Pith. sign in

REVIEW 4 major objections 3 minor 47 references

Quantum vs Classical Erasure: Equal Bounds but Unequal Costs

T0 review · 4 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper establishes that although Landauer's erasure bound is identical for classical and quantum bits, a classical majority-vote encoding can reach in a single parallel interaction the fidelity a qubit can only approach with many sequent

desk verdict A promising framework with a flawed central comparison: the qubit is denied the parallel control the classical bit gets, which drives the headline conclusion. read the letter →

arxiv 2607.27341 v1 pith:Q73LP3DG submitted 2026-07-29 quant-ph

Jan Neuser , Jake Xuereb , Pharnam Bakhshinezhad , Marcus Huber This is my paper · ORCID
classification quant-ph
keywords Landauerboundinformationerasurefinite-timethermodynamicsvirtualqubitmajority-voteencodingcontrolcomplexitythermaldissipationquantumcooling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

A bit erased in a classical many-body system and a bit erased in a single quantum two-level system obey the same Landauer bound: both require at least k_B T ln 2 plus an excess that grows with the target error. The paper argues that the similarity ends there. For finite resources, a qubit's fidelity after n cooling interactions is capped by F=1-(1-w1)^n/2, while a classical bit encoded by majority vote over N subsystems can reach the same or better fidelity in a single parallel interaction once N exceeds a threshold N_min(n). The authors conclude that classical encodings have a thermodynamic advantage in realistic finite-time, finite-control settings, and that the gap between practical erasure and Landauer's bound reflects the difficulty of approaching the bound, not a different fundamental limit.

What carries the argument

The argument runs on two constructions. First, the virtual qubit: a two-level subspace of the bath-plus-work-source Hilbert space whose population ratio defines a virtual temperature; each tripartite Hamiltonian is a partial swap between the system and one such virtual qubit, so n interactions thermalise the system towards that virtual temperature. Second, the majority-vote encoding: a classical bit is a set of N non-interacting two-level subsystems whose logical 0/1 is decided by the majority of individual outcomes, addressed by mutually commuting Hamiltonians in a single unitary U_C=exp(-i sum H_i tau_opt). The binomial tail bound F_C(N) >= 1 - exp(-N D(1/2||p)) converts many slightly cool

What would settle it

Construct an explicit qubit erasure protocol that couples to n virtual qubits in parallel using only tripartite interactions and total time tau_opt; if its attainable fidelity exceeds F_Q^max(n) for n swaps, the claimed separation fails. Equivalently, in an experiment with identical bath temperature and work-source purity, find a qubit with n swaps that reaches a fidelity above F_C^max(N_min-1) or reaches the classical bit's fidelity in less time.

Watch

Extended reading notes

Core claim

Working with energy-preserving tripartite swaps among system, thermal bath, and work source, the paper proves that the minimal dissipation for erasure to fidelity 1-epsilon is identical for a qubit and for a d-dimensional classical bit when the target population is spread uniformly over each logical subspace. In the asymptotic limit of infinitely many swaps, the two dissipations again coincide. The central quantitative result is a threshold: with work-source purity w1, a qubit allowed n perfectly timed swaps can at best reach F_Q^max(n)=1-(1-w1)^n/2 in the infinite-dissipation limit; a classical majority-vote bit of N qubit subsystems, each cooled by one swap and addressed simultaneously by

Load-bearing premise

The comparison assumes the qubit is limited to n sequential tripartite swaps while the classical bit is allowed one simultaneous global interaction with all N subsystems; if a qubit can be cooled by a collective coupling to many virtual qubits without that high control complexity, the claimed classical advantage could disappear.

