Physical complexity and black hole quantum computers

Michele Reilly; Seth Lloyd

REVIEW 4 major objections 3 minor 23 references

Physical complexity and black hole quantum computers

T0 review · 4 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Free energy, not logic or memory alone, is the physical currency of computation—and by that measure black hole quantum computers are physically intractable.

desk verdict The black-hole section is built on an inconsistent use of entropy that sinks the main new claim; the rest is a competent but mostly derivative summary of known physics. read the letter →

arxiv 2506.16527 v1 pith:KXIIMWIR submitted 2025-06-19 quant-ph cs.CC

Michele Reilly , Seth Lloyd This is my paper · ORCID

classification quant-phcs.CC

keywords physicalcomplexityfreeenergynegentropyerrorcorrectionblackholecomputationquantumspeedlimitthermodynamicsoferasurethermodynamicdepth

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the resources bounding all computation are thermodynamic, not merely logical: temporal complexity becomes an energy–time product, $\Phi\text{-TIME}=2Et/\pi\hbar$, spatial complexity becomes negentropy, $\Phi\text{-SPACE}=S_{\max}-S$, and once error correction is included both grow in proportion to the free energy consumed, which the paper calls FREE-$\Phi$. A sympathetic reader would take the central assertion to be that free energy is the single physical resource behind speed, memory, and error correction. Applied to black holes, this yields a result: exactly half of a black hole's energy is free energy available for computation, yet its total operations over its lifetime, $\sim M^4/m_P^4$, dwarf its information capacity, $\sim M^2/m_P^2$, so the entropy generated by errors cannot be accommodated. The paper concludes that black hole quantum computers are in principle powerful but physically intractable to program and correct.

What carries the argument

The central object is free physical complexity, FREE-$\Phi$, defined as the free energy used by a system over a computation. The argument runs through two identifications: $\Phi\text{-TIME}=2Et/\pi\hbar$ comes from the quantum speed limit on flipping a bit with energy $E$ in time $t$, and $\Phi\text{-SPACE}=S_{\max}-S$ identifies available memory with negentropy, also called thermodynamic depth. The error-correction step is the mechanism that forces $\Phi\text{-SPACE}\propto\Phi\text{-TIME}$: each operation at error rate $\epsilon$ injects $\epsilon\log(1/\epsilon)$ entropy, which must be pumped out at free-energy cost, so the physical resources of a noisy computation are governed by free energy. For black holes, the same machinery yields the lifetime operation count and the entropy-capacity bound that make error correction intractable.

What would settle it

Track the entropy budget in a unitary model of black hole evaporation with a radiation bath: if the bath can absorb $\epsilon\log(1/\epsilon)$ bits per operation at a rate no larger than the hole's entropy-emission rate while the hole remains programmable, then the per-operation error rate need not be as low as $m_P^2/M^2$. A concrete model exhibiting this would falsify the paper's intractability claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, physical computational complexity has three measures: temporal complexity $\Phi\text{-TIME}$ is the minimum energy–time product $2Et/\pi\hbar$ needed to perform a computation; spatial complexity $\Phi\text{-SPACE}$ is the negentropy $S_{\max}-S$ available as clean memory; and free physical complexity FREE-$\Phi$ is the free energy consumed, which combines the two. The load-bearing step is error correction: a bit-flip probability $\epsilon$ injects about $\epsilon\log(1/\epsilon)$ bits of entropy per operation, and the thermodynamic cost of erasing that entropy forces it out at free-energy cost $k_BT\,\epsilon\log(1/\epsilon)$, so physical memory requirements grow linearly with computation length no matter how small the error rate. For a black hole of mass $M$, the free energy is $F=Mc^2/2$, exactly half its energy, and the total number of operations over its lifetime is $\sim M^4/m_P^4$, while its entropy capacity is $\sim M^2/m_P^2$. Keeping the accumulated error entropy within that capacity forces a per-operation error rate below $\sim m_P^2/M^2$; the paper concludes that the physical spatial complexity of black hole error correction exceeds the black hole's own spatial complexity, making black hole quantum computers physically intractable.

