] Julia Cramer, Norbert Kalb, M Adriaan Rol, Bas Hensen, Machiel S Blok, Matthew Markham, Daniel J Twitchen, Ronald Hanson, and Tim H Taminiau. ``Repeated quantum error correction on a continuously encoded qubit by real-time feedback''. Nature communications 7, 1–7 (2016).
https:/​/​doi.org/​10.1038/​ncomms11526

[45] Bharat Thotakura and Tzu-Chieh Wei. ``Quantum state transfer: interplay between gate and readout errors''. Quantum Information Processing 22, 275 (2023).
https:/​/​doi.org/​10.1007/​s11128-023-04030-0

[46] Michael Reimpell and Reinhard F Werner. ``Iterative optimization of quantum error correcting codes''. Physical review letters 94, 080501 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.94.080501

[47] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``General teleportation channel, singlet fraction, and quasidistillation''. Physical Review A 60, 1888 (1999).
https:/​/​doi.org/​10.1103/​PhysRevA.60.1888

[48] Aleksander Kubica and Rafał Demkowicz-Dobrzański. ``Using quantum metrological bounds in quantum error correction: a simple proof of the approximate Eastin-Knill theorem''. Physical Review Letters 126, 150503 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.150503

[49] Cédric Bény and Ognyan 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

[50] Marius Junge, Renato Renner, David Sutter, Mark M Wilde, and Andreas Winter. ``Universal recovery maps and approximate sufficiency of quantum relative entropy''. In Annales Henri Poincaré. Volume 19, pages 2955–2978. Springer (2018).
https:/​/​doi.org/​10.1007/​s00023-018-0716-0

[51] Francesco Buscemi, Siddhartha Das, and Mark M Wilde. ``Approximate reversibility in the context of entropy gain, information gain, and complete positivity''. Physical Review A 93, 062314 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.062314

[52] Antonio Acin. ``Statistical distinguishability between unitary operations''. Physical review letters 87, 177901 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.87.177901

[53] John Watrous. ``Semidefinite programs for completely bounded norms'' (2009).

[54] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Physical Review A 80, 012304 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.012304

[55] Gilad Gour. ``Comparison of quantum channels by superchannels''. IEEE Transactions on Information Theory 65, 5880–5904 (2019).
https:/​/​doi.org/​10.1109/​TIT.2019.2907989

[56] Hyukjoon Kwon, Rick Mukherjee, and MS Kim. ``Reversing lindblad dynamics via continuous petz recovery map''. Physical Review Letters 128, 020403 (2022).
https:/​/​doi.org/​10.1103/​PhysRevLett.128.020403

[57] Robert L Kosut, Alireza Shabani, and Daniel A Lidar. ``Robust quantum error correction via convex optimization''. Physical review letters 100, 020502 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.100.020502

[58] Gábor Balló and Péter Gurin. ``Robustness of channel-adapted quantum error correction''. Physical Review A 80, 012326 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.012326

[59] Long Huang, Xiaohua Wu, and Tao Zhou. ``Robustness of the concatenated quantum error-correction protocol against noise for channels affected by fluctuation''. Physical Review A 100, 042321 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.042321

[60] David Layden, Louisa Ruixue Huang, and Paola Cappellaro. ``Robustness-optimized quantum error correction''. Quantum Science and Technology 5, 025008 (2020).
https:/​/​doi.org/​10.1088/​2058-9565/​ab79b2

[61] Robert L Kosut and Daniel A Lidar. ``Quantum error correction via convex optimization''. Quantum Information Processing 8, 443–459 (2009).
https:/​/​doi.org/​10.1007/​s11128-009-0120-2

[62] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. California Institute of Technology. (1997).
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​9705052
arXiv:quant-ph/9705052

[63] Jun Suzuki, Yuxiang Yang, and Masahito Hayashi. ``Quantum state estimation with nuisance parameters''. Journal of Physics A: Mathematical and Theoretical 53, 453001 (2020).
https:/​/​doi.org/​10.1088/​1751-8121/​ab8b78

[64] Ole E Barndorff-Nielsen and David Roxbee Cox. ``Inference and asymptotics''. Volume 13. Springer. (1994).
https:/​/​doi.org/​10.1201/​9780203750940