Editorial extensions

If this is right

  • For a given number n of allowed qubit swaps, any classical bit with more than N_min subsystems can match or beat the qubit's best possible fidelity in one interaction time tau_opt rather than n tau_opt.
  • Because the binomial error decays exponentially in N, adding subsystems is a direct substitute for longer interaction times, sharper control, or colder work sources.
  • The finite-error Landauer bound is the same for both encodings, so the practical shortfall from Landauer's limit is governed by control and time resources, not by the number of microstates.
  • Imperfect timing degrades the qubit's reachable fidelity by replacing w1 with (1-s^2)w1, whereas the classical bit retains its exponential error suppression in N.
  • With a finite-temperature work source, the qubit's reachable fidelity is capped by the virtual temperature; the classical bit can still reach high fidelity by increasing N.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the control asymmetry were relaxed—for example, if a qubit could be coherently coupled to many virtual qubits with a single global pulse of only modest complexity—the resource advantage of the classical encoding could shrink; the paper's own discussion of the (2n+1)-partite interaction needed for parallel qubit control flags this as the main boundary of the result.
  • The same exponential-concentration mechanism should apply to any redundant encoding with a binomial tail bound, not only majority vote; repetition codes and other classical error-correcting codes are natural testbeds.
  • The dissipation measure here charges only heat into the thermal bath; a fuller accounting that includes the preparation and maintenance cost of N subsystems, or the much larger energy scale of the classical bit, might partially offset the classical advantage.
  • A direct experimental probe would be sideband cooling of a single trapped ion (n pulses) versus collective cooling of a spin ensemble, measuring time and heat at fixed fidelity; the threshold N_min predicts where the classical protocol overtakes the qubit.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper compares the thermodynamics of erasing one bit of information encoded in a quantum two-level system versus a classical many-body majority-vote encoding. It claims that the asymptotic Landauer bound is the same for both, but that with finite resources—finite interaction time, finite work-source purity, and limited control—classical erasure is significantly cheaper and faster. The central quantitative result is an expression for the minimum number N of classical subsystems needed for a single parallel global interaction to reach the same asymptotic fidelity as n sequential swaps of a quantum qubit (Eq. (9)), together with trade-off relations and a robustness analysis against timing errors. The equal-bounds claim is supported by elementary entropy accounting; the classical advantage is derived from large-deviation estimates of majority voting.

Significance. If the central comparison is correct, the paper provides a conceptually interesting explanation of why practical erasure schemes fall short of the Landauer bound and identifies a fundamental thermodynamic advantage of classical encodings. The equal asymptotic bounds are not new, but framing them in a common virtual-qubit framework and contrasting finite-resource behavior is a useful contribution. The paper also makes explicit falsifiable scaling predictions, e.g., the N_min formula, and provides a supplement with derivations of the population dynamics and large-deviation bounds. However, the advertised quantum-versus-classical resource asymmetry rests on an asymmetric control assumption that is not stated in the Protocol Assumptions; this is a load-bearing issue that must be addressed before the main conclusion can be accepted.

major comments (4)
  1. [The Role of Time, Eqs. (6)-(7)] The comparison limits the qubit to sequential unitaries U_Q = ∏ U_i (Eq. (6)), while the classical bit is allowed the parallel unitary U_C = exp(-i ∑ H_i τ_opt) (Eq. (7)). The Protocol Assumptions only restrict to a single control frequency; they do not forbid applying all H_i simultaneously to a single qubit. The text dismisses the parallel qubit strategy as requiring a (2n+1)-partite interaction, but H_sum = ∑_i H_i is a sum of n tripartite couplings, each energy-preserving and at one frequency, not a single (2n+1)-partite interaction. In the single-excitation sector with n virtual qubits in the relevant state, the bright state couples with strength √n g, so a single interaction of duration π/(2√n g) fully transfers the qubit population in the ideal ΔE_B→∞ limit. Thus the qubit can reach at least F_Q^max(n) in time τ_opt/√n, undermining the claimed time and control advantage of the cla
  2. [Eq. (5)] Equation (5), the central dissipation expression ∆Q_B = ((F−1)p_i + F p_i′)/β log(F/(1−F W)), is asserted in the main text without derivation. The supplemental derives population dynamics and the infinite-swap limit, but does not derive this heat-dissipation formula. Since this expression underlies the dissipation-fidelity curves in Fig. 2 and the claimed trade-off between fidelity and dissipation, a derivation (or a precise reference) is essential. As written, the reader cannot verify the logarithmic divergence as F→1 or the claimed equality between the quantum and ladder cases.
  3. [Eq. (10)] Equation (10) is garbled as printed: FQ = 1− q^n/(1+e^{−βwΔE_S^Q} e^{−(β−βw)ΔE_B}) + q^n/2 is missing parentheses and the definition of q is ambiguous. This equation is used to argue that a finite-temperature work source constrains the reachable qubit fidelity. The reader cannot check this claim without a corrected formula and a clear statement of the regime of validity (e.g., which terms are kept in the large-gap limit).
  4. [Robustness of the protocol, Eq. (S27)-(S32)] The timing-error analysis in the supplement assumes that for the classical bit the timing errors for each subsystem are independent (ε_i drawn per subsystem), while for the qubit the error is common to all n swaps. This is a reasonable model for inhomogeneous coupling, but it is another asymmetry: a single global classical interaction with one clock would logically have a common error, not independent errors per subsystem. If the classical bit also suffers a common timing error, the majority-vote error suppression may degrade. The authors should state explicitly why independent per-subsystem errors are the relevant classical scenario and discuss the common-error case.
minor comments (3)
  1. [General] Throughout, the notation F is used both for fidelity and for the probability in Eq. (S13); please disambiguate. Also, Eq. (S14) uses p for the single-subsystem 0-population, while the main text uses p_i and w_1; this inconsistency makes the supplement harder to follow.
  2. [Introduction/Definitions] The phrase 'communal folklore' in the abstract is informal for a journal article; consider 'previous observations' or a reference. Also, the definition of Π_0 in the classical-bit section is stated for even d, but the majority-vote construction later assumes odd N; this is fine, but the relationship between the two encodings should be made explicit.
  3. [Fig. 2] In Fig. 2(a), the color gradient indicating protocol time is difficult to interpret because the legend does not give a quantitative scale. Please add a color bar with time units or a separate panel. In the caption, '10^11 + 1' is presumably meant as 10^11+1; the formatting is confusing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and the main comparison is an analytic model calculation, not a fitted or definitionally forced result.

