Thermodynamics of Computation

Speculative If the true thermodynamic floor for a real computation is not k_BT ln 2 per erased bit but that quantity plus a mismatch term and a speed term — which is what the stochastic-thermodynamics literature now asserts and nobody has measured on a working circuit — then the first thing this subject would deliver is a number nobody currently has: the actual minimum energy to run a specified logic circuit, at a specified speed, at a specified error rate. That is the condition and it is doing all the work; section 7 says what would establish it.

Established The magnitude, stated against the only baseline anyone quotes. Landauer’s bound at 300 K is 2.9 × 10⁻²¹ J per erased bit, about 0.018 eV. A 2025 review of the field states the gap to real machines as a factor of about a billion; the Institute’s own accounting in Ultra Efficient Computing Energy Systems puts it at roughly 10⁵ per gate-switching event and 10⁸ per delivered floating-point operation, and the disagreement is definitional rather than empirical — it depends entirely on what you call an operation. Frontier Between 10⁵ and 10⁹ is three to four orders of magnitude of unresolved accounting, and the question this page owns is how much of that range is physics and how much is engineering.

Speculative The prize is therefore a correction factor rather than a technology. If the corrected floor for a real circuit turns out to sit a factor of ten or a hundred above k_BT ln 2 per erased bit, then a large slice of the headroom the industry believes it has is not there, and the reachable engineering ceiling that brief reports — about 200× for CMOS, with even odds of stopping short — is closer to the physical limit than to the rhetorical one. Frontier If the corrected floor is within a small factor of the naive one, the opposite follows and the whole gap is engineering.

Handwave Who collects: nobody, directly, and this page does not pretend otherwise. The immediate beneficiaries of a correct floor are the people deciding whether to fund reversible logic, superconducting logic, or a thermodynamic-sampling architecture, and the value to them is the avoidance of a misdirected programme rather than a product. Handwave The condition, plainly: every measurement confirming Landauer’s bound so far has been made on a single bit, in an apparatus whose relaxation time is measured in seconds, running protocols lasting from 7.5 seconds to 16 minutes per bit — and section 5 is about why nobody has repeated one on anything that computes.

The thermodynamics of computation asks what the second law requires of a machine that processes information, as opposed to what a particular machine happens to cost. Its central claim is that logically irreversible operations carry an unavoidable thermodynamic price and logically reversible ones do not, and the price of erasing one bit at temperature T is k_BT ln 2 of heat to the environment.

Established This brief does not own the energy bill of computing. Ultra Efficient Computing Energy Systems owns joules per operation, watts per rack, facility overhead, terawatt-hours and every claim that a new device reduces them; it prices real hardware against the bound and finds the reachable CMOS ceiling at about 200× rather than the hundred million the bound naively suggests. Frontier The question left over, and the one this page owns, is whether k_BT ln 2 per erased bit is the right floor at all for a machine that runs fast, makes errors, corrects them, and never sits in thermal equilibrium.

Established The modern answer, from stochastic thermodynamics, is that the total entropy flow decomposes into three parts and only the first is Landauer’s. The decomposition is Landauer cost, which is the drop in Shannon entropy across the operation; mismatch cost, which is the penalty for feeding the device an input distribution different from the one its physical dynamics is optimised for; and a residual irreversible term. Frontier The second term is the interesting one, because it depends on the statistics of the inputs rather than on the logic, it is zero only in a coincidence, and it is not bounded by anything in Landauer’s argument.

Speculative Add to that a speed term. Landauer’s bound is saturated only quasistatically; every finite-time treatment adds a correction that grows as the inverse of the protocol duration, and at least one strong-coupling quantum treatment fixes the coefficient in terms of the Planckian time ℏ/k_BT. Handwave The three-term picture — erasure plus mismatch plus speed — is now standard in the theory literature and has never been evaluated end to end for any circuit anybody has built. That gap is this brief.

Frontier The framing this page tests is that Landauer’s bound is confirmed but irrelevant, and it survives with one large qualification. The confirmation is real and the irrelevance is real at today’s clock rates, where the fundamental speed penalty is a hundred-thousandth of the bound. Speculative The qualification is that the mismatch term, which nobody has measured, is the one with no small parameter attached, and a subject that has spent sixty years on the term it can calculate is not evidence that the other term is small.

Established The bound itself is not in dispute and its numerical value is settled. At 300 K, k_BT ln 2 is 2.9 × 10⁻²¹ J, about 0.018 eV. Established Nothing in the literature consulted disputes the arithmetic, and the 2025 review of the subject states it in exactly those terms.

Established The commission for this brief names as its anchor a single-bit erasure measured at the Landauer bound in an optical trap and independently confirmed within about 1%. Both halves need correcting, and the corrections are the most useful thing on this page.

