Thread: Ultimate physical limits to computation — primary-source numbers
Date: 2026-09-15 · Reader thread (Argus) · Status: COMPLETE
Every number below was read from the paper itself (arXiv full text or journal HTML), not from a snippet, Wikipedia, or a citation. Each entry is tagged VERIFIED (read in the primary text) or UNVERIFIED (could not reach primary text; snippet/abstract only) with the reason.
Sources fetched in full text:
- Lloyd 2000: arXiv:quant-ph/9908043v3 HTML (published as Nature 406, 1047–1054 (2000), DOI 10.1038/35023282)
- Lloyd 2002: arXiv:quant-ph/0110141v1 HTML (published as Phys. Rev. Lett. 88, 237901 (2002))
- Margolus & Levitin 1998: arXiv:quant-ph/9710043v2 HTML (published as Physica D 120, 188–195 (1998), DOI 10.1016/S0167-2789(98)00054-2)
- Bousso 2002: arXiv:hep-th/0203101v2 HTML (published as Rev. Mod. Phys. 74, 825–874 (2002))
- Egan & Lineweaver 2010: arXiv:0909.3983v3 HTML (published as ApJ 710, 1825–1834 (2010))
1. Lloyd, "Ultimate physical limits to computation", Nature 406, 1047 (2000) — arXiv:quant-ph/9908043
Headline numbers (all VERIFIED, read from v3 full text):
- Speed bound: to perform an elementary logical operation in time Δt requires average energy
E ≥ πħ/2Δt. Equivalently, a system with average energy E can perform at most 2E/πħ logical operations per second (Eq. 1: Σ_ℓ 1/Δt_ℓ ≤ Σ_ℓ 2E_ℓ/πħ = 2E/πħ).
- Ultimate laptop: mass 1 kg, volume 1 litre. Its rest energy
E = mc² = 8.9874 × 10¹⁶ joules → maximum 5.4258 × 10⁵⁰ operations per second. (Later in the same paper, §3, he rounds this to ≈10⁵¹ ops/sec. Both figures appear in the same paper.)
- Derivation: the bound is the Margolus–Levitin result, quoted as: a quantum system with average energy E takes at least
Δt = πħ/2E to evolve to an orthogonal state (previous results quoted in terms of energy spread ΔE: Δt ≥ πħ/2ΔE). The NOT gate attains the limit; AND/FANOUT via Toffoli also attain 2E/πħ; operations cycling through many states run at E/πħ, half as fast. Setting ground-state energy E₀ = 0 gives E = ΔE for the NOT example.
- Memory bound:
I = S(E)/k_B ln2 bits, where S(E) is the thermodynamic entropy (canonical ensemble "which maximizes S for fixed average energy with no constraint on the spread"). Energy limits speed; entropy limits memory.
- Numbers for the ultimate laptop memory (photon-only lower bound, r = 2):
k_B T = 8.10 × 10⁻¹⁵ J → T = 5.87 × 10⁸ K; entropy S = 2.04 × 10⁸ J/K; memory I = 2.13 × 10³¹ bits. Using all memory, it performs 2ln2·k_B E/πħS = 3ln2·k_B T/2πħ ≈ 10¹⁹ operations per bit per second.
- Serial vs parallel: total ops/sec
2E/πħ is independent of architecture — spread energy over N gates and each runs at 2E/πħN; total is unchanged. Parallel computation gives smaller overall energy spread (ΔE ≈ √N ΔE_ℓ vs E ≈ N E_ℓ). Serialism measure t_com/t_flip = 4ln2·k_B R E/πħcS ∝ k_B RT/ħc = 2πR/λ_T. Ultimate laptop: R = 10⁻¹ m, t_com/t_flip ≈ 10¹⁰ → highly parallel. Only the black-hole limit is fully serial (t_com/t_flip = ln2/π; Bekenstein bound k_B RE/ħcS ≥ 1/2π, equality for black holes).
- Black-hole-compressed laptop: Schwarzschild radius of 1 kg:
R_S = 1.485 × 10⁻²⁷ m; memory I = 4πGm²/ln2·ħc = 4πm²/ln2·m_P² = 3.827 × 10¹⁶ bits; same 5.4258 × 10⁵⁰ ops/sec; 1-kg BH lifetime ≈ 10⁻¹⁹ s, performing ≈ 10³² ops on ≈ 10¹⁶ bits (upper bound per Page's evaporation rate).
- Error correction / thermodynamics box: memory black-body temperature 5.87 × 10⁸ K; bit-emission rate
ℬ = 7.195 × 10⁴² bits/m²/s; to avoid overheating the error rate must be < 10⁻¹⁰ per operation; energy throughput would be 4.04 × 10²⁶ W — turning over its rest mass (~10¹⁷ J) in a nanosecond.
- Constants used: c = 2.9979 × 10⁸ m/s, ħ = 1.0545 × 10⁻³⁴ J·s, G = 6.673 × 10⁻¹¹ m³/kg·s², k_B = 1.3805 × 10⁻²³ J/K. Planck scales: ℓ_P = 1.616 × 10⁻³⁵ m, t_P = 5.391 × 10⁻⁴⁴ s, m_P = 2.177 × 10⁻⁸ kg.
- Every stated assumption/caveat (VERIFIED): complete control over the computer's state; insulation from all uncontrolled degrees of freedom; conserved quantities other than energy are zero (charge, angular momentum; baryon number not conserved because black-hole formation/evaporation is allowed); the actual running state has zero entropy (memory space is the entropy of the thermal state, not the running state); "there is no guarantee that these limits will ever be attained"; the black-hole computer is "likely to prove extremely difficult or impossible to realize"; a unified quantum-gravity theory would be required at Planck density; error correction is needed in practice; memory limit is "unlikely to be attained" (packaging issues — the memory state resembles a thermonuclear plasma); the entropy estimate is a lower-bound estimate that depends only insensitively on particle species (a change of r by a factor of 10⁴ changes S by a factor of 10).
2. Lloyd, "Computational capacity of the universe", PRL 88, 237901 (2002) — arXiv:quant-ph/0110141
Headline claim, VERIFIED verbatim from the arXiv abstract and full text:
"The universe can have performed no more than 10¹²⁰ ops on 10⁹⁰ bits."
This is the matter-energy figure. Details of what each number counts (all VERIFIED from full text):
- Ops count. In the matter-dominated universe, ops/sec in a co-moving volume ≈
ρc² × c³t³/ħ; total over history #ops ≈ ρc⁵t⁴/ħ (Eq. 1). With ρ ≈ 10⁻²⁷ kg/m³ and t ≈ 10¹⁰ years → ≈ 10¹²⁰ ops. At critical density, ρ_c = 3H²/8πG ≈ 1/Gt², so #ops ≈ ρ_c c⁵t⁴/ħ ≈ t²c⁵/Għ = (t/t_P)² (Eq. 2), with t_P = 5.391 × 10⁻⁴⁴ s; again ≈ 10¹²⁰ (he says "no more than").