Load-bearing premise

The black-hole intractability result assumes that all entropy generated by errors over the entire computation must be stored in the hole and can never exceed its instantaneous entropy capacity, $\sim M^2/m_P^2$; if the radiation emitted by the hole can carry that entropy away continuously, the conclusion collapses.

Editorial extensions

If this is right

Any physical computer running at finite error rate must consume fresh negentropy at a rate proportional to its number of operations, so fault tolerance is a thermodynamic requirement rather than a purely logical one.
Since $\Phi\text{-SPACE}$ and $\Phi\text{-TIME}$ are both proportional to FREE-$\Phi$, complexity classes gain a physical reading: polynomial versus exponential logical resource gaps correspond to gaps in free energy consumed.
Black holes can perform at most $\sim M^4/m_P^4$ operations over their lifetime while storing at most $\sim M^2/m_P^2$ bits, so a programmable hole requires per-operation error rates below $\sim m_P^2/M^2$; for all but microscopic holes this is unattainable in practice.
Biological information processing operates far closer to the thermodynamic limits than digital electronics do, so the roughly $10^5$ gap in energy efficiency is an architectural feature of current artificial intelligence, not a fundamental limit.
Because the universe is at critical density, the same formulas bound the total computation the universe can perform over its history.

Reading between the lines

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

The paper's intractability conclusion uses a lifetime-total entropy bound. If the radiation emitted by a black hole can dispose of error entropy continuously, as it disposes of other entropy, the relevant constraint is the disposal rate rather than the lifetime total, and the negative conclusion could weaken.
A testable extension is to measure how error entropy is flushed in near-term quantum error-correcting processors: if the spatial complexity cost can be paid continuously by the environment rather than stored in the computer, the linear growth $\Phi\text{-SPACE}\propto\Phi\text{-TIME}$ may be avoidable in open systems.
FREE-$\Phi$ offers a single number for comparing radically different substrates, such as silicon, neurons, chemical reaction networks, and black holes; it could serve as a benchmark for artificial intelligence by measuring free energy consumed per reliable bit-operation.
A unitary evaporation model with a radiation bath could be checked for whether the bath absorbs $\epsilon\log(1/\epsilon)$ per operation; if it does, the error-rate constraint becomes a rate condition rather than a total-capacity condition, directly testing the paper's central negative result.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The paper is a synthesis of Margolus–Levitin, Landauer, and black-hole thermodynamics, dressed up with new names (Φ-TIME, Φ-SPACE, FREE-Φ). The first two are just the standard energy–time product and negentropy; FREE-Φ is a relabeling of free energy, and its defining equation, FREE-Φ = T(S_eq − S(ρ))/πħ, is dimensionally wrong — it has units of frequency, not energy. That’s not a harmless typo; it’s the definition of their new measure.

What is genuinely useful: the point that error correction forces physical space complexity to grow linearly with time is correct and worth stating clearly, and the biological-vs-electronic ops/sec comparison is a nice back-of-the-envelope. But the bio-ops figure leans on an unpublished self-citation, which weakens that section.

The load-bearing problem is in Section 6. The black hole’s free energy is computed as F = Mc² − T S_BH = Mc²/2, treating the hole as if it were in its maximum-entropy thermal state. But the memory capacity they assign, O(M²/m_P²), is exactly that maximum entropy. Under their own Φ-SPACE = S_max − S, a thermal hole has zero memory; a programmed pure hole has O(M²) memory but F ≈ Mc², not Mc²/2. You cannot have both exactly-half free energy and O(M²) programmable bits from the same state. The intractability argument then compares total lifetime ops (∝M⁴) to instantaneous entropy capacity (∝M²), but if Hawking radiation can carry away error entropy continuously, the relevant bound is a disposal rate, not a lifetime total. The paper doesn’t justify the total-ops comparison.