Established The optical-trap experiment is real and it did not measure erasure at the bound. The apparatus was a single silica bead of radius 1.00 ± 0.05 µm in water, held in a double-well potential made by switching a 1024 nm trapping laser between two positions at 10 kHz, tracked by camera at 502 Hz. Established The mean dissipated heat falls with protocol duration as ln 2 + B/τ with B about 8.15 k_BT·s, and approaches the bound only in the long-time limit. Established At τ = 40 s the measured value was about 0.59 k_BT — below k_BT ln 2 — with error bars of ±0.15 k_BT, and a two-parameter fit gave an asymptote of 0.72 k_BT, described by the authors as close to k_BT ln 2. Frontier The reason a measurement can sit below the bound is stated in the same paper: the erasure never reaches a success rate higher than about 95%, and an erasure that sometimes fails is not the operation the bound is about.

Established The independent confirmation exists and it is not an optical trap. The 2014 high-precision test used a feedback trap implementing an anti-Brownian electrokinetic virtual potential — a fluorescent particle tracked by camera with electrical forces applied through electrodes — which is a different apparatus with different systematics, and that independence is the result’s real strength. Established Its asymptotic average work for full erasure was 0.71 ± 0.03 k_BT against ln 2 = 0.693, with a no-erasure control returning 0.05 ± 0.03 k_BT. Established That is agreement at the level of a few per cent with a four-per-cent uncertainty, not one per cent, and it is an extrapolation: the reported approach follows a τ⁻¹ law, with cycle times running from 7.5 seconds to 16 minutes against a diffusive timescale of about 15 s.

Speculative Restated, the anchor should read: two independent apparatus, of very different design, agree that the asymptotic cost of erasing one bit is k_BT ln 2 to within a few per cent, in protocols four orders of magnitude slower than a DRAM refresh and with erasure fidelities in the nineties. Frontier That is a good result and a narrow one, and every extrapolation from it to computing hardware crosses thirteen orders of magnitude in speed without an argument.

Established How far the best erasure experiments are from the finite-time optimum has been computed, and it is not close. The 2020 finite-time treatment tabulates five published erasure protocols against the minimum dissipation achievable in the same duration: the optical-trap experiment dissipates 5.67 times the minimum, and the others 3.96, 3.51, 3.11 and 2.89 times. Frontier So the field’s showcase demonstrations are simultaneously within a few per cent of the quasistatic bound and a factor of three to six above the bound that applies at the speed they actually ran. Speculative Both statements are true and the second is the one relevant to hardware.

Established The finite-time bound itself has a closed form. For a bit erased in duration τ with diffusion constant D, the minimum work satisfies ln 2 + Var(x)/(2Dτ) ≤ W_min/k_BT ≤ ln 2 + 2Var(x)/(Dτ), the lower bound achieved by an optimal-transport protocol that moves probability half as far as a local-equilibrium protocol does — which is the entire source of the factor of four between the two. Established Under full control, partial erasure with error probability ε in a flat well costs a minimum irreversible entropy of (1 − 2ε)²/τ, so the speed penalty falls as the erasure is allowed to be sloppier. Frontier The authors state the consequence plainly: the Landauer bound can be saturated only in the quasistatic limit, an assumption that is never valid for practical systems.

Frontier Beyond weak coupling, the speed penalty acquires a universal coefficient, and this is the single most useful number on this page for anyone arguing about clock rates. A 2023 treatment optimising over both energy and coupling gives heat Q ≥ k_BT(ln 2 + a·τ_Pl/τ) with a = 2.57946 and τ_Pl = ℏ/k_BT, the correction persisting even at arbitrarily strong system–bath coupling; in the weak-coupling regime the dissipation instead scales as π²/(4Γτ) with Γ the thermalisation rate. Established At 300 K the Planckian time is 25.5 fs, so this brief’s arithmetic on that bound gives a speed penalty of 6.6 × 10⁻⁵ k_BT at a one-nanosecond erasure, 6.6 × 10⁻³ k_BT at 10 ps, and 0.69 k_BT — a doubling of the total cost — only at about 95 fs. Frontier The fundamental speed penalty is therefore under one per cent of the Landauer bound at every clock rate any digital computer has ever run at, and the popular claim that computers are far above the bound because they are fast is wrong for the reason given.

Established What does bind, measurably, is the initial state. A 2026 measurement of erasure in silicon dynamic-random-access-memory cells, using single-electron charge counting, found that efficiency fell as the erasure error probability was reduced and that the Landauer limit was not reached even under effectively infinite-time erasure, because the cells cannot prepare their initial state in thermal equilibrium and therefore cannot be operated quasistatically. Frontier Ultra Efficient Computing Energy Systems reports that result as the correction to the claim that Landauer has been verified; this page takes the next step and names what it is an instance of. Speculative A device whose initial distribution is not the one its dynamics would relax to is paying mismatch cost, and mismatch cost is the term Landauer’s argument does not contain.

Established The mismatch term has a formula and it is a difference of relative entropies. The general decomposition writes total entropy flow as Landauer cost plus mismatch cost plus a residual, with mismatch cost E(p) = D(p‖q) − D(πp‖πq), where q is the input distribution the physical process is optimised for and π is the computation being performed. Established It vanishes when the actual input distribution equals that prior, or when the map preserves the measure, and otherwise it is strictly positive and bounds entropy production from below. Frontier A 2025 result establishes that for a Boolean circuit computing any non-constant function there is a strictly positive mismatch cost for every input distribution, so no choice of inputs makes it go away, and that the worst-case cost is bounded above by a quantity linear in circuit size. Speculative The same work states the point this brief is organised around: the Landauer bound, which accounts only for entropy changes, can be zero even when the dissipated heat is substantial.