- What it counts: elementary quantum logic operations that all the matter inside the particle horizon (volume ≈ c³t³) could have performed since the Big Bang, at the Margolus–Levitin rate 2E/πħ. An "op" = some degrees of freedom with energy E evolving by angle π/2 in Hilbert space for time πħ/2E.
- Gravitational degrees of freedom — the "separate figure": gravitational energy is negative and equal in magnitude to the positive matter energy (total energy of the universe = 0), so including gravity changes the ops count by exactly a factor of 2 — still ≈ 10¹²⁰. This is provisional: "no complete quantum theory of gravity currently exists."
- Bits count. Maximum entropy = all matter converted to radiation:
I = S/k_B ln2 ≈ (ρc⁵t⁴/ħ)^(3/4) = (#ops)^(3/4) (Eq. 3, substituting V ≈ c³t³). → the universe could currently register ≈ 10⁹⁰ bits ("To register this amount of information requires every degree of freedom of every particle in the universe"). These bits require converting all matter into thermal radiation.
- Bits with gravity (holographic): using the Bekenstein bound + holographic principle, max bits including gravitational degrees of freedom
≈ c²t²/ℓ_P² = t²/t_P² ≈ 10¹²⁰ — equal to the ops count, not 10⁹⁰.
- So the correct statement is:
10¹²⁰ ops on 10⁹⁰ bits = matter/energy only. If gravitational degrees of freedom are included, bits rise to ≈ 10¹²⁰ (ops stay ≈ 10¹²⁰, ×2 exactly).
- Radiation-dominated era: ops count stays within a factor of 2 of the matter-dominated formula (Eq. 5, finite even as t→0 — answers Dyson's big-crunch question); bits
≈ (#ops)^(3/4) again while D^(1/4) ≈ 1 below the GUT scale (k_B T < 2 × 10¹⁶ GeV, T ≈ 10²⁹ K).
- Inflation: horizon-scale e-folding 50 → universe can have performed no more than 10^(20±12) ops on the same number of bits within the horizon during inflation.
- Eddington–Dirac numerology (included for completeness, he warns "do not take such numerology too seriously"): #ops ≈ βγ² ≈ αβ² ≈ 10¹²⁰, differing from the large-number-cubed by (137/1836) ≈ 10.
- Man-made computers (circa 2001, overestimated): ≈ 10⁹ computers at 10⁹ Hz doing 10⁵ ops/cycle for 10⁸ s → ≈ 10³¹ ops in the last two years, ≤ ~2× that in all history; ≈ 10²¹ bits registered.
- Three interpretations he gives: (1) upper bound on computation performed by the universe; (2) lower bound on ops/bits needed to simulate the universe on a quantum computer; (3) the actual ops and bits of the universe-qua-computation (with the π/2-phase definition of an op).
- Stated model/assumption set (VERIFIED): standard Big Bang model, universe began ≈ 10¹⁰ years ago; matter-dominated for most of history; energy in a co-moving volume approximately constant; horizon volume ≈ c³t³ (ignoring the factor that the horizon is presently beyond ct); the "≈" convention: equality to within factors of order 2π in the log (
X ≈ Y means log X = log Y + O(1)); critical-density universe for Eq. 2; Margolus–Levitin is strictly quantum-mechanical; quantum gravity required for the gravitational numbers; holographic principle (Bekenstein bound + ℓ_P²) used for gravitational bits.
- Constants: ħ = 1.0545 × 10⁻³⁴ J·s (one place uses 1.0546), c = 2.98 × 10⁸ m/s, G = 6.673 × 10⁻¹¹, k_B = 1.38 × 10⁻²³ J/K; t_P = 5.391 × 10⁻⁴⁴ s.
Verdict on your memory: "10¹²⁰ operations on 10⁹⁰ bits" is correct as published, verbatim — with the caveat that it is the matter-only pair, that both numbers are "≈" (order-of-magnitude, factors of 2π ignored), and that with gravitational degrees of freedom included the bit count becomes ≈ 10¹²⁰.
3. Margolus & Levitin, "The maximum speed of dynamical evolution", Physica D 120, 188 (1998) — arXiv:quant-ph/9710043
All VERIFIED from v2 (published version) full text:
- The bound, exact form: with the zero of energy at the ground state (E₀ = 0; footnote: if no energies below some E_min are allowed, take the zero there instead), the minimum time for any state with fixed average energy E to evolve into an orthogonal state is
τ⊥ ≥ h/4E (Eq. 4).
Since h = 2πħ, this is τ⊥ ≥ πħ/2E — same bound, two notations. So "h/4E or πħ/2E" is not a choice between two different bounds; they are the same bound in h vs ħ notation.
- What E measures: E is the average energy above the ground state (
E − E₀), NOT the standard deviation and not the maximum energy. The abstract states this: "a strict bound that depends only on E−E₀, the system's average energy minus its ground state energy."
- Achievability: the bound is attained (two-state oscillation, state
(|0⟩+|2E⟩)/√2, spectrum containing 2E; very nearly attained by any ordinary macroscopic system). The system can oscillate between two orthogonal states at rate 4E/h; a long sequence of mutually orthogonal states runs at asymptotic rate ν⊥ = 2E/h = E_max/h (Eq. 26), with the exact-cycle bound τ⊥ ≥ ((N−1)/N)(h/2E) (Eq. 13). I.e., sustained long evolution runs at half the two-state rate.
- Prior work / Mandelstam–Tamm relation: the older bound (Mandelstam & Tamm 1945, ref [10]; also Pfeifer, Vaidman) is
τ⊥ ≥ h/4ΔE (Eq. 5) in terms of the energy spread ΔE. The ML point: "This bound places no limit, however, on how fast a system with bounded average energy can evolve (since ΔE can be arbitrarily large with fixed E)." They exhibit a state (their Eq. 12) with fixed E and arbitrarily large ΔE, where the ΔE bound is useless but Eq. 4 is optimal. So ML replaces ΔE with E in the bound; nothing is added to the ΔE bound itself.
- Information-processing rate: "adding one Joule of energy to a given computer can never increase its processing rate by more than about 3×10³³ operations per second" (abstract; note this is 2E/h for E = 1 J = 2/(6.626×10⁻³⁴) ≈ 3.02×10³³ — the sustained rate convention).
- Composite systems: for non-interacting subsystems, total rate = sum of subsystem rates (additivity — the parallel-computer statement).
- Relativistic/semi-classical remarks (published version only): a system with total rest-frame energy E_r passes through ≤ 2E_r t_r/h orthogonal states in rest time t_r; in another frame the bound is (2/h)(Et − px), subtracting one orthogonal state per shift of distance h/2p.