full rationale

The paper's load-bearing claims are the equality of Landauer bounds for classical and quantum bits and the finite-resource comparison of a sequential qubit protocol with a parallel majority-vote classical protocol. Neither reduces to its inputs. The equal-bounds statement is derived from the Reeb-Wolf dissipation inequality (Eq. S1) by optimizing the final distribution over microstates (Eqs. S2-S3), which is an independent entropy calculation. The finite-n qubit fidelity F_Q^max(n)=1-(1-w_1)^n/2 is obtained from the recursive population update Eqs. (S7)-(S8), and the required classical subsystem number N_min is obtained by equating F_C^max(N) from the binomial majority-vote expression (Eq. S15) with F_Q^max(n), then solving the transcendental equation to give Eq. (9)/(S19). This is an analytic comparison, not a fit of a parameter to the predicted quantity. No fitted parameter is relabeled as a prediction. The virtual-qubit formalism is cited to [25,29] as foundational and is re-derived in the supplement; the heavy self-citation for related cooling work is contextual and not load-bearing. The asymmetry between U_Q=prod U_i and U_C=e^{-i sum H_i tau_opt} is an explicit control-model choice, discussed in the text as the sequential/parallel trade-off; it may be physically contestable, but contesting an assumption is a modeling criticism, not evidence that the derivation is circular. The paper also identifies its own limitations (e.g., neglected rethermalization, timing overhead), which further indicates the derivation is not constructed to force the conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The assumptions are mostly domain restrictions about control and encoding; the main unlisted one — sequential vs parallel control — is the weakest load-bearing premise.

assumptions (5)
  • domain assumption Erasure can be modeled by energy-preserving tripartite interactions with a single control frequency, as in Eqs. (2)-(3).
    Protocol Assumptions state 'Limited control i.e. both the classical and quantum bits are erased utilising a single control frequency.' This is a modeling restriction, not a law.
  • domain assumption Bath and work source are macroscopic and refreshed after every swap, so their states are unchanged.
    Protocol Assumptions state 'The thermal bath and work source are macroscopic i.e. their states are unchanged by dissipation due to erasure'; asymptotic limit also refreshes the bath and work source.
  • domain assumption Classical bit is encoded via majority vote over N non-interacting qubit subsystems with a global control; N is odd.
    Section 'A Bit Encoded in a Collection of Subsystems' and Eq. (S13); this encoding is what produces exponential error suppression.
  • standard math Reeb-Wolf entropy-production inequality βΔQ ≥ −ΔS (Eq. S1) is valid for the considered processes.
    Used in the supplement to derive finite-error Landauer bounds; cited as Ref. [12].
  • standard math Large-deviation/Chernoff bound and Stirling approximation for binomial tails (Eqs. S14, S16).
    Used to show F_C(N)→1 exponentially and to derive N_min; standard results cited as Refs. [42,44].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum vs Classical Erasure: Equal Bounds but Unequal Costs." pith.science (2026). https://pith.science/paper/Q73LP3DG

