Pith. sign in

REVIEW 3 major objections 6 minor 38 references

The paper argues that a finite-difference momentum operator on an L-qubit grid, matched to the special-relativistic momentum, produces a mass-dependent generalised uncertainty principle whose minimum position uncertainty survives the contin

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 12:17 UTC pith:KFMLNBNU

load-bearing objection Low-energy GUP from a qubit grid is sound; the high-energy minimum-length claim is undone by a false commutator identity. the 3 major comments →

arxiv 2607.19602 v1 pith:KFMLNBNU submitted 2026-07-21 quant-ph gr-qc

Quantum Gravity Simulation: Quantum simulation with a minimum length based on the generalised uncertainty principle

Jack Keable-Elliott , David J. Bacon , Andrew Burbanks , Jaewoo Joo This is my paper
classification quant-ph gr-qc
keywords generalised uncertainty principlequantum simulationminimum lengthdiscretised momentumqubit gridspecial relativityquantum gravityfinite-difference method
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper seeks to show that the discretisation inherent in qubit-based quantum simulation is not just a numerical nuisance: it is a working model of a minimum length. On a grid with spacing ΔL, the usual Heisenberg bound is deformed, and in the low-energy regime the deformation has three regimes, including a single-point localisation at a finite momentum uncertainty. For high energies the authors define a deformed momentum as a power series in the grid momentum and match it to the special-relativistic momentum. That matching yields a mass-dependent GUP with parameter β_m = 3ℏc/(2Gm^2) and a minimum position uncertainty of order (√3/√2)ℏ/(mc), present even when ΔL→0. The paper's second high-energy branch shows that enforcing the standard HUP instead removes the minimum length but forces a modified energy-momentum relation.

Core claim

The central claim is that a first-quantised wavefunction on a 2^L-point lattice carries a built-in generalised uncertainty principle. The low-energy version reduces to the Heisenberg principle as ΔL/ℏ→0, but for finite ΔL yields three momentum-dependent regions; a wavepacket can reach zero position uncertainty at a finite momentum spread. The high-energy version begins from p^H_x = Σ κ_{n-1} p_x^n. Setting the κ to the special-relativistic series p/√(1−p²/m²c²) and assuming point-momentum behaviour (δp^(n)_x = 0) gives δx δp^H ≥ (ℏ/2)[1 + τ_2 (δp^H)² + ...], with τ_2 = 3/(2m²c²) − (ΔL)²/(2ℏ²). In the continuous limit ΔL→0 this is a quadratic GUP with β_m = 3ℏc/(2Gm²), so (δx)_min = (√3/√2)ℏ/

What carries the argument

The machinery is a pair of finite-difference operators on the 2^L-point grid: x = ΔL Σ l |l⟩⟨l| and p_x = −iℏ/(2ΔL)(A − A^T), with p_x² defined as the Laplacian rather than the algebraic square of p_x. From these, the paper derives generalised commutators [x,p_x^{2q}] = 2iℏ q p_x^{2q−1} and [x,p_x^{2q−1}] = iℏ[(2q−1)p_x^{2q−2} − q(ΔL)²/(2ℏ²) p_x^{2q}]. A deformed high-energy momentum is then written as a power series in p_x, and its coefficients are fixed by matching to the special-relativistic expansion; the point-momentum conjecture makes the expectation values factor as ⟨p^n_x⟩ = ⟨p_x⟩^n. These pieces convert grid fineness and particle mass into the τ_t corrections that shape the GUP.

Load-bearing premise

The minimum-length prediction rests on two idealisations the paper states explicitly: the grid shift operators commute (AA^T = A^T A = I) so the high-order commutator identities in SM4 are exact, and the relativistic particle has zero momentum spread (δp^(n)_x = 0) in SM5 so expectation values factor as ⟨p^n_x⟩ = ⟨p_x⟩^n; if either is relaxed, the predicted GUP and δx_min do not follow.

What would settle it

Compute [x, p_x^4] directly from the definitions in Eq. (2) on a grid with N = 2 points. Equation (14) predicts 4iℏ p_x^3; direct evaluation gives 2iℏ(p_x p_x² + p_x² p_x), which differs, so the high-energy GUP (16) and its minimum-length prediction are not consequences of the stated operators. A second check is to prepare a low-energy state with non-zero fourth cumulant and test whether the special-relativistic matched GUP lower bound still holds.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A 41-qubit electron simulation would have τ_2 > 0 and show a mass-dependent minimum position uncertainty around 10^-13 m, without needing Planck-scale hardware.
  • Low-energy simulations with finite ΔL exhibit three uncertainty regimes; a wavefunction can localise to a single grid point at a finite momentum spread, setting a practical resolution limit for simulators.
  • If the special-relativistic matched branch is correct, the minimum length depends on the particle's mass rather than being a universal Planck-scale constant, so lighter particles show larger minimum position spreads.
  • The HUP-preserving branch predicts a modified energy-momentum relation at high energy, which could be searched for as a dispersion-relation signature in simulated relativistic systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The derivation effectively replaces the lattice cutoff with a mass-dependent physical cutoff; if that swap is legitimate, any finite-resolution simulator—not only qubit grids—might exhibit the same GUP structure.
  • The point-momentum conjecture is testable directly: prepare Gaussian and sinc states with known higher cumulants and compare the observed (δx)(δp^H) against Eq. (21); realistic wavepackets should show corrections to the predicted bound.
  • A natural extension is to carry the same construction into two or three dimensions and to interacting multi-particle systems; the mass dependence of β_m would then appear as a species-dependent minimum resolution in simulations.
  • If the point-momentum conjecture is dropped, the coefficient matching in Eq. (19) becomes underdetermined, so an independent derivation of the κ_n would be needed to fix the predicted minimum length.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript proposes an L-qubit first-quantised lattice with grid spacing ΔL and derives a low-energy GUP from the central-difference momentum p_x and a Laplacian squared momentum p²_x. The derived commutator [x,p_x] = iℏ(I − (ΔL)²p²_x/2ℏ²) yields the uncertainty bound (6)-(7) with three regimes (8)-(10), including a zero-position-uncertainty bound and a finite (δx)_min in the large-⟨p_x⟩ regime. For high energies the authors define a deformed momentum p^H_x = Σ κ_{n−1}p^n_x, prove (in SM4) commutators (13)-(14), and choose the κ coefficients to match the special-relativistic momentum under the 'point-momentum' conjecture δp^{(n)}_x = 0. This yields a mass-dependent GUP with β_m = 3ℏc/(2Gm²) and a minimum length √(3/2)ℏ/(mc) persisting as ΔL→0, claimed to be visible in a 41-qubit electron simulation; an alternative HUP-preserving choice (22) is incompatible with special relativity.

