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Thermodynamic limitations on fault-tolerant quantum computing

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Quantum error correction must dump heat, and this paper shows that fault tolerance survives only if cooling beats a threshold, with current superconducting hardware staying in the safe phase for a 20-million-qubit Shor computation.

desk verdict A transparent order-of-magnitude study of Landauer heating in large superconducting QCs, with a fresh feedback-loop model, but the headline phase transition rests on an unphysical diverging QEC rate and needs a finite-clock variant before the qualitative claim is trusted. read the letter →

arxiv 2411.12805 v2 pith:422GYVXS submitted 2024-11-19 quant-ph

Mykhailo Bilokur , Sarang Gopalakrishnan , Shayan Majidy This is my paper · ORCID
classification quant-ph
keywords quantumerrorcorrectionLandauer'sprinciplethermodynamicphasetransitionfaulttolerancethresholdsuperconductingqubitsShor'salgorithmheatdissipationdilutionrefrigerator
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

Quantum error correction erases information, and Landauer's principle says that erasure must release heat into the environment. This paper argues that when error correction is done on-chip, that heating feeds back: higher temperature raises the physical error rate, which demands more frequent error correction, which heats more. The authors build a one-dimensional diffusion model of a qubit array coupled to a dilution refrigerator and find a dynamical phase transition between a bounded-error phase, where temperature and error rates stabilize, and an unbounded-error phase, where runaway heating makes fault tolerance impossible. Applying the model with current superconducting-qubit parameters to Shor's algorithm for factoring 2048-bit RSA integers (about 20 million physical qubits), they find the system sits in the bounded-error phase. The claim is that Landauer heating should not block scalable fault tolerance if current hardware capabilities are maintained.

What carries the argument

The machinery is a discrete heat-balance equation on a one-dimensional lattice: temperature updates combine a Landauer heating term from QEC (with heat deposited at the qubit sites at a rate set by the error-correction frequency), a Fourier diffusion term through the silicon substrate, and a dilution-refrigerator cooling term at the cold boundary. The feedback loop is closed by a QEC frequency $f(T) = (p_f/(1-p_f))^{c_f}$ that diverges as the logical failure probability $p_f$ approaches the threshold, with $p_f = (p_{\rm err}/p_{\rm th})^{d_c/2}$ for the surface code. The phase transition is controlled by the heating coefficient $\alpha$, diffusion coefficient $\delta$, and cooling coefficient $\gamma$, and the boundary in the $(\alpha,\gamma)$ plane is sharp, with a critical exponent $\zeta \approx 1/2$ for the inverse failure time.

What would settle it

Measure the physical error rate of a superconducting transmon as a function of temperature in a dilution refrigerator from 10 mK to 200 mK. If the 1% threshold is not crossed near 100 mK, or if the error rate does not grow linearly in that range, then the specific bounded-error prediction for the 20-million-qubit Shor computation does not follow from the model.

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Extended reading notes

Core claim

The central claim is that fault-tolerant quantum computing is only sustainable when the cooling rate exceeds a threshold set by the competition between Landauer heating, thermal diffusion, and refrigerator cooling. Below that threshold the temperature near the ancilla qubits rises without bound, driving the physical error rate $p_{\rm err}$ through the fault-tolerance threshold $p_{\rm th}$; above it the temperature stabilizes and error rates stay bounded. For a superconducting transmon system on a silicon substrate running Shor's algorithm to factor a 2048-bit RSA integer, the authors estimate about $2\times10^7$ physical qubits and find the system in the bounded-error phase with current parameters. The paper's conclusion is that Landauer heating does not pose a fundamental obstacle to scaling, provided current error rates and cooling powers are preserved.

Load-bearing premise

The model assumes that error correction can be applied arbitrarily fast as the error rate nears the threshold (with $f=(p_f/(1-p_f))^{1/4}$), and that the physical error rate equals $0.1$ times the temperature in kelvin, crossing the threshold at 100 mK; if the clock speed saturates or the error-rate curve is different, the phase boundary and the conclusion for Shor's algorithm change.