@misc{pith2026260727341,
  author       = {Pith},
  title        = {Pith review of: Quantum vs Classical Erasure: Equal Bounds but Unequal Costs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q73LP3DG}},
  note         = {Machine review of arXiv:2607.27341}
}
read the original abstract

Irreversibility has a fundamental thermodynamic cost, erasing information inevitably generates heat. This connection is quantified by the Landauer bound, which gives the minimum dissipation needed to erase a single bit of information. While this bound applies in both classical and quantum settings, it is saturated only in idealised limits of infinite resources. Here, we provide a unified first principles description of finite-resource erasure in both classical and quantum systems. We begin by proving the communal folklore that in the idealised regime the erasure cost of a bit encoded in a quantum or classical system is the same. Despite this, we show that their practical implementation differs substantially: achieving comparable erasure quality in quantum systems requires more control, larger accessible energy gaps and longer operation times. Classical protocols can achieve the erasure of a comparable quantum protocol under far weaker constraints which we expose in trade-off relations. Our results explain why practical erasure schemes fall short of Landauer's bound and show that classical systems enjoy several fundamental thermodynamic advantages.

Figures

Figures reproduced from arXiv: 2607.27341 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 2 canonical work pages

  1. [1]

    Thus ifN > Nmin we know that the classical system will be able to reach fidelities that are impossible for the qubit vianpopulation exchanges with an out-of-equilibrium reservoir

    +O(1),(9) which is the minimum number of subsystems the classical system needs to be comprised of in order to achieve the same asymptotic fidelity i.e.F max Q (n) =F max C (Nmin). Thus ifN > Nmin we know that the classical system will be able to reach fidelities that are impossible for the qubit vianpopulation exchanges with an out-of-equilibrium reservoi...

  2. [2]

    Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Devel- opment5, 183 (1961)

    R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Devel- opment5, 183 (1961)

    1961

  3. [3]

    C. H. Bennett, The thermodynamics of computation—a review, International Journal of Theoretical Physics21, 905 (1982)

    1982

  4. [4]

    Taranto, F

    P. Taranto, F. Bakhshinezhad, A. Bluhm, R. Silva, N. Friis, M. P. Lock, G. Vitagliano, F. C. Binder, T. De- barba, E. Schwarzhans, F. Clivaz, and M. Huber, Lan- dauer versus nernst: What is the true cost of cooling a quantum system?, PRX Quantum4, 010332 (2023)

    2023

  5. [5]

    Proesmans, J

    K. Proesmans, J. Ehrich, and J. Bechhoefer, Finite- time landauer principle, Physical Review Letters125, 10.1103/physrevlett.125.100602 (2020)

    work page doi:10.1103/physrevlett.125.100602 2020

  6. [6]

    Rolandi and M

    A. Rolandi and M. Perarnau-Llobet, Finite-time lan- dauer principle beyond weak coupling, Quantum7, 1161 (2023)

    2023

  7. [7]

    Van Vu and K

    T. Van Vu and K. Saito, Time-cost-error trade-off re- lation in thermodynamics: The third law and beyond, Phys. Rev. X15, 041029 (2025)

    2025

  8. [8]

    A. Ho, E. Erdil, and T. Besiroglu, Limits to the energy efficiency of cmos microprocessors (2023), arXiv:2312.08595 [cs.ET]

    arXiv 2023

Show all 47 references
  1. [9]

    Pop, Energy dissipation and transport in nanoscale devices, Nano Research3, 147 (2010)

    E. Pop, Energy dissipation and transport in nanoscale devices, Nano Research3, 147 (2010)

    2010

  2. [10]

    Carrasco-Codina, P

    M. Carrasco-Codina, P. Escofet, P. Hilaire, A. Soret, S. Nerenberg, V. Champain, G. Milburn, K. Theophilo, S. H. Li, I. Bautista, A. G´ omez, J. Miralles, S. Abadal, C. G. Almud´ ever, E. Alarc´ on, and R. Yehia, Energy effi- ciency of quantum computers (2026), arXiv:2605.15090

    2026 arXiv

  3. [11]

    Fellous-Asiani, J

    M. Fellous-Asiani, J. H. Chai, Y. Thonnart, H. K. Ng, R. S. Whitney, and A. Auff` eves, Optimizing resource effi- ciencies for scalable full-stack quantum computers, PRX Quantum4, 040319 (2023)

    2023

  4. [12]

