{"id":"563942c5-4061-4d51-976e-a122c7160eb1","arxiv_id":"2411.12805","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Landauer heating from quantum error correction creates a phase transition between stable and runaway error rates, and current superconducting qubit parameters for Shor's 2048-bit factoring sit in the stable phase.","lead":"This paper models heat produced by quantum error correction and asks whether it can overwhelm cooling as quantum computers scale. It finds a phase transition between stable and runaway heating, and estimates that current superconducting hardware parameters stay in the stable regime for a 2048-bit factoring task.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed unbounded-error phase depends entirely on the assumed divergent QEC frequency f = (p_f/(1-p_f))^{1/4}; with a finite or fixed syndrome clock the runaway disappears, so the phase transition may be an artifact of the ansatz.","rationale":"The reader's weakest assumption already identified the divergent QEC frequency as a key vulnerability; I sharpen that concern into the central load-bearing objection. The paper's novelty is the phase transition itself, and its existence requires an unbounded f(T). That divergence is not a consequence of thermodynamics or of the threshold theorem; it is imposed by hand in Sec. III. Real QEC has finite clock speed, and a bounded QEC rate removes the divergent heat source, giving a finite thermal fixed point and no unbounded-error phase. This does not necessarily invalidate the order-of-magnitude conclusion that current superconducting parameters keep the Shor-scale system in a stable low-temperature regime, because that conclusion may survive with a capped frequency. But the qualitative claim of a dynamical phase transition between bounded-error and unbounded-error phases would need to be reframed as a simple crossing of the fault-tolerance threshold, not a thermal runaway. Because the reader already demanded conditions on exactly this assumption, the appropriate action is to keep the CONDITIONAL verdict and add the finite-clock test as an explicit condition. I am not recommending REJECT because the paper is a transparent toy model and its quantitative estimate can, in principle, be salvaged; however, the core phase-transition claim must be tested against finite f_max before it can be accepted as a physical result.","tokens_in":11436,"tokens_out":8045,"duration_ms":91555,"concrete_test":"Rerun the numerical simulations behind Figs. 4 and 5 with the heating term in Eq. (8) modified to use f_cap(T) = min((p_f/(1-p_f))^{1/4}, f_max), sweeping f_max over a wide range (e.g., 10^-4 to 1 in units of rounds per time step) and also running the constant-clock case f(T) = f_clock. If the unbounded-error phase and the 1/τ divergence in Fig. 5 disappear for every finite f_max, the phase transition is an artifact of the divergent ansatz. As a secondary check, recompute the Sec. V Shor operating point under capped f to see whether the bounded-error conclusion remains stable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result is the dynamical phase transition between bounded-error and unbounded-error regimes (Sec. IV). Its mechanism is the feedback loop: QEC heat raises T, T raises p_err(T), and p_err increases the QEC frequency f(T), which adds more heat. The model closes this loop with the assumption, stated after Eq. (8), that f(T) = (p_f/(1-p_f))^{c_f} with c_f = 1/4, where p_f = (p_err/p_th)^{d/2}. This function diverges as p_err approaches p_th, and the unbounded-error phase is driven by exactly that divergence: heat deposition per unit time becomes infinite at threshold. Physical QEC hardware does not work this way. Syndrome extraction runs on a finite clock set by measurement and reset latencies; the surface-code cycle frequency is bounded by hardware speed, and the threshold theorem requires only a constant QEC rate below p_th rather than a rate that diverges at p_th. If f is replaced by a capped function f_cap = min(f(T), f_max) or by a fixed clock rate, the divergent heat source is removed. In the resulting lumped model, heating power grows at most linearly in T while the refrigerator term in Eq. (7) grows as T^2 at large T, so a finite fixed point exists for every cooling rate. The unbounded-error phase therefore collapses into ordinary logical failure above threshold, and the claimed thermodynamic phase transition is not established. The quantitative Shor conclusion of Sec. V may still survive, but the qualitative phase-transition claim rests on an unphysical divergence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11828,"tokens_out":6786,"duration_ms":75497,"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":[{"comment":"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.","section":"Sec. III, after Eq. (8); Sec. IV"},{"comment":"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.","section":"Eq. (4) and Sec. V A"},{"comment":"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.","section":"Sec. V A, paragraph beginning 'Various studies have related...'"