Frontier And the older result that most people get backwards: logical irreversibility does not by itself force dissipation. The standard treatment states that any function on a finite state space can in principle be implemented in a thermodynamically reversible manner with no irreversible entropy production, and that different implementations of the same input-output computation can have radically different thermodynamic consequences. Speculative Read carefully, that removes the circuit’s logic from the list of things that set its energy cost and replaces it with the circuit’s physical implementation and its input statistics — which is a much harder object to bound and the reason this subject is still open.

Established A second family of bounds constrains precision rather than erasure, and it has been tested on hardware. The thermodynamic uncertainty relation states that the relative variance of a thermodynamic current is bounded below by 2 divided by the entropy production, so low fluctuation is bought with dissipation. Established A 2025 experiment on a superconducting quantum processor verified the relation across systems of 4 to 130 qubits, with circuits up to depth 120 and 8,466 operations including 4,176 two-qubit gates, and found the bound saturating toward the classical form at high temperature. Frontier A 2025 entropic version derives Σ ≥ ln 2 − Λ, where Λ is an asymmetry measure bounded between 0 and ln 2. Speculative That ln 2 arrives from an argument about fluctuations, with no bit erased anywhere in it, is either a coincidence of units or the most interesting unexplained thing in this literature, and nobody has said which.

Frontier For error-corrected quantum computing the three-term structure has been written down explicitly, and it is reassuring in a way that is worth reporting. A 2026 analysis gives the minimum heat per error-correction cycle as k_BT ln 2 plus a strong-coupling term proportional to system–bath mutual information plus a finite-time term of order ℏ/τ, estimates the coupling correction at about 20% overhead for superconducting architectures, and puts the finite-time correction at 0.55% for a 100 ns reset and 5.5% for a 10 ns reset. Established Its aggregate conclusion is that total qubit-reset heat stays in the nanowatt range even for million-qubit machines, well inside dilution-refrigerator cooling power, and that the dominant heat load is control electronics and wiring. Speculative If that holds, the thermodynamic cost of error correction is not what limits Fault-Tolerant Quantum Computing, and a line of argument that has appeared repeatedly in that field is settled in the negative.

Speculative Finally, the commercial edge of the subject, which arrived in force in 2025 and 2026 and should be read with the flags on. Two companies are building hardware that computes by sampling from physical noise rather than by suppressing it. One demonstrated an eight-cluster resonator board performing matrix inversion in 2025 and has announced a silicon successor which, in the words of the reporting, has yet to be assessed by other experts; its prototype had to add noise by hand from a random-number generator, which costs energy the architecture is supposed to save. Frontier The other signed a non-binding letter of intent for up to $75 million with the United States Department of Commerce on 29 July 2026, names three chips — a prototype, a production sampling unit and a domestically fabricated iteration — and claims energy-efficiency gains of orders of magnitude on generative workloads (the company’s own figures). Handwave Its headline factor of 10,000 is explicitly from simulations of the hardware on small benchmarks rather than from measurement on silicon, and the most extreme figure in circulation — about 100 billion times less heat than a conventional network on a denoising task — is from a theoretical model, not a device.

Frontier Five questions are open, and they are ordered by how much the answer would move the floor.

Speculative Question one: how large is the mismatch cost for a real circuit? The theory says it is strictly positive for any non-constant Boolean function under every input distribution, and bounded above by something linear in circuit size when the conditional priors are bounded away from zero. Handwave Nobody has evaluated it numerically for any circuit that exists. Speculative The reason is that it requires knowing the prior distribution the physical implementation is optimised for, which is a property of the hardware nobody measures and no datasheet reports. Handwave Until that number exists for one adder, the statement that Landauer is the wrong floor is a theorem about a quantity of unknown size.

Frontier Question two: is there a thermodynamic cost of computation that is not a cost of erasure? The reversible-computing tradition says no in principle: any computation can be embedded in a logically reversible one and run arbitrarily close to reversibly. Speculative The circuit-thermodynamics results push the other way, giving strictly positive lower bounds that depend on circuit structure rather than on bits destroyed. Handwave These are not formally in contradiction — one is about what is possible, the other about what a given implementation costs — but they are routinely quoted against each other, and no worked example reconciles them.

Frontier Question three: what does reliability cost, exactly? Finite-time analysis gives partial erasure with error ε a speed penalty scaling as (1 − 2ε)²/τ, so sloppier erasure is cheaper at a given speed. Established The DRAM measurement found efficiency falling as error probability fell, which is the same trade-off appearing in a shipping memory technology. Speculative A 2021 result derives a closed-form lower bound on the work penalty as a joint function of protocol time and reset error; this brief could retrieve only its abstract and therefore states the existence of the bound and none of its content. Handwave What is missing is the composition rule: what the error-energy trade-off becomes once errors are corrected rather than tolerated, which is the regime every real computer is in.