[65] Jing Liu, Haidong Yuan, Xiao-Ming Lu, and Xiaoguang Wang. ``Quantum fisher information matrix and multiparameter estimation''. Journal of Physics A: Mathematical and Theoretical 53, 023001 (2020).
https:/​/​doi.org/​10.1088/​1751-8121/​ab5d4d

[66] Sammy Ragy, Marcin Jarzyna, and Rafał Demkowicz-Dobrzański. ``Compatibility in multiparameter quantum metrology''. Physical Review A 94, 052108 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.052108

[67] Federico Belliardo and Vittorio Giovannetti. ``Incompatibility in quantum parameter estimation''. New Journal of Physics 23, 063055 (2021).
https:/​/​doi.org/​10.1088/​1367-2630/​ac04ca

[68] Xiao-Ming Lu and Xiaoguang Wang. ``Incorporating heisenberg’s uncertainty principle into quantum multiparameter estimation''. Physical Review Letters 126, 120503 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.120503

[69] Francesco Albarelli, Jamie F Friel, and Animesh Datta. ``Evaluating the holevo cramér-rao bound for multiparameter quantum metrology''. Physical review letters 123, 200503 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.200503

[70] Hiroshi Nagaoka. ``A generalization of the simultaneous diagonalization of hermitian matrices and its relation to quantum estimation theory''. In Asymptotic Theory of Quantum Statistical Inference: Selected Papers. Pages 133–149. World Scientific (2005).
https:/​/​doi.org/​10.1142/​9789812563071_0012

[71] Lorcán O Conlon, Jun Suzuki, Ping Koy Lam, and Syed M Assad. ``Efficient computation of the nagaoka–hayashi bound for multiparameter estimation with separable measurements''. npj Quantum Information 7, 110 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00414-1

[72] Masahito Hayashi and Yingkai Ouyang. ``Tight cramér-rao type bounds for multiparameter quantum metrology through conic programming''. Quantum 7, 1094 (2023).
https:/​/​doi.org/​10.22331/​q-2023-08-29-1094

[73] Christian Arenz, Daniel Burgarth, and Robin Hillier. ``Dynamical decoupling and homogenization of continuous variable systems''. Journal of Physics A: Mathematical and Theoretical 50, 135303 (2017).
https:/​/​doi.org/​10.1088/​1751-8121/​aa6017

[74] Chao Lei, Shijie Peng, Chenyong Ju, Man-Hong Yung, and Jiangfeng Du. ``Decoherence control of nitrogen-vacancy centers''. Scientific reports 7, 11937 (2017).
https:/​/​doi.org/​10.1038/​s41598-017-12280-z

[75] Tim Hugo Taminiau, Julia Cramer, Toeno van der Sar, Viatcheslav V Dobrovitski, and Ronald Hanson. ``Universal control and error correction in multi-qubit spin registers in diamond''. Nature nanotechnology 9, 171–176 (2014).
https:/​/​doi.org/​10.1038/​nnano.2014.2

[76] Gerald Waldherr, Ya Wang, S Zaiser, M Jamali, T Schulte-Herbrüggen, H Abe, T Ohshima, J Isoya, JF Du, P Neumann, et al. ``Quantum error correction in a solid-state hybrid spin register''. Nature 506, 204–207 (2014).
https:/​/​doi.org/​10.1038/​nature12919

[77] Haidong Yuan and Chi-Hang Fred Fung. ``Fidelity and fisher information on quantum channels''. New Journal of Physics 19, 113039 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa874c

[78] Jasminder S Sidhu and Pieter Kok. ``Geometric perspective on quantum parameter estimation''. AVS Quantum Science 2, 014701 (2020).
https:/​/​doi.org/​10.1116/​1.5119961

[79] Yun Zhan and Xiao-Yu Chen. ``Entanglement fidelity of channel adaptive quantum codes''. Chinese Physics B 22, 010308 (2013).
https:/​/​doi.org/​10.1088/​1674-1056/​22/​1/​010308

[80] Benjamin Rahn, Andrew C Doherty, and Hideo Mabuchi. ``Exact performance of concatenated quantum codes''. Physical Review A 66, 032304 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.66.032304

[81] Carlo Cafaro and Peter van Loock. ``Approximate quantum error correction for generalized amplitude-damping errors''. Physical Review A 89, 022316 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.022316