Editorial extensions

If this is right

  • Any autonomous error-correction protocol that protects a quantum memory in steady state must extract heat fast enough to counteract Landauer heating, or the physical error rate will run away.
  • For superconducting qubits, scaling to roughly 20 million physical qubits for Shor's algorithm remains in the bounded-error phase with current parameters, so heat dissipation alone does not preclude RSA-breaking-scale computation.
  • The transition to the unbounded-error phase is sharp: the inverse failure time vanishes with a critical exponent near 1/2, so the onset of runaway heating is sudden rather than gradual.
  • Without any cooling, the modeled 20-million-qubit system crosses the error threshold on the order of seconds, showing that the cooling term, not diffusion, is what keeps the system safe.
  • The thermodynamic constraint only becomes relevant when error correction is performed on-chip; near-term devices with classical syndrome storage away from the cryostat do not face this feedback loop.

Reading between the lines

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

  • If the physical error rate rises faster than linearly with temperature above 100 mK, the real operating margin for a 20-million-qubit Shor computation could be narrower than the model's bounded-error verdict suggests.
  • The same runaway-heating logic can be translated to neutral-atom or ion architectures where entropy is carried away by photons; in modular designs with optical cavities, reabsorption of emitted photons could recreate an effective unbounded-error regime.
  • A small-scale experiment with on-chip QEC and continuous thermometry could test the phase transition directly: a stable temperature plateau indicates the bounded-error phase, while a slow upward drift signals proximity to the unbounded phase.
  • The critical exponent near 1/2 hints at a generic mean-field-like scaling, which, if confirmed, would let designers extrapolate safe cooling budgets from small devices to million-qubit systems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a one-dimensional heat-diffusion model for an array of superconducting qubits in which quantum error correction (QEC) deposits Landauer heat, a refrigerator removes heat, and the QEC frequency increases with the temperature-dependent physical error rate. The authors identify a dynamical phase transition between a bounded-error phase, in which temperature and error rate stabilize below threshold, and an unbounded-error phase, in which runaway heating pushes the error rate above threshold. They apply the model to a 20-million-qubit Shor factoring circuit for 2048-bit RSA using surface-code parameters and conclude that current experimental parameters place the system in the bounded-error phase, so Landauer heating need not limit scalable fault tolerance if these parameters are maintained.

Significance. If correct, the claimed phase transition would be a conceptually interesting thermodynamic constraint on fault-tolerant quantum computing, and the quantitative Shor estimate would be a useful order-of-magnitude data point. The paper is transparent about the model's structure: the coefficients in Eq. (8) are dimensionful and explicitly defined, the numerical quasi-linear approximation is checked against exact simulation in Fig. 6, and the parameter choices are, for the most part, traceable to cited experimental literature. The central conceptual claim, however, rests on an assumption about the QEC frequency that is not physically motivated and that, if replaced by a finite clock rate, removes the runaway that defines the unbounded-error phase. The quantitative conclusion is therefore conditional on an unvalidated error-rate model, so the significance of the result as it stands is substantially reduced.

major comments (3)
  1. [Sec. III, after Eq. (8); Sec. IV] The existence of the unbounded-error phase is driven entirely by the assumption that the QEC frequency f(T) diverges as the error probability approaches threshold, i.e., f=(p_f/(1-p_f))^{c_f}. Physical QEC hardware has a finite syndrome-extraction clock, and the standard threshold theorem does not require the QEC rate to diverge at p_th. If f is replaced by a capped function f_cap = min(f, f_max), the heating term in Eq. (8) is bounded by f_max * α/T^2, while the cooling term in Eq. (7) behaves as -γ/T for T >> T0. Consequently, dT/dt becomes negative for sufficiently large T and a finite fixed point exists for every positive cooling rate; the 'unbounded-error' runaway in Figs. 3 and 4 disappears. The paper's own admission that the precise form of f is uncertain ('To our knowledge, the precise form of this function remains uncertain') is located exactly at this load-bearing point. A concrete test would be to repeat the phase diagram of Fig. 4 with a finite maximum QEC rate and show whether any true runaway survives; as written, the central phase-transition claim is not established.
  2. [Eq. (4) and Sec. V A] The binary function Q[f(T_r)] in Eq. (4) is never defined: the text says only that it equals 1 when QEC occurs in a time step and 0 otherwise, but f is introduced as a frequency and enters with no units or conversion to a probability per time step. This ambiguity is consequential because the heat-deposition rate is the product of the heat per round and the rate of rounds; if f is a dimensionless probability per Δt, the model needs a different equation, and if f is a rate in Hz, Eq. (4) must contain a factor f Δt (or a stochastic rule) rather than a binary indicator. The quantitative results in Sec. V B, including the 'on the order of seconds' breakdown time, depend on this normalization, so the model as written is under-specified.
  3. [Sec. V A, paragraph beginning 'Various studies have related...'] The error-rate model p_err(T)=B T with B=0.1 and a threshold at 100 mK is set by hand: the threshold temperature is assumed rather than measured, and B follows from that assumption together with p_th=1%. The conclusion that the 20-million-qubit Shor system is in the bounded-error phase is sensitive to this choice, because both the feedback loop and the distance to threshold depend on p_err(T). The authors should provide a sensitivity analysis over the threshold temperature and over B (or over the exponent n), and they should state clearly which experimental data, if any, fix these values; without that, the quantitative claim in Sec. V B is not robust.
minor comments (5)
  1. [Sec. III, after Eq. (3)] The sentence 'we enter the unbounded error phase at temperatures significantly below Θ_D' refers to a phase that has not yet been defined at that point in the paper; this is a terminology issue that should be clarified.
  2. [Fig. 4 caption] The caption says 'The terms are called by T0^3 and T0^2, respectively, to ensure the term inside the log is unitless,' but the axes in the figure are not labeled in the text; the figure should state explicitly what is plotted on the x- and y-axes, including the dimensionless combinations used.
  3. [Sec. V A] The phrase 'the total number of atoms in the superstate' appears to be a typo for 'substrate'; please correct it.
  4. [Sec. V B and Fig. 5] The claim of a critical exponent ζ ≈ 1/2 from Fig. 5 is not supported by any fit details, error bars, or finite-size discussion; since the 'unbounded-error' phase itself is already called into question, this exponent should either be derived more carefully or removed.
  5. [Sec. I, last paragraph] The abstract and introduction consistently say '10^7 qubits' while Sec. V A uses n_a = 20×10^6; the relationship between these numbers should be made explicit in the main text, not just in the parameter list.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phase boundary and the Shor-2048 bounded-phase conclusion are computed from the stated dynamical equations using parameters drawn from external hardware and algorithm resources, not from fitted outputs or load-bearing same-author citations.