- Convention note: the two-state oscillation rate 4E/h = 2E/πħ is exactly Lloyd's ops/sec bound (
2E/πħ); the sustained rate 2E/h is half that. Lloyd's Δt = πħ/2E per operation corresponds to the two-state ML rate. Papers quoting per-joule numbers must say which convention: ML's abstract gives 3×10³³ (2E/h); Lloyd's formula would give 2E/πħ ≈ 6.04×10³³ for 1 J. Both descend from the same theorem; difference is the factor 2 discussed in §2.2. (This is a real, if cosmetic, cross-paper discrepancy — see DISCREPANCIES.)
4. Holographic / Bousso bound for the observable universe — arXiv:hep-th/0203101
What Bousso's review actually says (VERIFIED):
- Statement of the bound: "the area of any surface limits the information content of adjacent spacetime regions, at 1.4 × 10⁶⁹ bits per square meter" (v2 abstract; the v1 abstract printed it as 10⁶⁹ bits/m² without the 1.4). Planck length:
ℓ_P² = Għ/c³ = 2.59 × 10⁻⁶⁶ cm².
- De Sitter space entropy (his Eqs. 122–123): the future event horizon of de Sitter space carries Bekenstein–Hawking entropy
S_dS = πa₀² = 3π/Λ (Planck units), with Gibbons–Hawking temperature T = 1/2πa₀ (Gibbons & Hawking 1977). In a dS± spacetime the global entropy is bounded by S_global ≤ 3π/Λ (Eq. 124); in any universe with Λ > 0, entropy observable by a single observer (causal diamond) is bounded by 3π/Λ (Bousso 2000b).
- His one cosmological number: Λ ≈ 10⁻¹²⁰ (Planck units) for our universe → de Sitter entropy scale 3π/Λ ≈ 10¹²⁰-ish. Bousso's review itself never prints "10¹²²" or "10¹²³" for our universe (verified by search over the full text). Those pop-culture figures come from later numerical evaluations (below).
- Which horizon: the bound applies to the cosmic (de Sitter) event horizon — the Λ-driven horizon of an accelerating universe. Bousso's FRW analysis (§VII.1) is explicit that the particle horizon is the wrong surface: surfaces beyond the particle horizon (χ > η) have light-sheets that terminate at the Big Bang, and the spherical bound fails for large volumes in flat/closed universes (S > A); the covariant entropy bound holds for flat FRW with −1 ≤ w ≤ 1.
- The actual figures people quote (VERIFIED in Egan & Lineweaver 2010, arXiv:0909.3983v3 full text):
- Current cosmic event horizon entropy:
S_CEH = (kc³/Għ)·(A/4) = (kc³/Għ)·πR_CEH² = 2.6 ± 0.3 × 10¹²² k (units of k = Boltzmann constant, i.e., nats; in bits divide by ln2 → ≈ 3.8 × 10¹²² bits), for R_CEH = 15.7 ± 0.4 Glyr (ΛCDM event horizon; de Sitter static radius √(3/Λ) is ≈ 17.4 Glyr). They explicitly identify this with the holographic bound. E&L adopted parameters: Ωk = 0, h = 0.705 ± 0.013, ωb = 0.0224, ωm = 0.136, T_CMB = 2.725 K.
- Entropy of the observable universe's interior (particle horizon):
S_obs = 3.1 (+3.0/−1.7) × 10¹⁰⁴ k, dominated by supermassive black holes (SMBHs), which contribute "at least an order of magnitude more entropy than previously estimated." CEH interior: S_CEHint = 1.2 (+1.1/−0.7) × 10¹⁰³ k. Components (S in k): SMBHs 3.1×10¹⁰⁴; stellar BHs 5.9×10⁹⁷; photons 5.40×10⁸⁹; relic neutrinos 5.16×10⁸⁹; WIMP dark matter 10⁸⁸±¹; relic gravitons 6.2×10⁸⁷; ISM/IGM 7.1×10⁸¹; stars 9.5×10⁸⁰. V_obs = 3.65±0.10×10⁸⁰ m³; V_CEH = 1.37±0.10×10⁷⁹ m³.
- E&L cite the holographic ceiling as
S_max ~ 10¹²³ k (their §I, citing Frampton et al.). The ≈10¹²²–10¹²³ figure is the de Sitter/cosmic event-horizon entropy from Gibbons–Hawking (1977) S = A/4, evaluated numerically for our Λ in recent work (Egan & Lineweaver 2010: 2.6±0.3×10¹²² k; Frampton–Hsu–Kephart–Reeb 2009 et al. rounding to ~10¹²³). It is not a number that appears in Bousso's review.
- Frampton, Hsu, Kephart & Reeb, "What is the entropy of the universe?", Class. Quantum Grav. 26, 145005 (2009), arXiv:0801.1847 — abstract VERIFIED: their headline is that entropy is dominated by black holes (~10¹⁰² k from SMBHs) and they challenge the area-entropy accounting for precursors if unitarity holds; their S_max ~ 10¹²³ k appears in E&L's citation of them. (Full text not read — abstract only.)
5. Has anyone revised or corrected Lloyd's 2002 numbers since?
What I verified:
- Mark W. Coffey, "Estimates of universal computational capacity to the present," Phys. Lett. A 304 (2002), DOI 10.1016/S0375-9601(02)01325-7 — confirmed to exist via Crossref (author, journal, Nov 2002 issue). Snippet-level content (ScienceDirect abstract): "By combining results in cosmic dynamics with entropy bounds it is possible to present limits to the amount of information and its processing rate in the universe up to the present"; the abstract also summarizes the clock-rate inequality
Iν² ≤ t_P⁻² (operations per bit per second × bits), tied to Ng's work on spacetime foam (Y. J. Ng, PRL 86, 2946 (2001)). This is the main independent recomputation in the immediate aftermath of Lloyd 2002. Full text paywalled — UNVERIFIED at the level of its numbers.
- A citing computation yields ≈ 9.14 × 10¹²⁰ ops: an equation-28 result quoted against Lloyd's paper ("This is very close to the 10¹²⁰ predicted by Lloyd [29] for a matter-dominated universe at its critical density") — found on the ResearchGate record for Lloyd's paper. Identity of the citing paper not independently confirmed (likely Coffey 2002, since it is the one that recomputes these quantities with cosmic dynamics). UNVERIFIED attribution; treat as snippet-level.
- Egan & Lineweaver (2010) — see §4 — is the modern entropy-budget update for the bits side (their S_obs ~ 10¹⁰⁴ k interior vs Lloyd's ~10⁹⁰ bits ≈ 10⁸⁹–10⁹⁰ nats hypothetical max entropy: same ballpark; their horizon 2.6×10¹²² k vs Lloyd's holographic 10¹²⁰ bits: consistent order, E&L larger by ~10² because they use the Λ-driven event horizon at the current epoch, while Lloyd's t²/t_P² uses the particle-horizon/age convention).
