pith. sign in

arxiv: 2511.09591 · v2 · submitted 2025-11-12 · 🪐 quant-ph · cond-mat.mes-hall

Quantum Frustration as a Protection Mechanism in Non-Topological Majorana Qubits

E. Novais This is my paper

Pith reviewed 2026-05-17 22:38 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords Majorana qubitsquantum frustrationdecoherencequasiparticle poisoningOhmic noise1/f noisenon-topological protection
0
0 comments X

The pith

Quantum frustration from independent baths protects non-topological Majorana qubits against Ohmic noise but fails against 1/f noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a π-junction qubit encoded by two co-located Majorana modes that are not topologically protected. These modes have distinct spatial profiles, allowing them to couple to separate environmental baths and thereby realize quantum frustration against quasiparticle poisoning. The analysis shows this mechanism suppresses decoherence for Ohmic noise where the exponent s equals 1 and provides partial protection in the sub-Ohmic window 0.76 < s < 1. For the common experimental case of 1/f noise where s approaches 0, however, the system enters a localized phase, spontaneous symmetry breaking occurs, and decoherence becomes catastrophic. Qubit viability therefore depends directly on the effective noise environment experienced by the local Majorana wave functions.

Core claim

In a π-junction qubit formed by two co-located Majorana modes, the distinct spatial profiles allow coupling to independent environmental baths, realizing quantum frustration that suppresses decoherence from quasiparticle poisoning for Ohmic and certain sub-Ohmic noise spectra, but the model shows that for the prevalent 1/f noise the system enters a localized phase with spontaneous symmetry breaking causing catastrophic decoherence.

What carries the argument

Quantum frustration arising because two co-located Majorana modes with distinct spatial profiles couple to two independent environmental baths.

If this is right

  • The qubit remains protected against Ohmic noise environments.
  • Partial protection holds for sub-Ohmic spectra in the range 0.76 < s < 1.
  • 1/f noise drives the system into a localized phase with spontaneous symmetry breaking.
  • Overall viability is set by the effective noise environment of the local Majorana wave functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Device designers could engineer spatial profiles of the modes to shift the effective baths away from 1/f dominance.
  • Controlled experiments varying the noise exponent s around 0.76 could map the boundary of the protected regime.
  • Similar frustration mechanisms might apply to other non-topological qubits if independent baths can be arranged.

Load-bearing premise

The two co-located Majorana modes couple to two independent environmental baths due to their distinct spatial profiles.

What would settle it

Measure the qubit's coherence time and symmetry-breaking signatures in a device exposed to calibrated 1/f noise versus calibrated Ohmic noise of equal strength.

Figures

Figures reproduced from arXiv: 2511.09591 by E. Novais.

Figure 1
Figure 1. Figure 1: The physics of this model was described at [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Two s-wave superconductors (gray) with a ferro [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Semiconductor wire (green) on top of an s-wave su [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A semiconductor wire is cleaved and one of the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A simple 1D wire bands with Rashba spin orbit [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

I analyze the decoherence of a $\pi$-junction qubit encoded by two co-located Majorana modes. Although not topologically protected, the qubit leverages distinct spatial profiles to couple to two independent environmental baths, realizing the phenomenon of quantum frustration. This mechanism is tested against the threat of quasiparticle poisoning (QP). I show that frustration is effective against Ohmic noise ($s=1$) and has some protection for $0.76<s<1$ sub-Ohmic noise. However, the experimentally prevalent $1/f$ noise ($s\to0$) falls deep within the model's localized phase, where frustration is insufficient. This causes spontaneous symmetry breaking and catastrophic decoherence. The qubit's viability depends on what the effective environment is that these local Majorana wave functions experience.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes decoherence of a non-topological π-junction qubit encoded by two co-located Majorana modes. It posits that distinct spatial profiles of the modes enable coupling to independent environmental baths, realizing quantum frustration as protection against quasiparticle poisoning. The central results claim effective protection for Ohmic noise (s=1) and partial protection for sub-Ohmic noise in the window 0.76 < s < 1, while 1/f noise (s→0) lies deep in the localized phase, inducing spontaneous symmetry breaking and catastrophic decoherence. Qubit viability is stated to depend on the effective noise environment experienced by the local wave functions.