Speculative Question four: is ln 2 in the entropic uncertainty relation the same ln 2 as Landauer’s? One comes from the entropy of a binary distribution being destroyed; the other from the maximum asymmetry of a fluctuating current. Handwave No paper this brief could find asserts a connection, and no paper denies one. It is the kind of coincidence that is either trivial or the start of something, and the field has not looked.

Frontier Question five: do the thermodynamic-computing architectures actually evade any of this? They do not claim to violate Landauer’s bound; they claim that sampling from a physical distribution is cheaper than simulating one digitally, which is an algorithmic claim about which operations you need rather than a thermodynamic claim about what each costs. Speculative If the claim is right, the relevant bound is not per bit erased but per independent sample drawn, and no one has written down what that bound is. Handwave That missing bound is the most consequential unasked question in this subject, because a $75 million public commitment has now been made in the presence of it.

Frontier The obstacle here is instrumentation, and it is stark: every confirmation of the bound has been made on one bit, and no experiment has measured the total dissipation of a circuit against its own corrected bound.

Established Bottleneck one: zeptojoule calorimetry does not exist for working logic. The single-bit experiments measure work indirectly — from particle trajectories in a known potential, or from single-electron charge counting — and both methods require a system with one degree of freedom and a controllable landscape. Speculative A logic gate has neither. Handwave There is no published technique for measuring the heat released by a specified logical operation in a real device to within the 2.9 zJ that would make the comparison meaningful.

Frontier Bottleneck two: real devices cannot initialise in equilibrium. The 2026 DRAM result identifies this as the reason the bound was not reached even at unlimited time, and notes that the constraint extends to many electronic circuits sharing the same architecture. Speculative This is not a fabrication problem that better process control fixes; it is a consequence of storing information in a metastable state that is deliberately far from the thermal distribution, which is what makes memory memory. Handwave A device that initialised in equilibrium would forget.

Speculative Bottleneck three: the prior is unmeasurable with current methods. Mismatch cost is defined against the input distribution for which a physical process produces least entropy, and nothing in device characterisation identifies that distribution. Handwave Estimating it would require driving a gate with many input ensembles and inferring the minimiser from dissipation differences — which needs the calorimetry of bottleneck one.

Frontier Bottleneck four: the interesting corrections are small where measurement is easy and large where it is hard. The universal speed correction is 6.6 × 10⁻⁵ k_BT at a nanosecond — unmeasurable against a CMOS switching energy of 10⁴ to 10⁷ times the bound. Speculative It becomes comparable to the bound only near 100 fs, where no logic runs and no calorimeter works. Handwave The regime where this theory makes a testable prediction and the regime where instruments exist do not currently overlap, and closing that gap is the whole experimental programme.

Speculative Bottleneck five, and it is sociological: the bound is quoted far more often than it is used. A figure of 10⁵, 10⁶ or 10⁹ above Landauer appears in funding cases for reversible logic, superconducting logic, neuromorphic hardware and thermodynamic sampling, and the three figures refer to different operations. Frontier Ultra Efficient Computing Energy Systems resolves that accounting; what remains unresolved here is which of those ratios the corrected floor should be compared against, and nobody states it.

Frontier This subject waits on four things, none of which is being produced for it.

Established Single-charge and single-quantum metrology. The 2026 DRAM result exists because silicon cells were built that can count charges on a capacitor one electron at a time; that capability came from precision measurement rather than from this field. Speculative Precision Quantum Sensing develops the class of instrument that would have to be integrated with logic for the decisive experiment, and is not blocking, because it is being built for metrology whether or not anyone tests a thermodynamic bound with it.

Frontier Cryogenics and superconducting device physics. Every route to operating near the bound involves lowering T or lowering the switching energy toward it, which puts Superconducting Infrastructure and High Temperature Superconductors upstream. Speculative The dependency is real and the arithmetic is unkind: refrigeration cost is what decides whether a low-dissipation logic family is better or worse at the wall, and that accounting belongs to Ultra Efficient Computing Energy Systems rather than here.

Frontier Stochastic thermodynamics as a mathematical programme. The mismatch decomposition, the finite-time bounds and the uncertainty relations all came from the same twenty-year body of work on fluctuation theorems, and its open problems — strong coupling, non-Markovian baths, quantum coherence — are this brief’s open problems. Speculative Glass Transition Theory is the map’s nearest page on the same non-equilibrium statistical mechanics and supplies context rather than a prerequisite.

Speculative And a demand-side dependency that runs the other way. If the growth in computing load described in The Physical Stack of AI and Deep Learning Theory continues, the question of where the physical floor sits acquires a budget; if it does not, this remains a subject with excellent theory and no customer. Handwave That is an honest statement of why the field is funded at the level it is.

Frontier Six experiments, ranked by what a result would settle. The first is the decisive one and no part of it has been done.

Speculative E1, the decisive experiment: a device executing a nontrivial logical circuit whose measured total dissipation is compared against its own mismatch- and speed-corrected bound, and lands within a small constant factor of it. The specification is concrete. Take a circuit with more than one gate and a non-constant output — a full adder is sufficient and is the natural choice. Characterise the input distribution actually applied and the prior the implementation is optimised for, so the mismatch term can be evaluated rather than assumed. Measure the heat released per circuit evaluation to a resolution better than k_BT ln 2. Report the ratio of measured dissipation to the three-term bound. Frontier Every ingredient except the calorimetry exists, and the calorimetry is the reason this has never been done. Handwave Until that ratio exists for one circuit, every statement in this subject about how far real computers are from the floor is a statement about a floor nobody has computed for a real computer.