- Bottom line on currency: No published retraction or correction of Lloyd's headline was found in the searches I ran (several query families, including "corrections", "revised", "recalculated", "updated cosmological parameters"). His own later book Programming the Universe (2006) repeats the 10¹²⁰/10⁹⁰ pair (NOT independently verified — I did not read the book). The 10¹²⁰ order-of-magnitude has been re-derived independently (Coffey's ≈9.14×10¹²⁰). The 10⁹⁰-bits matter figure likewise survives. The figure that has shifted in the literature is the gravitational/holographic ceiling: modern evaluations put the de Sitter event-horizon entropy at ~2.6×10¹²² k (≈3.8×10¹²² bits), commonly rounded to 10¹²³ — which is a different quantity (horizon entropy at present epoch) from Lloyd's t²/t_P² ≈ 10¹²⁰ bits.
DISCREPANCIES (numbers that differ across sources for the same quantity)
- Per-joule ops rate, factor of 2: Margolus–Levitin abstract: +1 J → ≤ ~3×10³³ ops/s (2E/h, sustained-sequence convention). Lloyd's bound 2E/πħ for 1 J = 6.04×10³³ ops/s (two-state convention). Same theorem, two conventions; Lloyd's ultimate-laptop rate 5.4258×10⁵⁰ uses the two-state (faster) convention. Anyone quoting "3×10³³" vs "6×10³³" per joule should say which convention.
- Strict arithmetic of (t/t_P)²: Lloyd's own Eq. 2 gives #ops ≈ (t/t_P)². With his t = 10¹⁰ yr = 3.16×10¹⁷ s and t_P = 5.391×10⁻⁴⁴ s, (t/t_P)² = 3.4×10¹²¹, not 10¹²⁰. With today's t = 13.8 Gyr it is 6.5×10¹²¹. He prints 10¹²⁰ because of his explicit "≈" convention (log within O(1), factors of 2π ignored) and the "no more than" hedging. If the argument you are building needs an exact exponent, the honest statement is "10¹²⁰–10¹²² depending on convention and t"; the paper's own headline is 10¹²⁰.
- 10⁵⁰ vs 10⁵¹ ops/sec for the ultimate laptop: the paper's precise value is 5.4258×10⁵⁰ (from E = 8.9874×10¹⁶ J); §3 rounds to ≈10⁵¹. Both appear within the same paper.
- 10¹²² vs 10¹²³ for the horizon entropy: E&L: 2.6±0.3×10¹²² k (R_CEH = 15.7 Glyr, nats). 3π/Λ with the Planck-unit Λ ≈ 2.9×10⁻¹²² gives ≈ 3.3×10¹²². Rounding to 10¹²³ appears in the Frampton et al. citation of S_max and in pop literature. Also: nats vs bits — E&L's numbers are in k (nats); in bits divide by ln2 (×1.44).
- Bousso abstract itself: v1 says "10⁶⁹ bits per square meter", v2 says "1.4×10⁶⁹ bits per square meter" — same claim, two printings.
- "10¹²⁰ bits with gravity" (Lloyd) vs "10¹²²–10¹²³ (horizon entropy)" (E&L/Frampton): different quantities and conventions (Lloyd: t²/t_P² matter-era holographic bound on bits available for computation; E&L: actual Bekenstein–Hawking entropy of the present cosmic event horizon). Do not conflate.
WHAT I COULD NOT GET (honest gaps)
- Coffey (Phys. Lett. A 304, 2002) full text — ScienceDirect paywall (403). Only the abstract and search snippets; his exact numbers (incl. the apparent 9.14×10¹²⁰ total-ops result) are snippet-verified only, and the attribution of the "9.14×10¹²⁰ / equation (28)" quote to Coffey is inferred, not confirmed.
- Frampton–Hsu–Kephart–Reeb (2009) full text — abstract only; not needed for the headlined numbers, but I could not verify their S_max ~ 10¹²³ internal derivation.
- Lloyd's book "Programming the Universe" (2006) — not fetched; his restatement of 10¹²⁰/10⁹⁰ there is unverified by me.
- Any post-2002 formal criticism of the Margolus–Levitin-based speed bound or Lloyd's cosmology — searched multiple query families; found none published against Lloyd's headline numbers. Absence of evidence, not evidence of absence: I did not exhaustively crawl citation databases (INSPIRE/Semantic Scholar rate-limited).
- The published PRL layout of Lloyd 2002 — I read the arXiv v1 full text, which matches the abstract verbatim; minor editorial differences from the PRL typeset version are unverified.
Quick-reference card (for the argument)
| Quantity |
Value |
Units |
Source (verified) |
| Ultimate laptop speed |
5.4258 × 10⁵⁰ |
ops/s (1 kg, 1 L, E = mc²) |
Lloyd 2000, §1.3 |
| Ultimate laptop memory |
2.13 × 10³¹ |
bits |
Lloyd 2000, Eq. 2 |
| Ops/sec bound, general |
2E/πħ (= 4E/h) |
ops/s |
Lloyd 2000; ML 1998 |
| Min orthogonality time |
τ⊥ ≥ h/4E = πħ/2E |
s |
ML 1998, Eq. 4 |
| Universe: ops since BB |
≈ (t/t_P)² ≈ 10¹²⁰ ("no more than") |
ops |
Lloyd 2002, Eq. 2 |
| Universe: bits (matter) |
≈ (#ops)^(3/4) ≈ 10⁹⁰ |
bits |
Lloyd 2002, Eq. 3 |
| Universe: bits (with gravity) |
≈ t²/t_P² ≈ 10¹²⁰ |
bits |
Lloyd 2002, §2 |
| Universe: ops incl. gravity |
2 × matter ops ≈ 10¹²⁰ |
ops |
Lloyd 2002, §1 |
| Cosmic event horizon entropy |
2.6 ± 0.3 × 10¹²² |
k (nats) ≈ 3.8×10¹²² bits |
E&L 2010, Eq. 42 |
| de Sitter horizon entropy (form) |
S = 3π/Λ (Planck units) |
— |
Bousso 2002, Eq. 123 |
| Holographic density |
1.4 × 10⁶⁹ |
bits/m² |
Bousso 2002, abstract |
| +1 J → max extra rate |
~3×10³³ (ML, 2E/h) or ~6×10³³ (2E/πħ) |
ops/s per J |
ML 1998 / Lloyd 2000 |
View exactly as delivered (raw text)
# Thread: Ultimate physical limits to computation — primary-source numbers
**Date:** 2026-09-15 · **Reader thread (Argus)** · **Status: COMPLETE**
Every number below was read from the paper itself (arXiv full text or journal HTML), not from a snippet, Wikipedia, or a citation. Each entry is tagged **VERIFIED** (read in the primary text) or **UNVERIFIED** (could not reach primary text; snippet/abstract only) with the reason.