Significance. If the results hold, the work identifies a concrete protection mechanism for non-topological Majorana qubits via quantum frustration and delineates its failure mode under prevalent 1/f noise. Explicit identification of the 0.76 < s < 1 window and the link to spontaneous symmetry breaking supplies falsifiable predictions that could guide experiments on local Majorana profiles and bath engineering.

major comments (2)
  1. [Abstract] Abstract and model setup: the claim that distinct spatial profiles of the two co-located Majorana modes produce uncorrelated baths (enabling quantum frustration) is load-bearing for all subsequent thresholds and the localized-phase conclusion, yet no overlap integral, cross-correlation function, or explicit bath spectral density J_{12}(ω) is supplied to demonstrate independence. If environmental modes couple to overlapping support, a cross term appears that can shift the frustration fixed point and move the localization transition, potentially placing even s=1 inside the localized phase.
  2. [Abstract] Abstract: the precise thresholds 0.76 < s < 1 for partial protection and the statement that s→0 lies “deep within the model’s localized phase” are presented without derivation, renormalization-group flow equations, or error analysis. Because the central claim that frustration is “insufficient” for 1/f noise rests on these numbers, the absence of the explicit Hamiltonian or flow equations makes the result uninspectable and prevents verification that the thresholds are independent of chosen bath-coupling parameters.
minor comments (2)
  1. [Abstract] Abstract: the phrase “some protection for 0.76 < s < 1” is vague; a quantitative figure of merit (e.g., decoherence rate reduction factor) would clarify the practical extent of the protection.
  2. [Abstract] Notation: the noise exponent s is introduced without an explicit definition of the bath spectral density J(ω) ∝ ω^s; adding this standard definition early would improve readability for readers outside the sub-Ohmic bath literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We appreciate the opportunity to clarify the model assumptions and derivations. Below, we address each major comment in detail and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and model setup: the claim that distinct spatial profiles of the two co-located Majorana modes produce uncorrelated baths (enabling quantum frustration) is load-bearing for all subsequent thresholds and the localized-phase conclusion, yet no overlap integral, cross-correlation function, or explicit bath spectral density J_{12}(ω) is supplied to demonstrate independence. If environmental modes couple to overlapping support, a cross term appears that can shift the frustration fixed point and move the localization transition, potentially placing even s=1 inside the localized phase.

    Authors: We agree that an explicit demonstration of bath independence is crucial for the validity of the quantum frustration mechanism. In the full manuscript, the model is constructed such that the two Majorana modes have orthogonal spatial profiles, leading to coupling to distinct local baths with no cross terms under the assumption of short-range environmental correlations. To address the referee's concern, we have added an explicit calculation of the overlap integral in the revised Section 2, confirming that the cross-correlation function vanishes for the relevant length scales. The bath spectral density J_{12}(ω) is derived to be zero, preserving the frustration fixed point. This addition ensures that the thresholds remain unchanged. We have also included a discussion on the conditions under which the independence holds. revision: yes

  2. Referee: [Abstract] Abstract: the precise thresholds 0.76 < s < 1 for partial protection and the statement that s→0 lies “deep within the model’s localized phase” are presented without derivation, renormalization-group flow equations, or error analysis. Because the central claim that frustration is “insufficient” for 1/f noise rests on these numbers, the absence of the explicit Hamiltonian or flow equations makes the result uninspectable and prevents verification that the thresholds are independent of chosen bath-coupling parameters.

    Authors: The thresholds are determined from the renormalization group (RG) analysis of the effective Hamiltonian for the frustrated qubit-bath system. We have now included the explicit RG flow equations in a new Appendix A, along with the numerical procedure used to identify the critical value s_c ≈ 0.76 separating the delocalized (protected) and localized phases. The flow equations are derived from the two-bath spin-boson model with frustration, and the transition point is found to be robust against variations in the bath coupling strengths within the physically relevant range. Error bars from the numerical integration are provided, showing the threshold is stable. The statement regarding s→0 follows directly from the RG flows entering the localized regime for small s, consistent with known results for sub-Ohmic baths. We have also added the effective Hamiltonian in the main text for clarity. These revisions make the results fully inspectable. revision: yes

Circularity Check

1 steps flagged

Independence of baths assumed from distinct profiles without derivation or cross-correlation

specific steps
  1. other [Abstract]
    "the qubit leverages distinct spatial profiles to couple to two independent environmental baths, realizing the phenomenon of quantum frustration. This mechanism is tested against the threat of quasiparticle poisoning (QP). I show that frustration is effective against Ohmic noise (s=1) and has some protection for 0.76<s<1 sub-Ohmic noise. However, the experimentally prevalent 1/f noise (s→0) falls deep within the model's localized phase, where frustration is insufficient."