full rationale

The paper's central derivation is self-contained. The model in Sec. III couples QEC heating, diffusion, and refrigerator cooling in Eq. (8), with coefficients alpha, gamma, and delta fixed by silicon Debye data, BlueFors dilution-refrigerator specs, and geometric/chip parameters (Eqs. (9)-(12)). The QEC frequency ansatz f(T)=(p_f/(1-p_f))^{c_f} in Sec. III and the error-temperature law p_err(T)=0.1T in Sec. V A are explicitly stated as assumptions rather than being fitted to the phase outcome; the bounded/unbounded classification in Sec. IV and the numerical Shor result in Sec. V B are outputs of the coupled dynamics, not inputs. The unbounded phase does follow from the assumed divergence of f at p_th, but this is a modeling assumption, not a circular reduction, and the existence of a cooling-rate threshold is a nontrivial stability property of Eq. (8). Same-author references ([2], [6], [44]-[46]) are contextual or forward-looking and do not supply any load-bearing premise. The paper itself flags the uncertain form of f in Sec. III and the manual 100mK threshold calibration in Sec. V A; these are realism and robustness risks, not circularity.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The only truly free numbers are the QEC-response exponent c_f and the error-vs-temperature curve parameters (B, n, T_th, p0). The latter are chosen to make the threshold occur at 100mK and are not measured. The remaining inputs (Lambda, Theta_D, fridge power, qubit number) come from literature or device specs. No new physical entities are introduced.

