# Conditions for superdecoherence

Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Amsterdam, The Netherlands
QuSoft, CWI, Science Park 123, Amsterdam, The Netherlands

### Abstract

Decoherence is the main obstacle to quantum computation. The decoherence rate per qubit is typically assumed to be constant. It is known, however, that quantum registers coupling to a single reservoir can show a decoherence rate per qubit that increases linearly with the number of qubits. This effect has been referred to as superdecoherence, and has been suggested to pose a threat to the scalability of quantum computation. Here, we show that superdecoherence is absent when the spectrum of the single reservoir is continuous, rather than discrete. The reason of this absence, is that, as the number of qubits is increased, a quantum register inevitably becomes susceptible to an ever narrower bandwidth of frequencies in the reservoir. Furthermore, we show that for superdecoherence to occur in a reservoir with a discrete spectrum, one of the frequencies in the reservoir has to coincide exactly with the frequency the quantum register is most susceptible to. We thus fully resolve the conditions that determine the presence or absence of superdecoherence. We conclude that superdecoherence is easily avoidable in practical realizations of quantum computers.

In theory, quantum computers can solve problems that cannot be solved on any classical computer. Solving these problems would have far-reaching consequences in many fields, ranging from medicine to cybersecurity. It is, however, notoriously hard to build a quantum computer. This is because it is much harder to protect the fundamental building block of the quantum computer (the qubit) from noise, than it is to protect its classical counterpart (the bit) from noise. Some years ago, it was discovered that the situation might be even worse: the more qubits you add, the more each of these qubits suffers from noise, almost as if the other qubits amplify the noise. This effect is unknown to classical bits. We discovered that the more qubits you add, the more selective they become in the type of noise they are susceptible to. This is much like a fishbone radio antenna: the more vertical 'bones’ are added, the fewer different frequencies it receives. As we show, the net result is that the problem of increasing noise levels for larger systems can easily be avoided.

### ► References

[1] Maximilian A Schlosshauer. Decoherence and the quantum-to-classical transition. Springer-Verlag Berlin Heidelberg, 2007. 10.1007/​978-3-540-35775-9.
https:/​/​doi.org/​10.1007/​978-3-540-35775-9

[2] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, 2018. ISSN 2521-327X. 10.22331/​q-2018-08-06-79.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[3] Frank Arute et al. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779): 505–510, 2019. 10.1038/​s41586-019-1666-5.
https:/​/​doi.org/​10.1038/​s41586-019-1666-5

[4] Jiehang Zhang et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature, 551 (7682): 601, 2017. 10.1038/​nature24654.
https:/​/​doi.org/​10.1038/​nature24654

[5] W. G. Unruh. Maintaining coherence in quantum computers. Phys. Rev. A, 51: 992–997, Feb 1995. 10.1103/​PhysRevA.51.992.
https:/​/​doi.org/​10.1103/​PhysRevA.51.992

[6] G Massimo Palma, Kalle-Antti Suominen, and Artur K Ekert. Quantum computers and dissipation. Proc. R. Soc. Lond. A, 452 (1946): 567–584, 1996. 10.1098/​rspa.1996.0029.
https:/​/​doi.org/​10.1098/​rspa.1996.0029

[7] John H. Reina, Luis Quiroga, and Neil F. Johnson. Decoherence of quantum registers. Phys. Rev. A, 65: 032326, Mar 2002. 10.1103/​PhysRevA.65.032326.
https:/​/​doi.org/​10.1103/​PhysRevA.65.032326

[8] Heinz-Peter Breuer and Francesco Petruccione. The theory of open quantum systems. Oxford University Press, 2010. 0.1093/​acprof:oso/​9780199213900.001.0001.

[9] A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A. Fisher, Anupam Garg, and W. Zwerger. Dynamics of the dissipative two-state system. Rev. Mod. Phys., 59: 1–85, Jan 1987. 10.1103/​RevModPhys.59.1.
https:/​/​doi.org/​10.1103/​RevModPhys.59.1

[10] P. Zanardi and M. Rasetti. Noiseless quantum codes. Phys. Rev. Lett., 79: 3306–3309, Oct 1997a. 10.1103/​PhysRevLett.79.3306.
https:/​/​doi.org/​10.1103/​PhysRevLett.79.3306

[11] Lu-Ming Duan and Guang-Can Guo. Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment. Phys. Rev. A, 57: 737–741, Feb 1998. 10.1103/​PhysRevA.57.737.
https:/​/​doi.org/​10.1103/​PhysRevA.57.737

[12] Claudia Benedetti and Matteo GA Paris. Effective dephasing for a qubit interacting with a transverse classical field. International Journal of Quantum Information, 12 (02): 1461004, 2014. 10.1142/​S0219749914610048.
https:/​/​doi.org/​10.1142/​S0219749914610048