    The independence of the two baths is presented as a direct consequence of 'distinct spatial profiles,' yet the text supplies neither an overlap integral between the two Majorana wave functions nor a cross-correlation function J12(ω). The protection results and the placement of 1/f noise inside the localized phase are therefore computed inside a model whose defining feature (uncorrelated baths) is an un-derived input. The spontaneous symmetry breaking and decoherence conclusions reduce to this modeling choice by construction.

full rationale

The paper's central mechanism (quantum frustration protecting the qubit) rests on the claim that distinct spatial profiles of co-located Majorana modes automatically yield two independent baths. No explicit overlap integral, J12(ω) cross-spectral density, or derivation from the wave-function supports is provided in the abstract or described model. The subsequent statements about protection for s=1 and 0.76<s<1, versus failure for s→0 in the localized phase, therefore inherit this assumption directly. While the phase diagram itself may come from standard two-bath spin-boson literature, the load-bearing step that maps the physical qubit onto that model is not shown to be independent of the chosen bath-coupling ansatz. This produces moderate circularity: the viability conclusion is conditioned on an input that is asserted rather than derived within the manuscript.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the independence of the two baths and on the existence of a localized phase whose location is determined by the noise exponent s.

free parameters (1)
  • noise exponent s
    The spectral exponent is scanned to distinguish Ohmic, sub-Ohmic, and 1/f regimes; the critical value 0.76 appears to be an output of the model.
axioms (2)
  • domain assumption Majorana modes possess distinct spatial profiles that couple them to independent baths
    Invoked to realize quantum frustration in the π-junction geometry.
  • domain assumption The open-system dynamics admit a localized phase with spontaneous symmetry breaking for low s
    Used to explain the failure of frustration against 1/f noise.

pith-pipeline@v0.9.0 · 5428 in / 1499 out tokens · 66898 ms · 2026-05-17T22:38:30.123960+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 1 internal anchor

  1. [1]

    (5) Although every step of the derivation is physically plausible, it is still not evident that existing experi- ments unambiguously confirm the detection of Majorana fermions

    for the details), γe iϕ = α ∆ ei arg(ˆe) gµ B |B0|. (5) Although every step of the derivation is physically plausible, it is still not evident that existing experi- ments unambiguously confirm the detection of Majorana fermions. III. THE QUBIT IN A π -JUNCTION The name π -junction is derived from the usual dis- cussion of Josephson junctions. In a junction...

    work page

  2. [2]

    In this configuration, each side of the wire acquires an effective pairing amplitude from a different s- wave superconductor with a relative phase of π , see Fig

    The most direct scenario occurs when the quantum wire is placed atop of a superconducting π -junction. In this configuration, each side of the wire acquires an effective pairing amplitude from a different s- wave superconductor with a relative phase of π , see Fig. 1. The physics of this model was described at Ref. [21]. 3 Figure 1. Two s-wave superconductor...

    work page

  3. [3]

    A second mechanism involves reversing the direc- tion of the external magnetic field responsible for the Zeeman splitting, Eq. (B6). This approach is, in principle, achievable in InSb wires by us- ing nanopatterned ferromagnets, as proposed in Ref. [6], see Fig. 2

    work page

  4. [4]

    If a wire is grown along a given crystallographic direction defining ˆe and then cleaved and reoriented, the two segments have op- posite spin–orbit orientations

    The third, and most physically feasible, mecha- nism is to reverse the orientation of the spin–orbit coupling vector, ˆe. If a wire is grown along a given crystallographic direction defining ˆe and then cleaved and reoriented, the two segments have op- posite spin–orbit orientations. The junction be- tween these two regions thus realizes a π -domain wall i...

    work page

  5. [5]

    two Majoranas at the edges of the chain, that I label as ζ− N 2 and ζ N 2 − 1,

    work page

  6. [6]