So the central new claim — that black-hole quantum computers are intractable because error correction demands more negentropy than the hole has — fails on the paper’s own definitions. Sections 1–5 are a readable review of known results, but the black-hole application should not be taken at face value.

This deserves a serious referee only because the topic and author will attract attention and a referee can force the entropy bookkeeping to be cleaned up; as it stands, the conclusion doesn’t hold. I wouldn’t cite it.

Referee Report

4 major / 3 minor

Summary. This paper proposes physical analogues of computational time and space: Φ-TIME = 2Et/πħ (the number of operations permitted by the Margolus-Levitin bound) and Φ-SPACE = Smax−S (negentropy). It introduces FREE-Φ as the free energy used during computation, argues that error correction forces spatial physical complexity to grow linearly with temporal physical complexity, and applies these measures to black holes, concluding that exactly half of a black hole's energy is free energy but that error correction makes black hole quantum computers physically intractable because total lifetime operations scale as M^4/m_P^4 while storage capacity scales only as M^2/m_P^2. The paper also estimates biological information-processing rates and discusses implications for AI development.

Significance. The manuscript draws together well-established physical bounds (Margolus-Levitin, Bekenstein-Hawking) into a resource-theoretic perspective on computation, and the idea of free energy as a unified computational resource is potentially useful. If the black-hole analysis were sound, it would give a striking no-go result. However, the central FREE-Φ definition contains a dimensional inconsistency, the black-hole free-energy and memory-capacity claims rely on incompatible entropy assignments to the same state, and the intractability argument conflates lifetime accumulation with instantaneous capacity. These issues are load-bearing for the paper's main claims, so the paper cannot be accepted in its current form; the core ideas are likely salvageable with a careful revision that fixes the entropy bookkeeping and supplies the missing justifications.

major comments (4)

[Section 4, FREE-Φ definition] The displayed relation FREE-Φ = T(S_eq−S(ρ))/πħ is dimensionally inconsistent. With k_B = 1, S_eq and S(ρ) are dimensionless, T has dimensions of energy, and ħ has dimensions of energy×time, so the right-hand side has dimensions of inverse time, not energy. The correct relation between free energy and relative entropy is F(ρ)−F_eq = T(S_eq−S(ρ)); if instead FREE-Φ is intended to count operations per unit time, that should be derived from the Margolus-Levitin bound (e.g., (2/πħ)Ft) and labeled as a rate. Since FREE-Φ is advertised as representing the amount of free energy used, the formula as written would make FREE-Φ vanish for a thermal equilibrium state, which conflicts with the F = Mc^2/2 claim for a black hole in Section 6.1.
[Sections 6.1 and 3.1] The black-hole free-energy claim uses an entropy assignment that is incompatible with the memory-capacity claim. Section 6.1 states F = Mc^2 − TS = Mc^2/2, using S = S_BH, the Bekenstein-Hawking entropy, i.e., the maximum-entropy thermal state. But Section 3.1 defines usable memory space as Smax − S, so a state with S = S_BH has zero Φ-SPACE and cannot provide the O(M^2/m_P^2) bits of memory assumed for a programmed black hole. Conversely, if the black hole is programmed into a low-entropy state to have memory, its free energy is F ≈ Mc^2, not Mc^2/2. The paper must specify a single entropy assignment for the programmed state; the 'exactly half' claim and the O(M^2) memory claim cannot both hold for the same state.
[Section 6.2] The error-correction intractability argument compares total lifetime operations O(M^4/m_P^4) to the black hole's instantaneous entropy capacity O(M^2/m_P^2), concluding that the error rate must satisfy ε log(1/ε) ≲ m_P^2/M^2. This treats the hole's maximum-entropy capacity as a lifetime budget for error entropy and does not justify why error entropy cannot be expelled by Hawking radiation, which is the hole's normal entropy-disposal channel. The relevant constraint may instead be a rate constraint, comparing error-entropy production rate (∼ ε M in Planck units) with the Hawking entropy-emission rate (∼ 1/M in Planck units); such a rate-based derivation may yield a similar bound on ε, but it is not given in the manuscript. As written, the argument is incomplete and needs to be repaired.
[Sections 3.1 and 7] The quantitative claim that a human performs on the order of 10^20–10^22 bio-ops per second, comparable to global electronic computing, is supported only by reference [23], an unpublished self-citation. Because this number drives the paper's conclusions about biological efficiency, the 'currency of intelligence,' and the implications for AI, the authors should either provide a derivation in the present paper, cite a published source, or clearly label the figure as an order-of-magnitude estimate.