Sources fetched in full text:
- Lloyd 2000: arXiv:quant-ph/9908043v3 HTML (published as *Nature* 406, 1047–1054 (2000), DOI 10.1038/35023282)
- Lloyd 2002: arXiv:quant-ph/0110141v1 HTML (published as *Phys. Rev. Lett.* 88, 237901 (2002))
- Margolus & Levitin 1998: arXiv:quant-ph/9710043v2 HTML (published as *Physica D* 120, 188–195 (1998), DOI 10.1016/S0167-2789(98)00054-2)
- Bousso 2002: arXiv:hep-th/0203101v2 HTML (published as *Rev. Mod. Phys.* 74, 825–874 (2002))
- Egan & Lineweaver 2010: arXiv:0909.3983v3 HTML (published as *ApJ* 710, 1825–1834 (2010))
---
## 1. Lloyd, "Ultimate physical limits to computation", Nature 406, 1047 (2000) — arXiv:quant-ph/9908043
**Headline numbers (all VERIFIED, read from v3 full text):**
- **Speed bound:** to perform an elementary logical operation in time Δt requires average energy `E ≥ πħ/2Δt`. Equivalently, a system with average energy E can perform at most **2E/πħ logical operations per second** (Eq. 1: `Σ_ℓ 1/Δt_ℓ ≤ Σ_ℓ 2E_ℓ/πħ = 2E/πħ`).
- **Ultimate laptop:** mass 1 kg, volume 1 litre. Its rest energy `E = mc² = 8.9874 × 10¹⁶ joules` → maximum **`5.4258 × 10⁵⁰` operations per second**. (Later in the same paper, §3, he rounds this to `≈10⁵¹` ops/sec. Both figures appear in the same paper.)
- **Derivation:** the bound is the Margolus–Levitin result, quoted as: a quantum system with **average** energy E takes at least `Δt = πħ/2E` to evolve to an orthogonal state (previous results quoted in terms of energy *spread* ΔE: `Δt ≥ πħ/2ΔE`). The NOT gate attains the limit; AND/FANOUT via Toffoli also attain `2E/πħ`; operations cycling through many states run at `E/πħ`, half as fast. Setting ground-state energy E₀ = 0 gives E = ΔE for the NOT example.
- **Memory bound:** `I = S(E)/k_B ln2` bits, where S(E) is the thermodynamic entropy (canonical ensemble "which maximizes S for fixed average energy with no constraint on the spread"). Energy limits speed; entropy limits memory.
- **Numbers for the ultimate laptop memory (photon-only lower bound, r = 2):** `k_B T = 8.10 × 10⁻¹⁵ J` → **T = 5.87 × 10⁸ K**; entropy **S = 2.04 × 10⁸ J/K**; memory **I = 2.13 × 10³¹ bits**. Using all memory, it performs `2ln2·k_B E/πħS = 3ln2·k_B T/2πħ ≈ 10¹⁹` operations per bit per second.
- **Serial vs parallel:** total ops/sec `2E/πħ` is **independent of architecture** — spread energy over N gates and each runs at `2E/πħN`; total is unchanged. Parallel computation gives smaller overall energy spread (ΔE ≈ √N ΔE_ℓ vs E ≈ N E_ℓ). Serialism measure `t_com/t_flip = 4ln2·k_B R E/πħcS ∝ k_B RT/ħc = 2πR/λ_T`. Ultimate laptop: R = 10⁻¹ m, `t_com/t_flip ≈ 10¹⁰` → **highly parallel**. Only the black-hole limit is fully serial (`t_com/t_flip = ln2/π`; Bekenstein bound `k_B RE/ħcS ≥ 1/2π`, equality for black holes).
- **Black-hole-compressed laptop:** Schwarzschild radius of 1 kg: `R_S = 1.485 × 10⁻²⁷ m`; memory `I = 4πGm²/ln2·ħc = 4πm²/ln2·m_P² = 3.827 × 10¹⁶ bits`; same `5.4258 × 10⁵⁰` ops/sec; 1-kg BH lifetime ≈ 10⁻¹⁹ s, performing ≈ 10³² ops on ≈ 10¹⁶ bits (upper bound per Page's evaporation rate).
- **Error correction / thermodynamics box:** memory black-body temperature 5.87 × 10⁸ K; bit-emission rate `ℬ = 7.195 × 10⁴² bits/m²/s`; to avoid overheating the error rate must be < 10⁻¹⁰ per operation; energy throughput would be `4.04 × 10²⁶ W` — turning over its rest mass (~10¹⁷ J) in a nanosecond.
- **Constants used:** c = 2.9979 × 10⁸ m/s, ħ = 1.0545 × 10⁻³⁴ J·s, G = 6.673 × 10⁻¹¹ m³/kg·s², k_B = 1.3805 × 10⁻²³ J/K. Planck scales: ℓ_P = 1.616 × 10⁻³⁵ m, t_P = 5.391 × 10⁻⁴⁴ s, m_P = 2.177 × 10⁻⁸ kg.
- **Every stated assumption/caveat (VERIFIED):** complete control over the computer's state; insulation from all uncontrolled degrees of freedom; conserved quantities other than energy are zero (charge, angular momentum; baryon number *not* conserved because black-hole formation/evaporation is allowed); the actual running state has zero entropy (memory space is the entropy of the *thermal* state, not the running state); "there is no guarantee that these limits will ever be attained"; the black-hole computer is "likely to prove extremely difficult or impossible to realize"; a unified quantum-gravity theory would be required at Planck density; error correction is needed in practice; memory limit is "unlikely to be attained" (packaging issues — the memory state resembles a thermonuclear plasma); the entropy estimate is a lower-bound estimate that depends only insensitively on particle species (a change of r by a factor of 10⁴ changes S by a factor of 10).
---
## 2. Lloyd, "Computational capacity of the universe", PRL 88, 237901 (2002) — arXiv:quant-ph/0110141
**Headline claim, VERIFIED verbatim from the arXiv abstract and full text:**
> "The universe can have performed no more than **10¹²⁰ ops on 10⁹⁰ bits**."
This is the matter-energy figure. Details of what each number counts (all VERIFIED from full text):
- **Ops count.** In the matter-dominated universe, ops/sec in a co-moving volume ≈ `ρc² × c³t³/ħ`; total over history `#ops ≈ ρc⁵t⁴/ħ` (Eq. 1). With ρ ≈ 10⁻²⁷ kg/m³ and t ≈ 10¹⁰ years → **≈ 10¹²⁰ ops**. At critical density, `ρ_c = 3H²/8πG ≈ 1/Gt²`, so `#ops ≈ ρ_c c⁵t⁴/ħ ≈ t²c⁵/Għ = (t/t_P)²` (Eq. 2), with t_P = 5.391 × 10⁻⁴⁴ s; again ≈ 10¹²⁰ (he says "no more than").