Frontier E2, the cheap half of E1: measure the mismatch cost of a single gate by driving it with different input ensembles. Mismatch cost is a difference of relative entropies between the applied distribution and an unknown optimal prior, and the prior can in principle be found by scanning ensembles and locating the dissipation minimum. Speculative The measurement needs sensitivity to differences in dissipation rather than to absolute values, which is a far easier instrument than E1 requires. Speculative A single number — the mismatch cost of one NAND gate in units of k_BT ln 2 — would tell this field whether the term it cannot bound is a correction or the whole story.

Frontier E3: push a single-bit erasure into the regime where the universal speed bound is measurable. The strong-coupling result puts the correction at 2.58 Planckian times divided by protocol duration, which is 0.69 k_BT — equal to the entire Landauer cost — at about 95 fs at room temperature, and correspondingly longer at cryogenic temperatures where the Planckian time is longer. Speculative A superconducting or trapped-charge bit reset at a few picoseconds and at 10 mK is the plausible route, since the Planckian time at 10 mK is about 30,000 times its room-temperature value. Handwave Nobody has proposed this experiment in the literature this brief could reach, and stating it is the most speculative thing on this page.

Established E4: the DRAM measurement repeated across memory technologies. The single-electron result on silicon DRAM cells found the bound unreachable even at unlimited time because the initial state cannot be thermal, and stated that the conclusion extends to circuits sharing that architecture. Frontier The experiment is to repeat it on static RAM, on magnetic and resistive memory, and on a charge-trap cell, and report for each the asymptotic dissipation and the inferred non-equilibrium initial-state penalty. Speculative That would convert a single result into a map, and it needs no new physics.

Speculative E5: the sampling bound. If a thermodynamic sampling architecture is cheaper because it draws samples physically rather than simulating them, then the quantity to bound is the minimum dissipation per independent sample from a specified distribution, and the experiment is to measure it on the hardware that exists and compare it with the digital cost of the same sample. Frontier The reported figure of about 10,000× is from simulations of the hardware rather than measurements of it (the vendor’s own figures), and one published prototype had to supply its randomness from an external generator, which is the specific thing the measurement would expose. Handwave A public programme of up to $75 million has been committed in the absence of this measurement, and the measurement is not hard.

Speculative E6: the error-correction composition test. Finite-time theory prices erasure with a tolerated error rate; real machines detect and correct instead. Frontier The 2026 quantum analysis is the only place this brief found the composition written down, giving about 20% for strong coupling and 5.5% for a 10 ns reset, and concluding that reset heat is not what limits the machine. Speculative The classical version — measure the total dissipation of a redundantly encoded register with active correction, against an uncorrected one at matched logical error rate — has not been done, and would say whether error correction is thermodynamically cheap or whether it is where the real cost of reliable computing lives.

Established And the experiments that are not tests of this subject and are regularly reported as though they were. A new low-power chip is a measurement of engineering, not of a bound. Established A demonstration that some device dissipates less than another device tells you nothing about either one’s distance from its own floor unless the floor is computed for both. Frontier A simulation of a proposed architecture is a claim about a model. Speculative All three appear in this field’s funding literature described as evidence about thermodynamic limits, and none of them bears on any question in section 4.

Frontier The engineering requirement here is a measurement capability rather than a machine, and it has three parts.

Speculative Calorimetry at the zeptojoule scale on a device that computes. Resolution better than 2.9 × 10⁻²¹ J per operation, on a circuit rather than a single trapped object, with enough repetitions to separate a mean from a fluctuation. Frontier The nearest existing capabilities are single-electron charge counting integrated into silicon memory cells, and the trajectory-based work estimation used in the trap experiments. Handwave Neither scales to a multi-gate circuit and no published route does.

Speculative Controllable potential landscapes in solid state. The optimal finite-time protocols are optimal-transport solutions requiring full control of the potential during the operation; the trap experiments have that control and no transistor does. Frontier Adiabatic and reversible CMOS families are the existing attempt to buy some of it with clocked power supplies, and their accounting belongs to Ultra Efficient Computing Energy Systems rather than here. Speculative What this page needs from them is different: not lower energy, but a device whose potential is known well enough that the corrected bound can be computed for it.

Frontier Cryogenic operation, for a reason that is not the obvious one. Lowering temperature lowers k_BT ln 2 proportionally and lengthens the Planckian time in inverse proportion, so the fundamental speed correction becomes measurable at picoseconds rather than femtoseconds. Speculative That makes millikelvin superconducting hardware the only accessible regime where the universal finite-time bound is a testable prediction rather than a small number. Handwave Whether anyone will build it for that reason is doubtful, since the same hardware is fully committed to Fault-Tolerant Quantum Computing.