[13] S. M. Anton et al. Pure dephasing in flux qubits due to flux noise with spectral density scaling as $1/​{f}^{{\alpha}}$. Phys. Rev. B, 85: 224505, Jun 2012. 10.1103/​PhysRevB.85.224505.
https:/​/​doi.org/​10.1103/​PhysRevB.85.224505

[14] Carole Addis, Gregoire Brebner, Pinja Haikka, and Sabrina Maniscalco. Coherence trapping and information backflow in dephasing qubits. Phys. Rev. A, 89: 024101, Feb 2014. 10.1103/​PhysRevA.89.024101.
https:/​/​doi.org/​10.1103/​PhysRevA.89.024101

[15] T. Palm and P. Nalbach. Nonperturbative environmental influence on dephasing. Phys. Rev. A, 96: 032105, Sep 2017. 10.1103/​PhysRevA.96.032105.
https:/​/​doi.org/​10.1103/​PhysRevA.96.032105

[16] Jürgen T. Stockburger. Superdecoherence through gate control noise. https:/​/​arxiv.org/​abs/​quant-ph/​0701062, Jan 2007.
arXiv:quant-ph/0701062

[17] G. P. Berman, D. I. Kamenev, and V. I. Tsifrinovich. Collective decoherence of the superpositional entangled states in the quantum shor algorithm. Phys. Rev. A, 71: 032346, Mar 2005. 10.1103/​PhysRevA.71.032346.
https:/​/​doi.org/​10.1103/​PhysRevA.71.032346

[18] Fernando Galve, Antonio Mandarino, Matteo G. A. Paris, Claudia Benedetti, and Roberta Zambrini. Microscopic description for the emergence of collective dissipation in extended quantum systems. Scientific Reports, 7 (1), Feb 2017. 10.1038/​srep42050.
https:/​/​doi.org/​10.1038/​srep42050

[19] M A Cirone, G De Chiara, G M Palma, and A Recati. Collective decoherence of cold atoms coupled to a bose–einstein condensate. New Journal of Physics, 11 (10): 103055, Oct 2009. 10.1088/​1367-2630/​11/​10/​103055.
https:/​/​doi.org/​10.1088/​1367-2630/​11/​10/​103055

[20] Thomas Monz, Philipp Schindler, Julio T. Barreiro, Michael Chwalla, Daniel Nigg, William A. Coish, Maximilian Harlander, Wolfgang Hänsel, Markus Hennrich, and Rainer Blatt. 14-qubit entanglement: Creation and coherence. Phys. Rev. Lett., 106: 130506, Mar 2011. 10.1103/​PhysRevLett.106.130506.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.130506

[21] B. J. Dalton. Scaling of decoherence effects in quantum computers. Journal of Modern Optics, 50 (6-7): 951–966, 2003. 10.1080/​09500340308234544.
https:/​/​doi.org/​10.1080/​09500340308234544

[22] Boris Ischi, Michael Hilke, and Martin Dubé. Decoherence in a $n$-qubit solid-state quantum register. Phys. Rev. B, 71: 195325, May 2005. 10.1103/​PhysRevB.71.195325.
https:/​/​doi.org/​10.1103/​PhysRevB.71.195325

[23] Constantine A Balanis. Antenna theory: analysis and design. John Wiley & Sons, 2016. ISBN 978-1-118-64206-1.

[24] Iñigo Liberal, Iñigo Ederra, and Richard W. Ziolkowski. Quantum antenna arrays: The role of quantum interference on direction-dependent photon statistics. Phys. Rev. A, 97: 053847, May 2018. 10.1103/​PhysRevA.97.053847.
https:/​/​doi.org/​10.1103/​PhysRevA.97.053847

[25] David Morgan. Surface acoustic wave filters: With applications to electronic communications and signal processing. Academic Press, 2010. 10.1016/​B978-0-12-372537-0.X5000-6.
https:/​/​doi.org/​10.1016/​B978-0-12-372537-0.X5000-6

[26] Roland Doll. Decoherence of spatially separated quantum bits. Universität Augsburg PhD thesis, https:/​/​opus.bibliothek.uni-augsburg.de/​opus4/​frontdoor/​deliver/​index/​docId/​673/​file/​doll_diss.pdf, 2008.
https:/​/​opus.bibliothek.uni-augsburg.de/​opus4/​frontdoor/​deliver/​index/​docId/​673/​file/​doll_diss.pdf

[27] Daniel Gottesman. Stabilizer codes and quantum error correction. California Institute of Technology PhD thesis, https:/​/​arxiv.org/​abs/​quant-ph/​9705052, Jan 1997.
arXiv:quant-ph/9705052

[28] John Preskill. Lecture notes for physics 229: Quantum information and computation. http:/​/​www.theory.caltech.edu/​people/​preskill/​ph229/​notes/​chap7.pdf, 1998.
http:/​/​www.theory.caltech.edu/​people/​preskill/​ph229/​notes/​chap7.pdf