}],"minor_comments":[{"comment":"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.","section":"Sec. III, after Eq. (3)"},{"comment":"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.","section":"Fig. 4 caption"},{"comment":"The phrase 'the total number of atoms in the superstate' appears to be a typo for 'substrate'; please correct it.","section":"Sec. V A"},{"comment":"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.","section":"Sec. V B and Fig. 5"},{"comment":"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.","section":"Sec. I, last paragraph"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the topic is timely, but the main phase-transition claim is not yet supported because the divergent QEC-frequency assumption is unphysical and load-bearing. The authors might be able to revise by either (i) replacing the divergence with a finite maximum QEC rate and reframing the result as a crossover/bistability rather than a true dynamical phase transition, or (ii) providing an explicit physical mechanism that produces a diverging QEC rate at threshold. The quantitative Shor conclusion would additionally need a sensitivity analysis over the assumed p_err(T) model. I do not see citation or novelty concerns beyond what is stated in the main report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a transparent, well-parameterized toy model of Landauer heating in a scaled-up superconducting QC. The new piece is the feedback loop between QEC rate, temperature, and error rate, and the phase diagram that comes out of it. The quantitative conclusion—that a 20-million-qubit Shor machine sits in the bounded-error phase—is plausible and useful. But the headline phase transition is not robust: it relies on the assumed diverging QEC frequency f ~ (p_f/(1-p_f))^{1/4}, which blows up as the error rate approaches threshold. Real QEC runs on a finite clock. Cap f and the temperature runaway disappears; you're left with ordinary logical failure above threshold, not a thermodynamic phase transition. The paper acknowledges the form of f is unknown, but it doesn't probe how much the phase diagram depends on that choice.\n\nWhat I like: the model is simple enough to follow, the parameter choices are grounded in literature (Debye heat capacity, silicon phonon mean free path, BlueFors fridge specs), and the authors are upfront that this is an order-of-magnitude estimate. The quasi-linear approximation is sensible and validated against the full simulation. The discussion of alternative platforms (neutral atoms, ion) is thoughtful.\n\nSoft spots, in proportion: (1) The divergent f is the load-bearing assumption; a sensitivity analysis with capped f is essential. My reading of the stress-test note: its specific claim that the cooling term grows as T^2 at large T is wrong—Eq. (7) actually falls as 1/T because of the T^3 heat capacity—but the broader worry about f is correct. (2) perr(T)=0.1T with a 100mK threshold is set by hand, not measured; the conclusion that the Shor point is bounded should come with error bars on B and T_th. (3) The binary Q function in Eq. (4) is a bit underspecified, though the time-averaged heating rate is clear enough.\n\nWho's it for: people making hardware roadmaps for on-chip QEC, and anyone studying thermodynamic limits of quantum computing. It's a first pass, not a final word.\n\nRecommendation: send it to review. A competent referee can demand a finite-clock-speed variant and a sensitivity analysis; the paper as is is worth discussing, not desk-rejecting.","headline":"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.","tokens_in":12315,"tokens_out":5672,"would_cite":true,"duration_ms":54530,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["quantum error correction","Landauer's principle","thermodynamic phase transition","fault tolerance threshold","superconducting qubits","Shor's algorithm","heat dissipation","dilution refrigerator"],"falsifier":"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.","tokens_in":11228,"feed_emoji":"❄️","tokens_out":7155,"duration_ms":60868,"temperature":0.7,"pith_summary":"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.","feed_headline":"Cooling threshold decides if quantum error correction can scale","feed_subtitle":"A 20-million-qubit Shor computation stays in the safe, bounded-error phase with today's cooling and error rates.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes Landauer's principle, the basis for the heat generated by each ancilla reset in QEC.","marker":"[1]"},{"why":"Supply the threshold theorem that defines the physical error-rate threshold below which fault tolerance is possible.","marker":"[10, 11]"},{"why":"Provides the approximately 20-million-qubit resource estimate for factoring 2048-bit RSA integers used as the n_a parameter.","marker":"[15]"},{"why":"Sets the surface-code threshold p_th = 1% used to define the fault-tolerance boundary.","marker":"[31]"},{"why":"Gives the surface-code logical failure scaling p_f = (p_err/p_th)^{d_c/2} used in the QEC frequency model.","marker":"[32]"},{"why":"Support the linear temperature dependence of the physical error rate below about 100 mK used to fix perr(T).","marker":"[38-40]"},{"why":"Provides the BlueFors dilution refrigerator cooling parameters (84 ˙n3 and base temperature) used to set γ.","marker":"[30]"}],"fun_headline_variants":["Cooling rate sets the limit for fault-tolerant quantum computing","Heating from error correction decides if quantum computers can scale","Landauer heating sets a cooling threshold for fault-tolerant QC","20-million-qubit Shor computation stays in the safe error phase","Cooling beats Landauer heating to keep quantum error correction viable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Cooling rate sets the limit for fault-tolerant quantum computing","Heating from error correction decides if quantum computers can scale","Landauer heating sets a cooling threshold for fault-tolerant QC","20-million-qubit Shor computation stays in the safe error phase","Cooling beats Landauer heating to keep quantum error correction viable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000775,"raw_usage":{"total_tokens":3399,"prompt_tokens":883,"completion_tokens":2516,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":2432}},"tokens_in":499,"tokens_out":2516,"duration_ms":17050,"temperature":1.0,"reasoning_tokens":2432,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:12:25.013011+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Gidney and M","cited_arxiv_id":null,"evidence_quote":"Provides the approximately 20-million-qubit resource estimate for factoring 2048-bit RSA integers used as the n_a parameter."}],"review_version":1}