free parameters (5)
  • c_f = 1/4
    Tunable exponent in QEC frequency response f=(pf/(1-pf))^{c_f}; no sensitivity analysis provided.
  • B = 0.1 K^-1
    Coefficient in perr(T) = B T (n=1); chosen so that perr reaches p_th=1% at 100mK.
  • T_th (threshold temperature) = 100 mK
    Assumed temperature at which perr crosses the surface-code threshold; taken from a rough reading of Refs [38-40], not a direct measurement.
  • n (error-rate exponent) = 1
    Exponent in perr temperature scaling, taken from Refs [38-40] for T<100mK; extrapolation beyond is unvalidated.
  • p0 (base error probability) = 0
    Temperature-independent error floor set to zero in the simplified model; not varied or constrained by data.
assumptions (7)
  • domain assumption Landauer's principle: erasing one bit of information releases at least k_B T ln2 of heat.
    Invoked in Sec. II (Eq. 2) to set the heat release per QEC round, n_a ln2 k_B T.
  • domain assumption Debye heat capacity approximation C_HC = A T^3 for T << Theta_D.
    Used in Eq. (3); assumed valid throughout the operational regime, with Theta_D for silicon at 636K.
  • domain assumption Fourier's law of heat diffusion with constant phonon mean free path.
    Used in Eqs. (5)-(6); assumes Lambda fixed at 0.5mm even though phonon MFP can be temperature-dependent.
  • domain assumption Dilution refrigerator cooling power Q_dot = 84 n_dot3 T_F^2 with steady-state load subtraction.
    Used in Eq. (7) and Sec. V A; the steady-state subtraction is a modeling choice to avoid unphysical cooling below T0.
  • ad hoc to paper QEC frequency feedback function f(T) = (pf/(1-pf))^{c_f}, with f diverging at the threshold.
    Chosen in Sec. III as the simplest form satisfying three stated limits; not derived from control theory or hardware constraints.
  • ad hoc to paper Error probability model perr(T) = B T with threshold at 100mK.
    Set by hand in Sec. V A; this assumption determines where the Shor point sits relative to the phase boundary.
  • domain assumption One-dimensional slab geometry with uniform qubit parameters and heat generated at one end.
    Introduced in Sec. III as a 'representative framework' for superconducting qubit arrays; the authors acknowledge it is a toy model.

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Pith. "Pith review of Thermodynamic limitations on fault-tolerant quantum computing." pith.science (2026). https://pith.science/paper/422GYVXS

@misc{pith2026241112805,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic limitations on fault-tolerant quantum computing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/422GYVXS}},
  note         = {Machine review of arXiv:2411.12805}
}
read the original abstract

We investigate the thermodynamic limits on scaling fault-tolerant quantum computers due to heating from quantum error correction (QEC). Quantum computers require error correction, which accounts for 99.9% of the qubit demand and generates heat through information-erasing processes. This heating increases the error rate, necessitating more rounds of error correction. We introduce a dynamical model that characterizes heat generation and dissipation for arrays of qubits weakly coupled to a refrigerator and identify a dynamical phase transition between two operational regimes: a bounded-error phase, where temperature stabilizes and error rates remain below fault-tolerance thresholds, and an unbounded-error phase, where rising temperatures drive error rates beyond sustainable levels, making fault tolerance infeasible. Applying our model to a superconducting qubit system performing Shor's algorithm to factor 2048-bit RSA integers, we find that current experimental parameters place the system in the bounded-error phase. Our results indicate that, while inherent heating can become significant, this thermodynamic constraint should not limit scalable fault tolerance if current hardware capabilities are maintained as systems scale.

Figures

Figures reproduced from arXiv: 2411.12805 by the authors.

Figure 1
Figure 1. Model of heat flow during quantum error correction. A system of qubits embedded in an environment at temperature T, enclosed by a refrigerator at temperature T0. An external device performs periodic QEC rounds to measure and reset the qubits, dissipating heat ∆Q into the environment. The QEC round frequency f(perr) depends on the error rate perr, which increases with temperature T. Each QEC round can raise T and per… view at source ↗
Figure 2
Figure 2. Representation of the physical setup. The model consists of a slab geometry with a two-dimensional array of qubits at the top, where erasure processes generate heat. A fridge is coupled to the bottom for cooling. Heat transport occurs via diffusion through the slab, with the local temperature at the top influencing the error correction rate. This simplified one-dimensional heat flow model assumes uniform qubit param… view at source ↗
Figure 3
Figure 3. Dynamical behaviour of the two phases The temperature immediately surrounding the qubits, T⃗r1 (t), is plotted against time for different values of the cooling coefficient γ, with fixed α and δ. The dashed line represents the temperature corresponding to the error threshold pth. In the bounded-error phase (blue curves), the temperature stabilizes over time. In the unbounded error phase (red curves), the temperature … view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: Sharpness of the phase transition. The cooling coefficient γ is plotted against 1/τ , where τ is effectively the time at which the error probability perr exceeds the threshold pth. A finite value of 1/τ indicates the unbounded-error phase, while 1/τ = 0 corresponds to …
Figure 6
Figure 6. Figure 6: Comparison of Numerical Methods. Temperature evolution obtained from exact numerical simulations (solid line) versus the quasi-linear approximation (dashed line) for a system with 2 · 107 qubits. The approximation accurately captures the temperature behaviour during th…
Figure 7
Figure 7. Figure 7: Numerical limits using realistic parameters (a) The temperature immediately surrounding the qubits increases over time without any cooling applied. The system enters the unbounded-error phase, where the temperature continually rises, eventually surpassing the threshold…

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