minor comments (3)

[Section 3.1] The sentence 'the total number of bits of memory available to a system with entropy S is no greater than S−S_max' has the sign reversed; it should be S_max − S, consistent with the definition of Φ-SPACE.
[Section 6.1] In the sentence 'The lifetime of a black hole with mass M is M = 5120πG^2M^3/ħc^4', the left-hand side should be t_M (or τ), not M, since M is the mass.
[Table in Section 1] The table describes Φ-TIME as 'accumulated quantum phase,' but 2Et/πħ is a dimensionless count of operations, not a phase; a single Margolus-Levitin-saturating bit flip accumulates phase EΔt/ħ = π/2. The wording is misleading.

Circularity Check

1 steps flagged · score 4.0 of 10

The formal physical-complexity derivation is self-contained, but one load-bearing empirical claim (biological ops/sec) is supported only by an unpublished self-citation; black-hole entropy bookkeeping raises a separate consistency risk rather than a circular reduction.

self citation load bearing [Section 3.1 and Section 7 (Conclusion); reference [23]]
"An individual human being expends≈100 watts of energy (≈100 kilocalories/hour) on reproduction and metabolism Accordingly, cellular information processing in an individual human being is performing on the order of 10 20−22 bio-ops per second – approximately the same as the number of logical operations performed by all the electronic computers in the world (one zetaflop). [...] the biochemical information processing taking place in the living cells of a single human being is the same order of magnitude as the information processing being performed by all electronic computers on earth [23]."

The numerical bio-ops/sec figure and the comparison to all electronic computers are asserted in the body and then attributed in the Conclusion to [23], an unpublished manuscript by the same authors. The figure is not derived from the paper's equations and no external, reproducible measurement is provided, so the biological-efficiency conclusion rests on a self-citation for its key empirical input. This does not affect the formal Φ-TIME/Φ-SPACE derivation, but it makes the biological/AI claim circular in the sense that the authors' own unpublished number is used as if it were independent support.

full rationale

The core derivation of physical complexity is not circular: Φ-TIME = 2Et/πħ is an application of the externally established Margolus-Levitin quantum speed limit; Φ-SPACE = S_max − S is a stated definition; the error-correction scaling Φ-SPACE ~ t ε log(1/ε) follows by counting entropy injected per errored bit; and the black-hole ops and capacity numbers are standard Bekenstein-Hawking/Margolus-Levitin arithmetic. These parts are self-contained and do not assume the conclusions. The main circularity burden is the empirical 10^20−22 bio-ops/sec figure, which is supported only by an unpublished self-citation ([23]); this affects the biological/AI implications but not the formal complexity results. A separate consistency concern that is not a circular reduction: §6.1 computes black-hole free energy F = Mc²/2 with S = S_BH, while §3.1 defines available memory as S_max − S; if the hole is programmed (low S) the memory exists but F is not Mc²/2, and if S = S_BH then Φ-SPACE = 0. This is a correctness/consistency risk, not a case of the prediction being equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

The paper introduces no fitted parameters. Its claims rest on standard results in quantum speed limits, thermodynamics of computation, error correction, and black hole thermodynamics, plus order-of-magnitude estimates for biological information processing. The main new entities are the named measures FREE-Phi and bio-op, both of which are relabelings or modeling conventions rather than novel physical postulates.