    Meier, M

    F. Meier, M. Huber, P. Erker, and J. Xuereb, Au- tonomous quantum processing unit: an autonomous thermal computing machine & its physical limitations, Reports on Progress in Physics89, 077601 (2026)

    2026

  5. [13]

    Reeb and M

    D. Reeb and M. M. Wolf, An improved landauer principle with finite-size corrections, New Journal of Physics16, 103011 (2014)

    2014

  6. [14]

    Buffoni and M

    L. Buffoni and M. Campisi, Cooperative quantum in- formation erasure, Quantum7, 961 (2023), 2206.10230 [quant-ph]

    2023 arXiv

  7. [15]

    Taranto, F

    P. Taranto, F. Bakhshinezhad, P. Sch¨ uttelkopf, F. Clivaz, and M. Huber, Exponential improvement for quantum cooling through finite-memory effects, Physical Review Applied14, 054005 (2020), 2004.00323 [quant-ph]

    2020 arXiv

  8. [16]

    Taranto, P

    P. Taranto, P. Lipka-Bartosik, N. A. Rodr ´ ıguez- Briones, M. Perarnau-Llobet, N. Friis, M. Huber, and P. Bakhshinezhad, Efficiently Cooling Quantum Systems with Finite Resources: Insights from Thermodynamic Geometry, Physical Review Letters134, 070401 (2025), 2404.06649 [quant-ph]

    2025 arXiv

  9. [17]

    Xuereb, B

    J. Xuereb, B. Stratton, A. Rolandi, J. He, M. Huber, and P. Bakhshinezhad, Cooling a qubit using n others, PRX Quantum6, 040368 (2025)

    2025

  10. [18]

    H. J. D. Miller, G. Guarnieri, M. T. Mitchison, and J. Goold, Quantum fluctuations hinder finite-time infor- mation erasure near the Landauer limit, Physical Review Letters125, 160602 (2020), 2007.01882 [quant-ph]

    2020 arXiv

  11. [19]

    Dago and L

    S. Dago and L. Bellon, Fooling the Landauer bound with a demon biased thermal bath (2026), 2602.06560 [cond- mat]

    2026

  12. [20]

    M. B. Plenio and V. Vitelli, The physics of forget- ting: Landauer’s erasure principle and information the- ory, Contemporary Physics42, 25–60 (2001)

    2001

  13. [21]

    Koomey, S

    J. Koomey, S. Berard, M. Sanchez, and H. Wong, Im- plications of historical trends in the electrical efficiency of computing, IEEE Annals of the History of Computing 33, 46 (2011)

    2011

  14. [22]

    Van Vu and K

    T. Van Vu and K. Saito, Finite-time quantum landauer principle and quantum coherence, Physical Review Let- ters128(2022)

    2022

  15. [23]

    Y.-Z. Zhen, D. Egloff, K. Modi, and O. Dahlsten, Uni- versal bound on energy cost of bit reset in finite time, Phys. Rev. Lett.127, 190602 (2021)

    2021

  16. [24]

    Lipka-Bartosik and M

    P. Lipka-Bartosik and M. Perarnau-Llobet, Minimiz- ing dissipation via interacting environments: Quadratic convergence to landauer bound, Phys. Rev. Lett.135, 170404 (2025)

    2025

  17. [25]

    Aimet, M

    S. Aimet, M. Tajik, G. Tournaire, P. Sch¨ uttelkopf, J. Sabino, S. Sotiriadis, G. Guarnieri, J. Schmiedmayer, and J. Eisert, Experimentally probing landauer’s princi- ple in the quantum many-body regime, Nature Physics 21, 1326 (2025)

    2025

  18. [26]

    Brunner, N

    N. Brunner, N. Linden, S. Popescu, and P. Skrzypczyk, Virtual qubits, virtual temperatures, and the foundations of thermodynamics, Phys. Rev. E85, 051117 (2012)

    2012

  19. [27]

    M. T. Mitchison, Quantum thermal absorption machines: refrigerators, engines and clocks, Contemporary Physics 60, 164–187 (2019)

    2019

  20. [28]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge Univer- sity Press, 2010). 8

    2010

  21. [29]

    In the supplemental material we provide the derivation of the finite-error Landauer bound, the virtual-qubit cool- ing dynamics, the equality in dissipation for an equally spaced ladder as classical system, the majority vote scal- ing in subsystem number, the minimum subsystem...