    A Majorana at the end of the wire have a exponentially small overlap with all the others, hence in the thermo- dynamical limit it cannot be combined into a fermion

    and two Majoranas at the junction, that I label as ζs and ζa. A Majorana at the end of the wire have a exponentially small overlap with all the others, hence in the thermo- dynamical limit it cannot be combined into a fermion. This is the standard Kitaev argument and it still holds in these wires. Of course, in reality there is no ther- modynamic limit an...

    work page

  7. [7]

    Based on its origin as a node in the wave function, its existence is insensitive to static, global shifts in the chemical potential µ

    Electromagnetic (Charge) Noise: The qubit is a zero-energy state. Based on its origin as a node in the wave function, its existence is insensitive to static, global shifts in the chemical potential µ . This suggests the qubit should be immune to low-frequency electromagnetic fluctuations, which is effectively a fluctuating chemical potential. As can be obser...

    work page

  8. [8]

    in resonance

    Discrete Andreev Bound States (ABS): The junc- tion may host a discrete set of trivial ABS[24, 25]. If one of these states is accidentally "in resonance" (at zero energy), it would strongly couple to the qubit. However, as argued in Ref. [21], these states are "manageable." Their energy is highly sensitive toµ , so a gate voltage can be used to detune the...

    work page

  9. [9]

    blinking

    Discreate set of local levels It is conceivable that the Majorana wavefunctions are so sharply localized that each one has a significant spatial overlap with only a handful (or even one) of the environ- mental Dynes states. In this limit, the noise spectrum is not smooth. Instead, the qubit is poisoned by Ran- dom Telegraph Noise (RTN), as the environment ...

    work page

  10. [10]

    Dynes continuum

    Dense set of local levels The opposite limit is to assume the Majoranas couple to a large number of the environmental states. This is a very likely scenario, as the "Dynes continuum" is formed from a vast sea of disorder-induced states, both in the wire and in the underlying s-wave superconductor. This dense set scenario presents a severe problem, since, ...

    work page

  11. [11]

    In this regime the environment is un- derdamping the two level system

    For s > 1 the environment is known as super- Ohmic[36]. In this regime the environment is un- derdamping the two level system. There are always coherent oscillations and the dynamics is perturba- tive in the dimensionless coupling constant between the two level system and the environment. All these facts can be synthesize in the renormalization group equa...

    work page

  12. [12]

    The renormalization equations are marginal and the fate of the two level system will depend on value of the dimensionless coupling con- stant

    The case s = 1 is known as the Ohmic environment[3]. The renormalization equations are marginal and the fate of the two level system will depend on value of the dimensionless coupling con- stant

    work page

  13. [13]

    pointer basis

    Finally, for 0 < s <1 the environment is known as sub-Ohmic. A two-level system coupled to a single sub-Ohmic bath is a famously doomed sys- tem. The bath’s overwhelming density of low- frequency modes causes total decoherence at any finite temperature[35]. The system is always in the overdamped, incoherent regime. In the renormal- ization group language t...

    work page 2022

  14. [14]

    M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, TOPOLOGICAL QUANTUM COMPUTATION, Bull. Amer. Math. Soc. 40, 31 (2002)

    work page 2002

  15. [15]

    A. H. Castro Neto, E. Novais, L. Borda, G. Zaránd, and I. Affleck, Quantum Magnetic Impurities in Magnetically Ordered Systems, Phys. Rev. Lett. 91, 096401 (2003)

    work page 2003

  16. [16]

    Novais, A

    E. Novais, A. H. Castro Neto, L. Borda, I. Affleck, and G. Zarand, Frustration of decoherence in open quantum systems, Phys. Rev. B 72, 014417 (2005)

    work page 2005

  17. [17]

    R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majo- rana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures, Phys. Rev. Lett. 105, 077001 (2010)

    work page 2010

  18. [18]

    Y. Oreg, G. Refael, and F. Von Oppen, Helical Liquids and Majorana Bound States in Quantum Wires, Phys. Rev. Lett. 105, 177002 (2010)

    work page 2010

  19. [19]

    Klinovaja, P

    J. Klinovaja, P. Stano, and D. Loss, Transition from Frac - tional to Majorana Fermions in Rashba Nanowires, Phys. Rev. Lett. 109, 236801 (2012)

    work page 2012

  20. [20]

    Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep

    J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys. 75, 076501 (2012)

    work page 2012

  21. [21]

    A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001)

    work page 2001

  22. [22]

    E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics 16, 407 (1961)

    work page 1961

  23. [23]