assumptions (8)

standard math Margolus-Levitin theorem bounds operation rate: any process flipping a bit in time Delta t requires average energy E >= pi hbar / (2 Delta t).
Used in Section 3.3 to define Phi-TIME = 2Et/pi hbar and to bound black hole ops/sec (Section 6.1). It is a proven quantum speed limit.
domain assumption Landauer's principle: erasing a bit costs at least k_B T ln 2 and dumps one bit of entropy.
Invoked in Sections 2 and 3.2 to connect error correction entropy to free energy cost.
domain assumption Error correction requires a continuous supply of fresh zero-entropy ancilla bits; used ancillas are discarded.
Central to the claim that Phi-SPACE grows linearly with Phi-TIME; based on Bennett's reversible computation and standard quantum error correction, Sections 2 and 3.2.
domain assumption Bekenstein-Hawking entropy S = 4 pi M^2 / m_P^2 and temperature T = m_P^2 c^2 / (8 pi M).
Used in Section 6.1 to compute black hole free energy F = Mc^2/2 and total operations over the hole's lifetime.
domain assumption Black hole evaporation is unitary.
Stated in Section 6.1 as the condition for programmed information to emerge in Hawking radiation; acknowledged as an assumption.
domain assumption The universe at critical density can be treated as a black hole for complexity purposes.
Stated in Section 6, following reference [2]; this extrapolates black hole results to the whole universe.
domain assumption Human biological information processing runs at roughly 10^20 to 10^22 operations per second.
Used in Section 3.1 and Conclusion to compare biological and digital computation; based on reference [23], an unpublished self-citation.
standard math Shannon entropy of a bit-flip error is approximately epsilon log(1/epsilon).
Used in Section 3.2 to estimate entropy production per error. This is a standard information-theoretic result.

invented entities (2)

FREE-Phi (free physical complexity)
purpose: To serve as a unified complexity measure combining time and space costs.
Defined as the free energy used during computation, it renames a standard thermodynamic quantity and makes no new falsifiable prediction.
bio-op
purpose: To count biological molecular operations (bond formation/breaking) as computational steps for comparing life to computers.
Defined in Section 3.1; the conversion of biological processes to operations is an order-of-magnitude modeling choice, not a measurable new entity.

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Cite this review

Pith. "Pith review of Physical complexity and black hole quantum computers." pith.science (2026). https://pith.science/paper/KXIIMWIR

@misc{pith2026250616527,
  author       = {Pith},
  title        = {Pith review of: Physical complexity and black hole quantum computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXIIMWIR}},
  note         = {Machine review of arXiv:2506.16527}
}

read the original abstract

The ultimate limits of computation are not just logical, but physical. We investigate the physical resources -- time, energy, entropy, and free energy -- required to perform computational work. We apply the resulting measures of physical complexity to conventional electronic computers, to quantum computers, to biological systems, to black holes, and to the universe itself, with implications for artificial intelligence development where biological efficiency limits suggest new computational paradigms beyond current digital architectures.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [23]

Lloyd S and Reilly M 2025 Natural Intelligence: The Information Processing Power of Life

work page 2025

[2] [1]

Lloyd S 2000 Ultimate Physical Limits to ComputationNature4061047-1054

work page 2000

[3] [2]

Lloyd S 2002 Computational Capacity of the UniversePhysical Review Letters88237901

work page 2002

[4] [3]

Landauer R 1961 Irreversibility and heat generation in the computing processIBM Journal of Research and Development5183–191

work page 1961

[5] [4]

Lecerf Y. 1963 Machines de Turing r´ eversiblesComptes Rendus Hebdomadaires des S´ eances de L’acad´ emie des Sciences2572597–2600 (Also available at http://www.cise.ufl.edu/ mpf/rc/Lecerf/lecerf.html)

work page 1963

[6] [5]

Bennett C H 1973 Logical reversibility of computation.IBM Journal of Research and Development 17525–532

work page 1973

[7] [6]

Bennett C H 1982 The thermodynamics of computation—a reviewInternational Journal of Theoretical Physics21905–940

work page 1982

[8] [7]

Fredkin E and Toffoli T 1982 Conservative logicInternational Journal of Theoretical Physics21 219–253

work page 1982