  22. [30]

    Clivaz, R

    F. Clivaz, R. Silva, G. Haack, J. B. Brask, N. Brunner, and M. Huber, Unifying paradigms of quantum refrig- eration: A universal and attainable bound on cooling, Physical Review Letters123(2019)

    2019

  23. [31]

    L. L. Yan, T. P. Xiong, K. Rehan, F. Zhou, D. F. Liang, L. Chen, J. Q. Zhang, W. L. Yang, Z. H. Ma, and M. Feng, Single-atom demonstration of the quantum lan- dauer principle, Phys. Rev. Lett.120, 210601 (2018)

    2018

  24. [32]

    Leibfried, R

    D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys.75, 281 (2003)

    2003

  25. [33]

    B¨ aumer, M

    E. B¨ aumer, M. Perarnau-Llobet, P. Kammerlander, H. Wilming, and R. Renner, Imperfect Thermalizations Allow for Optimal Thermodynamic Processes, Quantum 3, 153 (2019)

    2019

  26. [34]

    Xuereb, P

    J. Xuereb, P. Erker, F. Meier, M. T. Mitchison, and M. Huber, Impact of imperfect timekeeping on quantum control, Physical Review Letters131(2023)

    2023

  27. [35]

    A. Levy, R. Alicki, and R. Kosloff, Quantum refrigerators and the third law of thermodynamics, Physical Review E85(2012)

    2012

  28. [36]

    Masanes and J

    L. Masanes and J. Oppenheim, A general derivation and quantification of the third law of thermodynamics, Na- ture Communications8, 14538 (2017)

    2017

  29. [37]

    Fiorelli, M

    E. Fiorelli, M. M¨ uller, I. Lesanovsky, and F. Carollo, Mean-field dynamics of open quantum systems with col- lective operator-valued rates: validity and application, New Journal of Physics25, 083010 (2023)

    2023

  30. [38]

    Leccabue and V

    F. Leccabue and V. Sagredo,Mag- netism, Magnetic Materials and Their Ap- plications(WORLD SCIENTIFIC, 1996) https://www.worldscientific.com/doi/pdf/10.1142/3164

    1996 doi

  31. [39]

    Taur and T

    Y. Taur and T. H. Ning,Fundamentals of Modern VLSI Devices, 2nd ed. (Cambridge University Press, 2009)

    2009

  32. [40]

    Taranto, P

    P. Taranto, P. Lipka-Bartosik, N. A. Rodr ´ ıguez- Briones, M. Perarnau-Llobet, N. Friis, M. Huber, and P. Bakhshinezhad, Efficiently cooling quantum systems with finite resources: Insights from thermodynamic ge- ometry, Physical Review Letters134(2025)

    2025

  33. [41]

    Janovitch, M

    M. Janovitch, M. Brunelli, and P. P. Potts, Active quan- tum reservoir engineering: Using a qubit to manipulate its environment (2025), arXiv:2505.16898 [quant-ph]

    2025 arXiv

  34. [42]

    A. Usui, W. Niedenzu, and M. Huber, Simplifying the de- sign of multilevel thermal machines using virtual qubits, Phys. Rev. A104, 042224 (2021)

    2021

  35. [43]

    R. B. Ash,Information Theory(Dover Publications, 1990)

    1990

  36. [44]

    R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. Chantrell, Atomistic spin model simulations of magnetic nanomaterials, Journal of Physics: Condensed Matter26, 103202 (2014)

    2014

  37. [45]

    Gyorfi, P

    L. Gyorfi, P. Harremoes, and G. Tusnady, Some re- finements of large deviation tail probabilities (2012), arXiv:1205.1005 [math.ST]. 1 Supplemental Material: Quantum vs Classical Erasure: Equal Bounds but Unequal Costs Proof of the Equivalent Landauer Bound of Classical and Qu...

    2012 arXiv

  38. [46]

    + log log(1−w1) log(1−w2 1) log (1−w 2

  39. [47]

    We model this by timing each interaction byτ= π 2g + ε g , where we drawεfrom a normal distributionε=N(0, s 2), with mean 0 and varianceswiths≪1

    +O 1 n .(S19) Effects of Imperfect Timekeeping or Missing Knowledge of the Couplings of the Protocol In this section we assume that we do not have access to perfect knowledge of the coupling constantgor equally we could have an imperfect clock. We model this by timing each int...

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.