Established The nearest page is Ultra Efficient Computing Energy Systems and the boundary is worth stating precisely, because the two pages share a constant and almost nothing else. That brief owns the demand side: joules per operation, the distance of real silicon from the bound at several definitions of an operation, the reachable CMOS ceiling of about 200×, facility overhead and the terawatt-hours. Frontier This page owns whether the constant it measures against is the right constant. The arbitration rule where they meet: if the question is what a computation costs, it belongs there; if the question is what it could not cost less than, it belongs here.

Frontier Fault-Tolerant Quantum Computing is where the corrected bound has actually been applied, and the answer — nanowatt-scale reset heat for a million qubits, with control wiring dominating — retires a thermodynamic objection that has circulated in that field for a decade. Speculative It is also the only place where somebody has written the three-term bound down for a real machine, which is a comment on where the money is rather than on where the physics is.

Speculative Biological Computing shares this page’s question in its sharpest form, since cells compute at energies far closer to k_BT than any engineered device and are the standing existence proof that the gap is not a law of nature. Handwave They also make errors, correct them, and never sit in equilibrium, which makes them the natural test case for every correction term on this page and a poorly instrumented one.

Frontier Further out: Superconducting Infrastructure and High Temperature Superconductors for the low-dissipation device families and the refrigeration that decides them; Precision Quantum Sensing for the metrology the decisive experiment needs; Quantum Materials for the fabrication; The Physical Stack of AI and Deep Learning Theory for the load that has made this an economic question rather than a curiosity; and Glass Transition Theory for the neighbouring non-equilibrium statistical mechanics.

Frontier This field’s institutional problem is that its central number is quoted by people who are not measuring it, in support of programmes whose cases rest on it.

Speculative Requirement one: a reporting convention for dissipation per logical operation. A claim that a device is some factor above Landauer means nothing without the definition of the operation, the temperature, the error rate and whether the figure includes the clocking and control energy. Established The same ambiguity produces the 10⁵-to-10⁹ spread this page opened with, from sources that do not actually disagree. Handwave No standards body owns this, and the convention would fit on one page.

Speculative Requirement two: independent measurement of thermodynamic-computing claims before public money is committed. A non-binding letter of intent for up to $75 million was signed in July 2026 with a company whose headline efficiency figure comes from simulations of its own hardware on small benchmarks. Frontier That is not an accusation; it is a description of the evidence base, and it is the company’s own description. Speculative The remedy is a measured energy-per-sample figure from an independent laboratory, which section 7 specifies and which nobody has commissioned.

Frontier Requirement three: somebody has to own the calorimetry. The decisive experiment needs a capability that no single group has an incentive to build, since it produces a bound rather than a device. Speculative This is the classic shape of a national-laboratory instrument: expensive, slow, of no commercial value, and the only thing that would settle the question. Handwave The cheapest institutional act available is also the one with no constituency — funding one metrology group for a decade to measure the heat released by an adder.

Speculative Requirement four: publication of the negative results. The 2026 DRAM measurement is a negative result — the bound was not reached — and it is among the most informative papers in this subject. Handwave A field whose confirmations are publishable and whose failures-to-reach are not would have produced sixty years of agreement and no knowledge.

Frontier The ethical content here is narrow and specific: a physical constant is being used as a warrant for investment, and the warrant is weaker than it reads.

Speculative The pattern is consistent across three device programmes. Reversible logic, superconducting logic and thermodynamic sampling have each been argued for on the basis of the distance between current hardware and Landauer’s bound. Established That distance is real. Frontier It is also not the available headroom, because the relevant floor for a real circuit includes terms nobody has evaluated, and because the reachable engineering ceiling reported next door is about 200× rather than 10⁵ or 10⁹. Handwave Presenting the ratio to the naive bound as the opportunity is the commonest rhetorical move in this area and it overstates the case by orders of magnitude that are not in dispute.

Speculative There is also a smaller point about attribution. The showcase confirmations of Landauer’s principle are careful, honest experiments whose authors state their limitations clearly — the success rate capped near 95%, the asymptotic fit, the ±0.15 k_BT error bars, the tens of seconds per bit. Established Those qualifications are in the papers and absent from almost every secondary account. Handwave The distortion is downstream of the experimentalists, which is worth saying because they are the ones usually cited for it.

Speculative And the distributive observation. Whatever the floor turns out to be, the benefit of knowing it accrues to whoever is deciding between architectures at scale, which is a handful of firms and two or three states. Frontier The cost of not knowing it is borne by public research budgets that fund device programmes whose case is a ratio. Handwave That asymmetry is the practical argument for a publicly funded measurement, and it is an argument about procurement rather than about physics.

Speculative If the corrected floor for real computation sits within a small factor of k_BT ln 2 per erased bit, then computation is thermodynamically almost free and the energy cost of information processing is a temporary engineering fact. Frontier On that reading the long-run constraint on a computing civilization is waste heat rejection and materials rather than energy input, and the interesting limits are the ones about temperature and radiating area rather than about joules per operation.

Speculative If instead the mismatch term is large for any device that stores information in metastable states — which is every memory technology anyone has built — then there is a floor well above Landauer’s that applies to physical computers rather than to idealised ones, and it is a floor nobody has located. Handwave The distance between these two futures is one measurement, and this page’s central complaint is that it is a cheap measurement nobody is making.