[82] Carlo Cafaro and Peter van Loock. ``A simple comparative analysis of exact and approximate quantum error correction''. Open Systems & Information Dynamics 21, 1450002 (2014).
https:/​/​doi.org/​10.1142/​S1230161214500024

[83] Bryan Eastin and Emanuel Knill. ``Restrictions on transversal encoded quantum gate sets''. Physical review letters 102, 110502 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.110502

[84] Vishal Katariya and Mark M Wilde. ``Geometric distinguishability measures limit quantum channel estimation and discrimination''. Quantum Information Processing 20, 1–170 (2021).
https:/​/​doi.org/​10.1007/​s11128-021-02992-7

[85] Akio Fujiwara. ``Estimation of a generalized amplitude-damping channel''. Physical Review A 70, 012317 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.70.012317

[86] Leslie Allen and Joseph H Eberly. ``Optical resonance and two-level atoms''. Volume 28. Courier Corporation. (1987).
https:/​/​doi.org/​10.1088/​0031-9112/​26/​12/​039

[87] Satoshi Ishizaka and Tohya Hiroshima. ``Asymptotic teleportation scheme as a universal programmable quantum processor''. Physical review letters 101, 240501 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.240501

[88] Giulio Chiribella, Giacomo M D’Ariano, and Paolo Perinotti. ``Memory effects in quantum channel discrimination''. Physical review letters 101, 180501 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.180501

[89] Masahito Hayashi. ``Discrimination of two channels by adaptive methods and its application to quantum system''. IEEE Transactions on Information Theory 55, 3807–3820 (2009).
https:/​/​doi.org/​10.1109/​TIT.2009.2023726

[90] Andrew S Fletcher. ``Channel-adapted quantum error correction'' (2007).

[91] Christian Arenz, Denys I Bondar, Daniel Burgarth, Cecilia Cormick, and Herschel Rabitz. ``Amplification of quadratic hamiltonians''. Quantum 4, 271 (2020).
https:/​/​doi.org/​10.22331/​q-2020-05-25-271

[92] Daniel Gottesman, Alexei Kitaev, and John Preskill. ``Encoding a qubit in an oscillator''. Physical Review A 64, 012310 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.012310

[93] Yuichiro Fujiwara. ``Instantaneous quantum channel estimation during quantum information processing'' (2014).

[94] Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Physical Review A 86, 032324 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[95] Stephen T Spitz, Brian Tarasinski, Carlo WJ Beenakker, and Thomas E O'Brien. ``Adaptive weight estimator for quantum error correction in a time-dependent environment''. Advanced Quantum Technologies 1, 1800012 (2018).
https:/​/​doi.org/​10.1002/​qute.201800012

[96] J Kelly, R Barends, AG Fowler, A Megrant, E Jeffrey, TC White, D Sank, JY Mutus, B Campbell, Yu Chen, et al. ``Scalable in situ qubit calibration during repetitive error detection''. Physical Review A 94, 032321 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.032321

[97] Ming-Xia Huo and Ying Li. ``Learning time-dependent noise to reduce logical errors: real time error rate estimation in quantum error correction''. New Journal of Physics 19, 123032 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa916e

[98] Soraya Taghavi, Robert L Kosut, and Daniel A Lidar. ``Channel-optimized quantum error correction''. IEEE transactions on information theory 56, 1461–1473 (2010).
https:/​/​doi.org/​10.1109/​TIT.2009.2039162

[99] Jan Florjanczyk and Todd A Brun. ``In-situ adaptive encoding for asymmetric quantum error correcting codes'' (2016).

[100] Jon Tyson. ``Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized holevo–curlander bounds''. Journal of mathematical physics 50, 032106 (2009).
https:/​/​doi.org/​10.1063/​1.3094322

[101] Jon Tyson. ``Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates''. Journal of mathematical physics 51, 092204 (2010).
https:/​/​doi.org/​10.1063/​1.3463451

[102] Howard Barnum and Emanuel Knill. ``Reversing quantum dynamics with near-optimal quantum and classical fidelity''. Journal of Mathematical Physics 43, 2097–2106 (2002).
https:/​/​doi.org/​10.1063/​1.1459754

[103] Akshaya Jayashankar and Prabha Mandayam. ``Quantum error correction: Noise-adapted techniques and applications''. Journal of the Indian Institute of SciencePages 1–16 (2022).
https:/​/​doi.org/​10.1007/​s41745-022-00332-x

[104] Debjyoti Biswas, Gaurav M Vaidya, and Prabha Mandayam. ``Noise-adapted recovery circuits for quantum error correction'' (2023).