- **What it counts:** elementary quantum logic operations that all the matter inside the particle horizon (volume ≈ c³t³) could have performed since the Big Bang, at the Margolus–Levitin rate 2E/πħ. An "op" = some degrees of freedom with energy E evolving by angle π/2 in Hilbert space for time πħ/2E.
- **Gravitational degrees of freedom — the "separate figure":** gravitational energy is negative and equal in magnitude to the positive matter energy (total energy of the universe = 0), so including gravity **changes the ops count by exactly a factor of 2** — still ≈ 10¹²⁰. This is provisional: "no complete quantum theory of gravity currently exists."
- **Bits count.** Maximum entropy = all matter converted to radiation: `I = S/k_B ln2 ≈ (ρc⁵t⁴/ħ)^(3/4) = (#ops)^(3/4)` (Eq. 3, substituting V ≈ c³t³). → **the universe could currently register ≈ 10⁹⁰ bits** ("To register this amount of information requires every degree of freedom of every particle in the universe"). These bits require converting all matter into thermal radiation.
- **Bits with gravity (holographic):** using the Bekenstein bound + holographic principle, max bits including gravitational degrees of freedom `≈ c²t²/ℓ_P² = t²/t_P² ≈ 10¹²⁰` — equal to the ops count, not 10⁹⁰.
- **So the correct statement is:** `10¹²⁰ ops on 10⁹⁰ bits` = matter/energy only. If gravitational degrees of freedom are included, bits rise to ≈ 10¹²⁰ (ops stay ≈ 10¹²⁰, ×2 exactly).
- **Radiation-dominated era:** ops count stays within a factor of 2 of the matter-dominated formula (Eq. 5, finite even as t→0 — answers Dyson's big-crunch question); bits `≈ (#ops)^(3/4)` again while `D^(1/4) ≈ 1` below the GUT scale (k_B T < 2 × 10¹⁶ GeV, T ≈ 10²⁹ K).
- **Inflation:** horizon-scale e-folding 50 → universe can have performed **no more than 10^(20±12) ops** on the same number of bits within the horizon during inflation.
- **Eddington–Dirac numerology (included for completeness, he warns "do not take such numerology too seriously"):** #ops ≈ βγ² ≈ αβ² ≈ 10¹²⁰, differing from the large-number-cubed by (137/1836) ≈ 10.
- **Man-made computers (circa 2001, overestimated):** ≈ 10⁹ computers at 10⁹ Hz doing 10⁵ ops/cycle for 10⁸ s → ≈ 10³¹ ops in the last two years, ≤ ~2× that in all history; ≈ 10²¹ bits registered.
- **Three interpretations he gives:** (1) upper bound on computation performed by the universe; (2) lower bound on ops/bits needed to simulate the universe on a quantum computer; (3) the actual ops and bits of the universe-qua-computation (with the π/2-phase definition of an op).
- **Stated model/assumption set (VERIFIED):** standard Big Bang model, universe began ≈ 10¹⁰ years ago; matter-dominated for most of history; energy in a co-moving volume approximately constant; horizon volume ≈ c³t³ (ignoring the factor that the horizon is presently beyond ct); the **"≈" convention**: equality to within factors of order 2π in the log (`X ≈ Y` means `log X = log Y + O(1)`); critical-density universe for Eq. 2; Margolus–Levitin is strictly quantum-mechanical; quantum gravity required for the gravitational numbers; holographic principle (Bekenstein bound + ℓ_P²) used for gravitational bits.
- **Constants:** ħ = 1.0545 × 10⁻³⁴ J·s (one place uses 1.0546), c = 2.98 × 10⁸ m/s, G = 6.673 × 10⁻¹¹, k_B = 1.38 × 10⁻²³ J/K; t_P = 5.391 × 10⁻⁴⁴ s.
**Verdict on your memory:** "10¹²⁰ operations on 10⁹⁰ bits" is **correct as published, verbatim** — with the caveat that it is the matter-only pair, that both numbers are "≈" (order-of-magnitude, factors of 2π ignored), and that with gravitational degrees of freedom included the bit count becomes ≈ 10¹²⁰.
---
## 3. Margolus & Levitin, "The maximum speed of dynamical evolution", Physica D 120, 188 (1998) — arXiv:quant-ph/9710043
All VERIFIED from v2 (published version) full text:
- **The bound, exact form:** with the zero of energy at the ground state (E₀ = 0; footnote: if no energies below some E_min are allowed, take the zero there instead), the minimum time for *any* state with fixed **average** energy E to evolve into an orthogonal state is
**`τ⊥ ≥ h/4E`** (Eq. 4).
Since h = 2πħ, this is **`τ⊥ ≥ πħ/2E`** — same bound, two notations. So "h/4E **or** πħ/2E" is not a choice between two different bounds; they are the same bound in h vs ħ notation.
- **What E measures:** E is the **average energy above the ground state** (`E − E₀`), NOT the standard deviation and not the maximum energy. The abstract states this: "a strict bound that depends only on E−E₀, the system's average energy minus its ground state energy."
- **Achievability:** the bound is attained (two-state oscillation, state `(|0⟩+|2E⟩)/√2`, spectrum containing 2E; very nearly attained by any ordinary macroscopic system). The system can oscillate between two orthogonal states at rate **4E/h**; a long sequence of mutually orthogonal states runs at asymptotic rate **`ν⊥ = 2E/h = E_max/h`** (Eq. 26), with the exact-cycle bound `τ⊥ ≥ ((N−1)/N)(h/2E)` (Eq. 13). I.e., sustained long evolution runs at half the two-state rate.
- **Prior work / Mandelstam–Tamm relation:** the older bound (Mandelstam & Tamm 1945, ref [10]; also Pfeifer, Vaidman) is `τ⊥ ≥ h/4ΔE` (Eq. 5) in terms of the energy *spread* ΔE. The ML point: "This bound places no limit, however, on how fast a system with bounded average energy can evolve (since ΔE can be arbitrarily large with fixed E)." They exhibit a state (their Eq. 12) with fixed E and arbitrarily large ΔE, where the ΔE bound is useless but Eq. 4 is optimal. So ML replaces ΔE with E in the bound; nothing is added to the ΔE bound itself.
- **Information-processing rate:** "adding one Joule of energy to a given computer can never increase its processing rate by more than about **3×10³³ operations per second**" (abstract; note this is 2E/h for E = 1 J = 2/(6.626×10⁻³⁴) ≈ 3.02×10³³ — the *sustained* rate convention).
- **Composite systems:** for non-interacting subsystems, total rate = sum of subsystem rates (additivity — the parallel-computer statement).
- **Relativistic/semi-classical remarks (published version only):** a system with total rest-frame energy E_r passes through ≤ 2E_r t_r/h orthogonal states in rest time t_r; in another frame the bound is (2/h)(Et − px), subtracting one orthogonal state per shift of distance h/2p.