Frontier There is a more immediate consequence and it is about how a technology gets funded. Public money has now been committed to a computing architecture whose efficiency case rests on simulations, in a field whose foundational constant is routinely quoted at the wrong order of magnitude. Speculative Whether that commitment is vindicated is not knowable from here; what is knowable is that the measurement which would inform it costs a fraction of a per cent of the commitment. Handwave A civilization that can afford to build the thing and not to measure it is making a choice about evidence, and this page records the choice rather than predicting its outcome.

Speculative The slowest consequence is epistemic. Living systems compute at energies far closer to the bound than engineered ones, and if the corrected floor is high then biology is not the existence proof it is usually taken for, because cells may be paying mismatch cost too. Handwave That reframing would matter to Biological Computing and to every argument that brains demonstrate what is physically possible, and it is currently unsupported in either direction.

These horizons track what would have to be measured for this page’s assessment to change, because no deployment depends on the answer:

• 10 yr: Frontier Expect the single-electron erasure measurement to be repeated across memory technologies, turning the 2026 DRAM result from a single point into a map of how far each architecture sits from its own asymptotic floor. Speculative Expect the first numerical evaluation of a mismatch cost for a real gate, most plausibly from a simulation of a well-characterised device rather than from calorimetry. Frontier Expect independent measurement of a thermodynamic sampling unit’s energy per sample, because a public commitment of this size usually produces one. Speculative Do not expect the decisive experiment on this horizon; the calorimetry does not exist and nobody is building it.
• 25 yr: Speculative This is the window in which a circuit-scale dissipation measurement against a computed three-term bound becomes plausible, most likely in superconducting hardware at millikelvin temperatures where both the bound and the instruments are more favourable. Speculative A result within a small constant factor would establish that the corrections are corrections and that the remaining gap to real hardware is engineering. Handwave A result orders of magnitude above the computed bound would be the more interesting outcome and would mean the three-term decomposition is missing a term.
• 50 yr: Handwave On the theory side the plausible development is a bound on dissipation per independent sample, which the sampling architectures need and nobody has written. Speculative On the hardware side the plausible development is that the question stops being asked, because the reachable engineering ceiling is reached, demand growth settles the energy argument on its own terms, and the distance to the thermodynamic floor becomes a curiosity rather than a target.
• 100 / 250+ yr: Handwave Beyond useful forecasting, and the defensible statement is about shape. Handwave A constant that is exactly known, experimentally confirmed on one degree of freedom, and never once compared against a working machine either eventually gets that comparison or is quietly retired as the wrong quantity. Handwave Any specific claim about the energy cost of computation at that horizon is an assertion, and this page does not make one.

• Depends on No typed dependency edge to another brief on this map, and the absence is informative. Ultra Efficient Computing Energy Systems is the nearest page and is downstream rather than upstream: it prices hardware against the constant this page is auditing, and its conclusions stand whichever way the audit goes, because its binding variable is a demand curve. Precision Quantum Sensing and Superconducting Infrastructure supply the instruments and the cold, and neither is blocking, because both are being built for other customers.
• Requires (not on this map) Six conditions, in the order of the tokens above. The calorimeter. Heat resolution better than 2.9 × 10⁻²¹ J per operation on a circuit rather than a single trapped particle; everything else on this page waits on it, and no published technique reaches it. A measured mismatch cost. One number, for one gate, in units of the Landauer bound, which would settle whether the term the theory cannot bound is a correction or the dominant cost. A sampling bound. The minimum dissipation per independent sample from a specified distribution, which is the quantity the thermodynamic-computing architectures should be judged against and which nobody has derived. Error correction, priced. Finite-time theory prices erasure with a tolerated error rate; every real machine detects and corrects instead, and the composition rule exists in the literature only for one quantum architecture. The convention. A one-page definition of what “a factor above Landauer” means — which operation, what temperature, what error rate, whether clocking is included — without which the 10⁵-to-10⁹ spread cannot be read. The metrology. Single-charge counting of the kind already demonstrated in silicon memory cells, integrated with logic rather than with storage.
• Enables A corrected floor would put a hard number under the device-programme arguments in Ultra Efficient Computing Energy Systems and would tell Fault-Tolerant Quantum Computing whether the thermodynamic objection to error correction is closed in general or only for superconducting qubits. No typed enabling edge is claimed, because a bound that has never been evaluated for a working circuit enables nothing on a schedule, and the more likely outcome is that it tightens an argument rather than releasing a technology.
• Adjacent Ultra Efficient Computing Energy Systems for the demand side and the engineering ceiling; Fault-Tolerant Quantum Computing for the one architecture where the three-term bound has been written down; Biological Computing for the substrates that already operate near k_BT; Superconducting Infrastructure, High Temperature Superconductors and Quantum Materials for low-dissipation devices and their fabrication; Precision Quantum Sensing for the metrology; Glass Transition Theory for the neighbouring non-equilibrium statistical mechanics; and Deep Learning Theory and The Physical Stack of AI for the load that made this an economic question.