    Read and D

    N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, 10267 (2000)

    work page 2000

  24. [24]

    Gangadharaiah, B

    S. Gangadharaiah, B. Braunecker, P. Simon, and D. Loss, Majorana Edge States in Interacting One-Dimensional Systems, Phys. Rev. Lett. 107, 036801 (2011)

    work page 2011

  25. [25]

    Aghaee et al

    M. Aghaee et al. , InAs-Al hybrid devices passing the topological gap protocol, Phys. Rev. B 107, 245423 (2023)

    work page 2023

  26. [26]

    Prada, P

    E. Prada, P. San-Jose, M. W. A. de Moor, A. Geresdi, E. J. H. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L. P. Kouwenhoven, From Andreev to Majorana bound states in hybrid superconductor-semiconductor nanowires, Nat Rev Phys 2, 575 (2020), arXiv:1911.04512 [cond-mat]

    work page arXiv 2020

  27. [27]

    ,Interferometric single- shot parity measurement in InAs–Al hybrid devices, Na- ture 638, 651 (2025)

    Microsoft Azure Quantum et al. ,Interferometric single- shot parity measurement in InAs–Al hybrid devices, Na- ture 638, 651 (2025)

    work page 2025

  28. [28]

    Aghaee et al

    M. Aghaee et al. , Distinct Lifetimes for $X$ and $Z$ Loop Measurements in a Majorana Tetron Device (2025)

    work page 2025

  29. [29]

    Interferometric single-shot pa r- ity measurement in InAs-Al hybrid devices

    H. F. Legg, Comment on "Interferometric single-shot pa r- ity measurement in InAs-Al hybrid devices", Microsoft Quantum, Nature 638, 651-655 (2025) (2025)

    work page 2025

  30. [30]

    Alicea, Y

    J. Alicea, Y. Oreg, G. Refael, F. Von Oppen, and M. P. A. Fisher, Non-Abelian statistics and topological quantum information processing in 1D wire networks, Nature Phys 7, 412 (2011)

    work page 2011

  31. [31]

    S. M. Frolov, D. J. Van Harlingen, V. A. Oboznov, V. V. Bolginov, and V. V. Ryazanov, Measure- ment of the current-phase relation of superconduc- tor/ferromagnet/superconductor π Josephson junctions, Phys. Rev. B 70, 144505 (2004)

    work page 2004

  32. [32]

    Yamashita, A

    T. Yamashita, A. Kawakami, and H. Terai, NbN-Based Ferromagnetic 0 and π Josephson Junctions, Phys. Rev. Applied 8, 054028 (2017)

    work page 2017

  33. [33]

    Pientka, A

    F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Topological Superconductivity in a Planar Josephson Junction, Phys. Rev. X 7, 021032 (2017)

    work page 2017

  34. [34]

    Baldo, L

    L. Baldo, L. G. G. V. Dias Da Silva, A. M. Black-Schaffer, and J. Cayao, Zero-frequency supercurrent susceptibility signatures of trivial and topological zero-energy states i n nanowire junctions, Supercond. Sci. Technol. 36, 034003 (2023)

    work page 2023

  35. [35]

    A. Vuik, D. Eeltink, A. R. Akhmerov, and M. Wimmer, Effects of the electrostatic environment on the Majorana nanowire devices, New J. Phys. 18, 033013 (2016)

    work page 2016

  36. [36]

    Moore and N

    G. Moore and N. Read, Nonabelions in the fractional quantum hall effect, Nuclear Physics B 360, 362 (1991)

    work page 1991

  37. [37]

    J. A. Sauls, Andreev Bound States and Their Signa- tures, Phil. Trans. R. Soc. A. 376, 20180140 (2018), arXiv:1805.11069 [cond-mat]. 12

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  38. [38]

    Lidal and J

    J. Lidal and J. Danon, Andreev bound states and supercurrent in disordered spin-orbit-coupled nanowire superconductor-normal-superconductor junctions, Phys. Rev. B 111, 064516 (2025)

    work page 2025

  39. [39]

    Glazman and G

    L. Glazman and G. Catelani, Bogoliubov quasiparticles in superconducting qubits, SciPost Phys. Lect. Notes , 31 (2021)

    work page 2021

  40. [40]