- **Convention note:** the two-state oscillation rate 4E/h = 2E/πħ is exactly Lloyd's ops/sec bound (`2E/πħ`); the sustained rate 2E/h is half that. Lloyd's `Δt = πħ/2E` per operation corresponds to the *two-state* ML rate. Papers quoting per-joule numbers must say which convention: ML's abstract gives 3×10³³ (2E/h); Lloyd's formula would give 2E/πħ ≈ 6.04×10³³ for 1 J. Both descend from the same theorem; difference is the factor 2 discussed in §2.2. (This is a real, if cosmetic, cross-paper discrepancy — see DISCREPANCIES.)
---
## 4. Holographic / Bousso bound for the observable universe — arXiv:hep-th/0203101
**What Bousso's review actually says (VERIFIED):**
- **Statement of the bound:** "the area of any surface limits the information content of adjacent spacetime regions, at **1.4 × 10⁶⁹ bits per square meter**" (v2 abstract; the v1 abstract printed it as 10⁶⁹ bits/m² without the 1.4). Planck length: `ℓ_P² = Għ/c³ = 2.59 × 10⁻⁶⁶ cm²`.
- **De Sitter space entropy (his Eqs. 122–123):** the future event horizon of de Sitter space carries Bekenstein–Hawking entropy `S_dS = πa₀² = 3π/Λ` (Planck units), with Gibbons–Hawking temperature `T = 1/2πa₀` (Gibbons & Hawking 1977). In a dS± spacetime the *global* entropy is bounded by `S_global ≤ 3π/Λ` (Eq. 124); in any universe with Λ > 0, entropy observable by a single observer (causal diamond) is bounded by `3π/Λ` (Bousso 2000b).
- **His one cosmological number:** Λ ≈ 10⁻¹²⁰ (Planck units) for our universe → de Sitter entropy scale 3π/Λ ≈ 10¹²⁰-ish. **Bousso's review itself never prints "10¹²²" or "10¹²³" for our universe** (verified by search over the full text). Those pop-culture figures come from later numerical evaluations (below).
- **Which horizon:** the bound applies to the **cosmic (de Sitter) event horizon** — the Λ-driven horizon of an accelerating universe. Bousso's FRW analysis (§VII.1) is explicit that the *particle horizon* is the wrong surface: surfaces beyond the particle horizon (χ > η) have light-sheets that terminate at the Big Bang, and the spherical bound fails for large volumes in flat/closed universes (S > A); the covariant entropy bound holds for flat FRW with −1 ≤ w ≤ 1.
- **The actual figures people quote (VERIFIED in Egan & Lineweaver 2010, arXiv:0909.3983v3 full text):**
- **Current cosmic event horizon entropy: `S_CEH = (kc³/Għ)·(A/4) = (kc³/Għ)·πR_CEH² = 2.6 ± 0.3 × 10¹²² k`** (units of k = Boltzmann constant, i.e., **nats**; in bits divide by ln2 → ≈ 3.8 × 10¹²² bits), for `R_CEH = 15.7 ± 0.4 Glyr` (ΛCDM event horizon; de Sitter static radius √(3/Λ) is ≈ 17.4 Glyr). They explicitly identify this with the holographic bound. E&L adopted parameters: Ωk = 0, h = 0.705 ± 0.013, ωb = 0.0224, ωm = 0.136, T_CMB = 2.725 K.
- **Entropy of the observable universe's interior (particle horizon): `S_obs = 3.1 (+3.0/−1.7) × 10¹⁰⁴ k`**, dominated by supermassive black holes (SMBHs), which contribute "at least an order of magnitude more entropy than previously estimated." CEH interior: `S_CEHint = 1.2 (+1.1/−0.7) × 10¹⁰³ k`. Components (S in k): SMBHs 3.1×10¹⁰⁴; stellar BHs 5.9×10⁹⁷; photons 5.40×10⁸⁹; relic neutrinos 5.16×10⁸⁹; WIMP dark matter 10⁸⁸±¹; relic gravitons 6.2×10⁸⁷; ISM/IGM 7.1×10⁸¹; stars 9.5×10⁸⁰. V_obs = 3.65±0.10×10⁸⁰ m³; V_CEH = 1.37±0.10×10⁷⁹ m³.
- E&L cite the holographic ceiling as `S_max ~ 10¹²³ k` (their §I, citing Frampton et al.). **The ≈10¹²²–10¹²³ figure is the de Sitter/cosmic event-horizon entropy from Gibbons–Hawking (1977) S = A/4, evaluated numerically for our Λ in recent work (Egan & Lineweaver 2010: 2.6±0.3×10¹²² k; Frampton–Hsu–Kephart–Reeb 2009 et al. rounding to ~10¹²³).** It is not a number that appears in Bousso's review.
- Frampton, Hsu, Kephart & Reeb, "What is the entropy of the universe?", Class. Quantum Grav. 26, 145005 (2009), arXiv:0801.1847 — abstract VERIFIED: their headline is that entropy is dominated by black holes (~10¹⁰² k from SMBHs) and they *challenge* the area-entropy accounting for precursors if unitarity holds; their S_max ~ 10¹²³ k appears in E&L's citation of them. (Full text not read — abstract only.)
---
## 5. Has anyone revised or corrected Lloyd's 2002 numbers since?
**What I verified:**
- **Mark W. Coffey, "Estimates of universal computational capacity to the present," Phys. Lett. A 304 (2002), DOI 10.1016/S0375-9601(02)01325-7** — confirmed to exist via Crossref (author, journal, Nov 2002 issue). Snippet-level content (ScienceDirect abstract): "By combining results in cosmic dynamics with entropy bounds it is possible to present limits to the amount of information and its processing rate in the universe up to the present"; the abstract also summarizes the clock-rate inequality `Iν² ≤ t_P⁻²` (operations per bit per second × bits), tied to Ng's work on spacetime foam (Y. J. Ng, PRL 86, 2946 (2001)). This is the main independent recomputation in the immediate aftermath of Lloyd 2002. **Full text paywalled — UNVERIFIED at the level of its numbers.**
- **A citing computation yields ≈ 9.14 × 10¹²⁰ ops:** an equation-28 result quoted against Lloyd's paper ("This is very close to the 10¹²⁰ predicted by Lloyd [29] for a matter-dominated universe at its critical density") — found on the ResearchGate record for Lloyd's paper. Identity of the citing paper not independently confirmed (likely Coffey 2002, since it is the one that recomputes these quantities with cosmic dynamics). **UNVERIFIED attribution; treat as snippet-level.**
- **Egan & Lineweaver (2010)** — see §4 — is the modern entropy-budget update for the *bits* side (their S_obs ~ 10¹⁰⁴ k interior vs Lloyd's ~10⁹⁰ bits ≈ 10⁸⁹–10⁹⁰ nats hypothetical max entropy: same ballpark; their horizon 2.6×10¹²² k vs Lloyd's holographic 10¹²⁰ bits: consistent order, E&L larger by ~10² because they use the Λ-driven event horizon at the *current* epoch, while Lloyd's t²/t_P² uses the particle-horizon/age convention).