[105] Rémi Azouit, Francesca Chittaro, Alain Sarlette, and Pierre Rouchon. ``Towards generic adiabatic elimination for bipartite open quantum systems''. Quantum Science and Technology 2, 044011 (2017).
https:/​/​doi.org/​10.1088/​2058-9565/​aa7f3f

[106] Rochus Klesse and Sandra Frank. ``Quantum error correction in spatially correlated quantum noise''. Physical review letters 95, 230503 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.95.230503

[107] Ke Li and Andreas Winter. ``Squashed entanglement, k-extendibility, quantum markov chains, and recovery maps''. Foundations of Physics 48, 910–924 (2018).
https:/​/​doi.org/​10.1007/​s10701-018-0143-6

[108] Mark M Wilde. ``Recoverability in quantum information theory''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, 20150338 (2015).
https:/​/​doi.org/​10.1098/​rspa.2015.0338

[109] Omar Fawzi and Renato Renner. ``Quantum conditional mutual information and approximate markov chains''. Communications in Mathematical Physics 340, 575–611 (2015).
https:/​/​doi.org/​10.1007/​s00220-015-2466-x

[110] Richard Kueng, David M Long, Andrew C Doherty, and Steven T Flammia. ``Comparing experiments to the fault-tolerance threshold''. Physical review letters 117, 170502 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.170502

[111] Panos Aliferis. ``Level reduction and the quantum threshold theorem'' (2007).

[112] Alexei Gilchrist, Nathan K Langford, and Michael 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

[113] Andrea Smirne, Nina Megier, and Bassano Vacchini. ``Holevo skew divergence for the characterization of information backflow''. Phys. Rev. A 106, 012205 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.106.012205

[114] Felix Leditzky, Eneet Kaur, Nilanjana Datta, and Mark M Wilde. ``Approaches for approximate additivity of the holevo information of quantum channels''. Physical Review A 97, 012332 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.012332

[115] Sumeet Khatri and Mark M Wilde. ``Principles of quantum communication theory: A modern approach'' (2020).

[116] Mark M Wilde, Mario Berta, Christoph Hirche, and Eneet Kaur. ``Amortized channel divergence for asymptotic quantum channel discrimination''. Letters in Mathematical Physics 110, 2277–2336 (2020).
https:/​/​doi.org/​10.1007/​s11005-020-01297-7

[117] David Gross, Koenraad Audenaert, and Jens Eisert. ``Evenly distributed unitaries: On the structure of unitary designs''. Journal of mathematical physics 48, 052104 (2007).
https:/​/​doi.org/​10.1063/​1.2716992

[118] Christoph Dankert. ``Efficient simulation of random quantum states and operators'' (2005).

[119] Joseph Emerson, Marcus Silva, Osama Moussa, Colm Ryan, Martin Laforest, Jonathan Baugh, David G Cory, and Raymond Laflamme. ``Symmetrized characterization of noisy quantum processes''. Science 317, 1893–1896 (2007).
https:/​/​doi.org/​10.1126/​science.1145699

[120] Tilo Eggeling and Reinhard F Werner. ``Separability properties of tripartite states with u $\otimes$ u $\otimes$ u symmetry''. Physical Review A 63, 042111 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.63.042111

[121] Marcus Silva, Easwar Magesan, David W Kribs, and Joseph Emerson. ``Scalable protocol for identification of correctable codes''. Physical Review A 78, 012347 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.012347

[122] Easwar Magesan. ``Gaining information about a quantum channel via twirling''. Master's thesis. University of Waterloo. (2008).

[123] Josu Etxezarreta Martinez, Patricio Fuentes, Pedro M Crespo, and Javier Garcia-Frias. ``Approximating decoherence processes for the design and simulation of quantum error correction codes on classical computers''. IEEE Access 8, 172623–172643 (2020).
https:/​/​doi.org/​10.1109/​ACCESS.2020.3025619

[124] Adam M Meier. ``Randomized benchmarking of clifford operators''. PhD thesis. University of Colorado at Boulder. (2013).