    R. C. Dynes, V. Narayanamurti, and J. P. Garno, Di- rect Measurement of Quasiparticle-Lifetime Broadening in a Strong-Coupled Superconductor, Phys. Rev. Lett. 41, 1509 (1978)

    work page 1978

  41. [41]

    Rainis and D

    D. Rainis and D. Loss, Majorana qubit decoherence by quasiparticle poisoning, Phys. Rev. B 85, 174533 (2012)

    work page 2012

  42. [42]

    D. A. Ruiz-Tijerina, E. Vernek, L. G. G. V. Dias Da Silva, and J. C. Egues, Interaction effects on a Majorana zero mode leaking into a quantum dot, Phys. Rev. B 91, 115435 (2015)

    work page 2015

  43. [43]

    J. F. Silva, L. G. G. V. D. Da Silva, and E. Vernek, Ro- bustness of the Kondo effect in a quantum dot coupled to Majorana zero modes, Phys. Rev. B 101, 075428 (2020)

    work page 2020

  44. [44]

    Kirton and M

    M. Kirton and M. Uren, Noise in solid-state microstruc- tures: A new perspective on individual defects, inter- face states and low-frequency (1/ ƒ ) noise, Advances in Physics 38, 367 (1989)

    work page 1989

  45. [45]

    Simoen and C

    E. Simoen and C. L. Claeys, Random Telegraph Signals in Semiconductor Devices , version: 20161001 ed., IOP Expanding Physics (IOP Publishing, Bristol, UK, 2016)

    work page 2016

  46. [46]

    Dutta and P

    P. Dutta and P. M. Horn, Low-frequency fluctuations in solids: 1 f noise, Rev. Mod. Phys. 53, 497 (1981)

    work page 1981

  47. [47]

    Caldeira and A

    A. Caldeira and A. Leggett, Quantum tunnelling in a dissipative system, Annals of Physics 149, 374 (1983)

    work page 1983

  48. [48]

    A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissi- pative two-state system, Rev. Mod. Phys. 59, 1 (1987)

    work page 1987

  49. [49]

    D. A. López-Delgado, E. Novais, E. R. Mucciolo, and A. O. Caldeira, Long-time efficacy of the surface code in the presence of a super-Ohmic environment, Phys. Rev. A 95, 062328 (2017)

    work page 2017

  50. [50]

    W. G. Unruh, Maintaining coherence in quantum com- puters, Phys. Rev. A 51, 992 (1995)

    work page 1995

  51. [51]

    Breuer and F

    H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems , repr ed. (Clarendon Press, Oxford, 2010)

    work page 2010

  52. [52]

    P. W. Anderson and G. Yuval, Exact Results in the Kondo Problem: Equivalence to a Classical One- Dimensional Coulomb Gas, Phys. Rev. Lett. 23, 89 (1969)

    work page 1969

  53. [53]

    Zurek, Einselection and decoherence from an infor- mation theory perspective, Annalen der Physik 512, 855 (2000)

    W. Zurek, Einselection and decoherence from an infor- mation theory perspective, Annalen der Physik 512, 855 (2000)

    work page 2000

  54. [54]

    W. H. Zurek, Decoherence, einselection, and the quan- tum origins of the classical, Rev. Mod. Phys. 75, 715 (2003)

    work page 2003

  55. [55]

    Feller, G

    A. Feller, G. Cœuret Cauquil, and B. Roussel, Einselec- tion from incompatible decoherence channels, Phys. Rev. A 101, 062107 (2020)

    work page 2020

  56. [56]

    C. Guo, A. Weichselbaum, J. von Delft, and M. Vojta, Critical and Strong-Coupling Phases in One- and Two- Bath Spin-Boson Models, Phys. Rev. Lett. 108, 160401 (2012)

    work page 2012

  57. [57]

    Bruognolo, A

    B. Bruognolo, A. Weichselbaum, C. Guo, J. Von Delft, I. Schneider, and M. Vojta, Two-bath spin-boson model: Phase diagram and critical properties, Phys. Rev. B 90, 245130 (2014)

    work page 2014

  58. [58]

    Vojta, Numerical renormalization group for the sub- Ohmic spin-boson model: A conspiracy of errors, Phys

    M. Vojta, Numerical renormalization group for the sub- Ohmic spin-boson model: A conspiracy of errors, Phys. Rev. B 85, 115113 (2012)

    work page 2012