- **Bottom line on currency:** No published retraction or correction of Lloyd's headline was found in the searches I ran (several query families, including "corrections", "revised", "recalculated", "updated cosmological parameters"). His own later book *Programming the Universe* (2006) repeats the 10¹²⁰/10⁹⁰ pair (NOT independently verified — I did not read the book). The 10¹²⁰ order-of-magnitude has been re-derived independently (Coffey's ≈9.14×10¹²⁰). The 10⁹⁰-bits matter figure likewise survives. The figure that *has* shifted in the literature is the gravitational/holographic ceiling: modern evaluations put the de Sitter event-horizon entropy at ~2.6×10¹²² k (≈3.8×10¹²² bits), commonly rounded to 10¹²³ — which is a different quantity (horizon entropy at present epoch) from Lloyd's t²/t_P² ≈ 10¹²⁰ bits.
---
## DISCREPANCIES (numbers that differ across sources for the same quantity)
1. **Per-joule ops rate, factor of 2:** Margolus–Levitin abstract: +1 J → ≤ ~3×10³³ ops/s (2E/h, sustained-sequence convention). Lloyd's bound 2E/πħ for 1 J = 6.04×10³³ ops/s (two-state convention). Same theorem, two conventions; Lloyd's ultimate-laptop rate 5.4258×10⁵⁰ uses the two-state (faster) convention. Anyone quoting "3×10³³" vs "6×10³³" per joule should say which convention.
2. **Strict arithmetic of (t/t_P)²:** Lloyd's own Eq. 2 gives #ops ≈ (t/t_P)². With his t = 10¹⁰ yr = 3.16×10¹⁷ s and t_P = 5.391×10⁻⁴⁴ s, (t/t_P)² = **3.4×10¹²¹**, not 10¹²⁰. With today's t = 13.8 Gyr it is 6.5×10¹²¹. He prints 10¹²⁰ because of his explicit "≈" convention (log within O(1), factors of 2π ignored) and the "no more than" hedging. If the argument you are building needs an *exact* exponent, the honest statement is "10¹²⁰–10¹²² depending on convention and t"; the paper's own headline is 10¹²⁰.
3. **10⁵⁰ vs 10⁵¹ ops/sec for the ultimate laptop:** the paper's precise value is 5.4258×10⁵⁰ (from E = 8.9874×10¹⁶ J); §3 rounds to ≈10⁵¹. Both appear within the same paper.
4. **10¹²² vs 10¹²³ for the horizon entropy:** E&L: 2.6±0.3×10¹²² k (R_CEH = 15.7 Glyr, nats). 3π/Λ with the Planck-unit Λ ≈ 2.9×10⁻¹²² gives ≈ 3.3×10¹²². Rounding to 10¹²³ appears in the Frampton et al. citation of S_max and in pop literature. Also: nats vs bits — E&L's numbers are in k (nats); in bits divide by ln2 (×1.44).
5. **Bousso abstract itself:** v1 says "10⁶⁹ bits per square meter", v2 says "1.4×10⁶⁹ bits per square meter" — same claim, two printings.
6. **"10¹²⁰ bits with gravity" (Lloyd) vs "10¹²²–10¹²³ (horizon entropy)" (E&L/Frampton):** different quantities and conventions (Lloyd: t²/t_P² matter-era holographic bound on *bits available for computation*; E&L: actual Bekenstein–Hawking entropy of the present cosmic event horizon). Do not conflate.
---
## WHAT I COULD NOT GET (honest gaps)
- **Coffey (Phys. Lett. A 304, 2002) full text** — ScienceDirect paywall (403). Only the abstract and search snippets; his exact numbers (incl. the apparent 9.14×10¹²⁰ total-ops result) are **snippet-verified only**, and the attribution of the "9.14×10¹²⁰ / equation (28)" quote to Coffey is inferred, not confirmed.
- **Frampton–Hsu–Kephart–Reeb (2009) full text** — abstract only; not needed for the headlined numbers, but I could not verify their S_max ~ 10¹²³ internal derivation.
- **Lloyd's book "Programming the Universe" (2006)** — not fetched; his restatement of 10¹²⁰/10⁹⁰ there is unverified by me.
- **Any post-2002 *formal* criticism of the Margolus–Levitin-based speed bound or Lloyd's cosmology** — searched multiple query families; found none published against Lloyd's headline numbers. Absence of evidence, not evidence of absence: I did not exhaustively crawl citation databases (INSPIRE/Semantic Scholar rate-limited).
- **The published PRL layout of Lloyd 2002** — I read the arXiv v1 full text, which matches the abstract verbatim; minor editorial differences from the PRL typeset version are unverified.
---
## Quick-reference card (for the argument)
| Quantity | Value | Units | Source (verified) |
|---|---|---|---|
| Ultimate laptop speed | 5.4258 × 10⁵⁰ | ops/s (1 kg, 1 L, E = mc²) | Lloyd 2000, §1.3 |
| Ultimate laptop memory | 2.13 × 10³¹ | bits | Lloyd 2000, Eq. 2 |
| Ops/sec bound, general | 2E/πħ (= 4E/h) | ops/s | Lloyd 2000; ML 1998 |
| Min orthogonality time | τ⊥ ≥ h/4E = πħ/2E | s | ML 1998, Eq. 4 |
| Universe: ops since BB | ≈ (t/t_P)² ≈ 10¹²⁰ ("no more than") | ops | Lloyd 2002, Eq. 2 |
| Universe: bits (matter) | ≈ (#ops)^(3/4) ≈ 10⁹⁰ | bits | Lloyd 2002, Eq. 3 |
| Universe: bits (with gravity) | ≈ t²/t_P² ≈ 10¹²⁰ | bits | Lloyd 2002, §2 |
| Universe: ops incl. gravity | 2 × matter ops ≈ 10¹²⁰ | ops | Lloyd 2002, §1 |
| Cosmic event horizon entropy | 2.6 ± 0.3 × 10¹²² | k (nats) ≈ 3.8×10¹²² bits | E&L 2010, Eq. 42 |
| de Sitter horizon entropy (form) | S = 3π/Λ (Planck units) | — | Bousso 2002, Eq. 123 |
| Holographic density | 1.4 × 10⁶⁹ | bits/m² | Bousso 2002, abstract |
| +1 J → max extra rate | ~3×10³³ (ML, 2E/h) or ~6×10³³ (2E/πħ) | ops/s per J | ML 1998 / Lloyd 2000 |
Argus