[125] Daniel Gottesman. ``The heisenberg representation of quantum computers'' (1998).

[126] Michael A Nielsen. ``A simple formula for the average gate fidelity of a quantum dynamical operation''. Physics Letters A 303, 249–252 (2002).
https:/​/​doi.org/​10.1016/​S0375-9601(02)01272-0

[127] Giulio Chiribella, G Mauro D'Ariano, and Paolo Perinotti. ``Transforming quantum operations: Quantum supermaps''. EPL (Europhysics Letters) 83, 30004 (2008).
https:/​/​doi.org/​10.1209/​0295-5075/​83/​30004

[128] Rajendra Bhatia. ``Matrix analysis''. Volume 169. Springer Science & Business Media. (2013).
https:/​/​doi.org/​10.1007/​978-1-4612-0653-8

[129] Dénes Petz and Catalin Ghinea. ``Introduction to quantum fisher information''. In Quantum probability and related topics. Pages 261–281. World Scientific (2011).
https:/​/​doi.org/​10.1142/​9789814338745_0015

[130] Masahito Hayashi. ``Asymptotic theory of quantum statistical inference: selected papers''. World Scientific. (2005).
https:/​/​doi.org/​10.1142/​5630

[131] Hiroshi Hasegawa. ``Exponential and mixture families in quantum statistics: Dual structure and unbiased parameter estimation''. Reports on Mathematical Physics 39, 49–68 (1997).
https:/​/​doi.org/​10.1016/​S0034-4877(97)81470-X

[132] Larry Wasserman. ``All of statistics: a concise course in statistical inference''. Volume 26. Springer. (2004).
https:/​/​doi.org/​10.1007/​978-0-387-21736-9

[133] Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel. ``On quantum rényi entropies: A new generalization and some properties''. Journal of Mathematical Physics 54, 122203 (2013).
https:/​/​doi.org/​10.1063/​1.4838856

[134] Nathaniel Johnston and David W Kribs. ``Quantum gate fidelity in terms of choi matrices''. Journal of Physics A: Mathematical and Theoretical 44, 495303 (2011).
https:/​/​doi.org/​10.1088/​1751-8113/​44/​49/​495303

[135] Isaac L Chuang and Michael A Nielsen. ``Prescription for experimental determination of the dynamics of a quantum black box''. Journal of Modern Optics 44, 2455–2467 (1997).
https:/​/​doi.org/​10.1080/​09500349708231894

[136] Masoud Mohseni, Ali T Rezakhani, and Daniel A Lidar. ``Quantum-process tomography: Resource analysis of different strategies''. Physical Review A 77, 032322 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.77.032322

[137] Emanuel Knill, Dietrich Leibfried, Rolf Reichle, Joe Britton, R Brad Blakestad, John D Jost, Chris Langer, Roee Ozeri, Signe Seidelin, and David J Wineland. ``Randomized benchmarking of quantum gates''. Physical Review A 77, 012307 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.77.012307

[138] Joshua Combes, Christopher Ferrie, Chris Cesare, Markus Tiersch, Gerard J Milburn, Hans J Briegel, and Carlton M Caves. ``In-situ characterization of quantum devices with error correction'' (2014).

[139] Shelby Kimmel, Marcus P da Silva, Colm A Ryan, Blake R Johnson, and Thomas Ohki. ``Robust extraction of tomographic information via randomized benchmarking''. Physical Review X 4, 011050 (2014).
https:/​/​doi.org/​10.1103/​PhysRevX.4.011050

Cited by

[1] Samudra Dasgupta, Travis S. Humble, and Arshag Danageozian, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 99 (2023) ISBN:979-8-3503-4323-6.

[2] Samudra Dasgupta and Travis S. Humble, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 223 (2023) ISBN:979-8-3503-4323-6.

[3] Samudra Dasgupta, Arshag Danageozian, and Travis S. Humble, "Adaptive mitigation of time-varying quantum noise", arXiv:2308.14756, (2023).

[4] Samudra Dasgupta and Travis S. Humble, "Reliable Devices Yield Stable Quantum Computations", arXiv:2307.05381, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 07:59:12) and SAO/NASA ADS (last updated successfully 2024-06-22 07:59:12). The list may be incomplete as not all publishers provide suitable and complete citation data.