Established “Landauer’s principle has been experimentally verified.” It has been verified for a single bit, in two independent apparatus, asymptotically, at protocol durations of tens of seconds to minutes. Established The optical-trap result approaches the bound as ln 2 + B/τ with B about 8.15 k_BT·s and reports 0.59 ± 0.15 k_BT at a 40-second protocol with an erasure success rate never above about 95%; the feedback-trap result reports an asymptotic 0.71 ± 0.03 k_BT against ln 2 = 0.693. Speculative “Verified” is correct for the quantity tested and misleading for the quantity people take it to license, which is the cost of an operation in a machine.

Established “Computers are a billion times above the Landauer limit, so there is a billion times of headroom.” Two errors in one sentence. Established The ratio depends on what is called an operation — roughly 10⁵ for a gate-switching event and 10⁸ for a delivered floating-point operation, as Ultra Efficient Computing Energy Systems sets out, against the figure of about a billion given in the 2025 review — and most of a chip’s energy goes into charging interconnect capacitance and into leakage, neither of which is thermodynamically required to dissipate anything. Frontier And the floor being compared against is the wrong one, because it omits the mismatch and speed terms. Speculative The headroom question belongs next door; the floor question belongs here; conflating them is how the ratio became a funding argument.

Frontier “The bound is unreachable because real computers are fast.” This is the intuitive objection and it is quantitatively wrong at every clock rate in use. Established The universal finite-time correction beyond weak coupling is 2.57946 Planckian times divided by the protocol duration, and the Planckian time at 300 K is 25.5 fs. Established On this brief’s arithmetic that is 6.6 × 10⁻⁵ k_BT at one nanosecond and under one per cent of the bound even at 10 ps. Speculative Speed only doubles the cost at about 95 fs, which no logic family approaches, so whatever is keeping real hardware away from the bound, it is not the fundamental speed limit.

Established “Finite-time effects do not matter in practice, then.” The universal bound is not the operative one for a given device; the device’s own relaxation time is. Established In the optical-trap experiment the coefficient of the 1/τ term is about 8.15 k_BT·s, so reaching within ten per cent of the bound required protocols longer than about two minutes. Frontier The tabulated comparison of five published erasure protocols against the finite-time optimum at their own durations gives ratios of 2.89 to 5.67. Speculative The honest statement is that finite-time cost is enormous relative to how slowly these experiments ran and negligible relative to how fast a transistor is, and the two facts belong to different quantities.

Frontier “Erasing bits is what costs energy.” The standard treatment states that any function on a finite state space can in principle be implemented thermodynamically reversibly, so logical irreversibility does not by itself force dissipation, and that different implementations of the same computation can differ radically in cost. Established The circuit result goes further: for any non-constant Boolean function there is a strictly positive mismatch cost under every input distribution, so a circuit has a positive floor even where the Landauer term is zero. Speculative The cost is a property of the implementation and the input statistics, not of the truth table, which is the single most commonly reversed claim in this subject.

Speculative “Reversible computing gets round Landauer.” In the idealised sense it does, and that has been understood since the 1970s. Frontier What it does not get round is the finite-time term, which is what actually prices an adiabatic circuit, and the mismatch term, which depends on inputs. Handwave A reversible circuit run at a nanosecond pays a speed penalty that is negligible against k_BT ln 2 and enormous against zero, and since zero is what reversibility promises, the relevant comparison is the one nobody makes.

Speculative “Thermodynamic computing beats the Landauer bound.” Its proponents do not claim this and the claim should not be attributed to them. Frontier What is claimed is that drawing samples from a physical distribution is cheaper than simulating the draw digitally, with a headline factor of about 10,000 from simulations of the hardware on small benchmarks (the company’s own figures) and a more extreme figure of about 100 billion from a theoretical model of a denoising task. Handwave Neither is a silicon measurement, one published prototype had to buy its randomness from an external generator, and the bound the architecture should be judged against — dissipation per independent sample — has not been written down by anyone.

Frontier “Thermodynamics limits quantum error correction.” The one worked analysis says it does not: about 20% overhead from strong coupling, 0.55% from a 100 ns reset and 5.5% from a 10 ns reset, with total reset heat in the nanowatt range for a million qubits and the dominant load in control electronics and wiring. Speculative That is one paper on one architecture, and this page flags it as such — but it is the only quantitative answer in existence, and the qualitative objection has been repeated for a decade without one.

Frontier And the correction this page makes to its own commission. The anchor named for this brief is single-bit erasure measured at the Landauer bound in an optical trap, and independently within about 1%. Established The optical-trap measurement did not sit at the bound: it reported about 0.59 k_BT with ±0.15 k_BT error bars at a 40-second protocol, an asymptotic fit of 0.72 k_BT, and an erasure that succeeded at best about 95% of the time. Established The independent confirmation was not an optical trap and was not at one per cent: it used a feedback trap with a virtual electrokinetic potential and reported 0.71 ± 0.03 k_BT against 0.693, which is agreement at the level of a few per cent with a four-per-cent uncertainty. Speculative The corrected anchor is stronger than the one commissioned, because the value of the pair is that two unrelated apparatus agree, not that either is precise — and it makes the point this brief exists for, which is that the confirmed quantity is the quasistatic cost of one bit and the interesting quantity is the cost of a circuit that runs.