[29] Claudia Benedetti, Fahimeh Salari Sehdaran, Mohammad H. Zandi, and Matteo G. A. Paris. Quantum probes for the cutoff frequency of ohmic environments. Phys. Rev. A, 97: 012126, Jan 2018. 10.1103/​PhysRevA.97.012126.
https:/​/​doi.org/​10.1103/​PhysRevA.97.012126

[30] R Bulla, NH Tong, and M Vojta. Numerical renormalization group for bosonic systems and application to the sub-ohmic spin-boson model. Physical Review Letters, 91 (17), Oct 2003. ISSN 0031-9007. 10.1103/​PhysRevLett.91.170601.
https:/​/​doi.org/​10.1103/​PhysRevLett.91.170601

[31] M Vojta, NH Tong, and R Bulla. Quantum phase transitions in the sub-Ohmic spin-boson model: Failure of the quantum-classical mapping. Physical Review Letters, 94 (7), Feb 2005. ISSN 0031-9007. 10.1103/​PhysRevLett.94.070604.
https:/​/​doi.org/​10.1103/​PhysRevLett.94.070604

[32] Frithjof B. Anders, Ralf Bulla, and Matthias Vojta. Equilibrium and nonequilibrium dynamics of the sub-Ohmic spin-boson model. Physical Review Letters, 98 (21), May 2007. ISSN 0031-9007. 10.1103/​PhysRevLett.98.210402.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.210402

[33] Stefan K Kehrein and Andreas Mielke. On the spin-boson model with a sub-ohmic reservoir. Physics Letters A, 219 (5-6): 313–318, 1996. 10.1016/​0375-9601(96)00475-6.
https:/​/​doi.org/​10.1016/​0375-9601(96)00475-6

[34] Joris Kattemölle and Jasper van Wezel. Dynamical fidelity susceptibility of decoherence-free subspaces. Phys. Rev. A, 99: 062340, Jun 2019. 10.1103/​PhysRevA.99.062340.
https:/​/​doi.org/​10.1103/​PhysRevA.99.062340

[35] Lu-Ming Duan and Guang-Can Guo. Preserving coherence in quantum computation by pairing quantum bits. Phys. Rev. Lett., 79: 1953–1956, Sep 1997. 10.1103/​PhysRevLett.79.1953.
https:/​/​doi.org/​10.1103/​PhysRevLett.79.1953

[36] Paolo Zanardi and Mario Rasetti. Error avoiding quantum codes. Modern Physics Letters B, 11 (25): 1085–1093, 1997b. 10.1142/​S0217984997001304.
https:/​/​doi.org/​10.1142/​S0217984997001304

[37] D. A. Lidar, I. L. Chuang, and K. B. Whaley. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett., 81: 2594–2597, Sep 1998. 10.1103/​PhysRevLett.81.2594.
https:/​/​doi.org/​10.1103/​PhysRevLett.81.2594

[38] Daniel A Lidar and K Birgitta Whaley. Decoherence-free subspaces and subsystems. In Irreversible quantum dynamics, pages 83–120. Springer, 2003. 10.1007/​3-540-44874-8_5.
https:/​/​doi.org/​10.1007/​3-540-44874-8_5

[39] J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley. Theory of decoherence-free fault-tolerant universal quantum computation. Phys. Rev. A, 63: 042307, Mar 2001. 10.1103/​PhysRevA.63.042307.
https:/​/​doi.org/​10.1103/​PhysRevA.63.042307

[40] Roland Doll, Martijn Wubs, Peter Hänggi, and Sigmund Kohler. Incomplete dephasing of $n$-qubit entangled w states. Phys. Rev. B, 76: 045317, Jul 2007. 10.1103/​PhysRevB.76.045317.
https:/​/​doi.org/​10.1103/​PhysRevB.76.045317

[41] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum metrology. Physical Review Letters, 96 (1), Jan 2006. 10.1103/​physrevlett.96.010401.
https:/​/​doi.org/​10.1103/​physrevlett.96.010401

[42] V. Giovannetti. Quantum-enhanced measurements: Beating the standard quantum limit. Science, 306 (5700): 1330–1336, Nov 2004. 10.1126/​science.1104149.
https:/​/​doi.org/​10.1126/​science.1104149

[43] A. Ferraro, S. Olivares, and M. G. A. Paris. Gaussian states in continuous variable quantum information. https:/​/​arxiv.org/​abs/​quant-ph/​0503237, Mar 2005.
arXiv:quant-ph/0503237

[44] Gerardo Adesso, Sammy Ragy, and Antony R Lee. Continuous variable quantum information: Gaussian states and beyond. Open Systems & Information Dynamics, 21 (01n02): 1440001, 2014. 10.1142/​s1230161214400010.
https:/​/​doi.org/​10.1142/​s1230161214400010

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