Significance. If established, the high-energy result would be a striking, falsifiable bridge from lattice simulation to GUP phenomenology, with an explicit qubit count (L=41 for an electron) at which the minimum length appears. The paper has real strengths: the low-energy derivation (Eqs. (4)-(11), SM1) is self-contained and algebraically correct; the operator definitions are explicit; and the QuTiP-based numerics with pseudocode (Algorithms 1-2) make the low-energy results reproducible, with data matching the analytic curves. However, the central high-energy claim is not a derived consequence of the framework. It rests on a false commutator identity (Major Comment 1) and on an unproven conjecture (Major Comment 2); the operator series defining the relativistic momentum is also not convergent on the discrete spectrum (Major Comment 3). The significance of the paper in its present form is therefore confined to the low-energy lattice GUP, a modest but sound result.

major comments (3)
  1. [SM4, Eq. (14)] SM4 proves Eqs. (13)-(14) under the explicit assumption A A^T = A^T A = I, but for A defined in Eq. (2), A A^T = I − |2^L⟩⟨2^L| and A^T A = I − |1⟩⟨1|, so [A,A^T] ≠ 0. The error is directly checkable: for L=1, p_x² = −(ℏ/2ΔL)² I and therefore [x,p_x⁴] = 0, contradicting Eq. (14) with q=2, which gives 4iℏp_x³ ≠ 0. Even if the commutation assumption were made, the proof identifies (p²_x)^q with (p_x)^{2q}, conflating the Laplacian (3) with the algebraic square that the paper itself distinguishes earlier; the Leibniz reduction cannot be collapsed to q[x,p²_x](p²_x)^{q−1}. Thus Eq. (16) and everything downstream — the high-energy GUP (20), β_m and the minimum length (21), and the abstract's headline prediction — are not validly derived.
  2. [Eq. (19), SM5] The coefficient matching at Eq. (19) invokes the 'strong conjecture' δp^{(n)}_x = ⟨p^n_x⟩ − ⟨p_x⟩^n = 0 for all n, so that ⟨p^n_x⟩ is replaced by ⟨p_x⟩^n. SM5 does not justify this: its conclusion that δp^{(n)}_x = 0 implies ⟨p^n_x⟩ = ⟨p_x⟩^n simply assumes the lower-order central moments vanish and restates the definition. For the Gaussian and sinc states used in the simulations δp_x ≠ 0 by construction, so the conjecture is inapplicable to the states actually tested. Without the conjecture the κ_n in Eq. (15) are free parameters; matching them to the SR series (17) is a choice, not a derivation. The abstract's unconditional claim that the framework 'predicts' a minimum length therefore overstates what has been shown; Eq. (21) is conditional on an unsupported assumption.
  3. [Eq. (18), SM6] The relativistic momentum operator in Eq. (18) is defined by a formal Taylor series in p²_x/(m²c²). On the L-qubit grid the Laplacian (3) has eigenvalues up to 4ℏ²/ΔL²; for the advertised 41-qubit electron ΔL = 2⁻⁴¹ m ≈ 1.18 ℏ/(m_ec), so p²_x/(m²c²) reaches ≈ 2.9 and the series (1−z)^{−1/2} diverges on part of the Hilbert space. The statement that the series 'converges slowly' concerns expectation values for ⟨p_x⟩ ≲ mc, not operator-level convergence. The definition of p^H_x used in Algorithm 2 is therefore not well founded, independently of Major Comment 1.
minor comments (6)
  1. [Abstract, §Low-energy GUP] Comparing ΔL with ℏ mixes units of length and action; the numerics set ℏ=0.1. Recommend natural units or dimensionless ratios.
  2. [Eqs. (13)-(14)] The symbol p^{2q}_x is ambiguous — algebraic power of p_x or power of the Laplacian p²_x? The low-energy section's emphasis on this distinction makes the ambiguity consequential; the convention should be fixed.
  3. [Around Eq. (11)] The δx=0 bound is only an inequality; no state is shown to attain it, and the Fig. 1 data approach but do not reach it. Please soften 'can exhibit a single-point localisation'.
  4. [Planck-length paragraph; SM1] Typos: 'acheive' should be 'achieve'; 'opearator' should be 'operator'.
  5. [Fig. 2] The L=40 and L=42 curves in the inset are difficult to distinguish; add markers or use separate panels.
  6. [References] Ref. [29] is incomplete ('Physical Review, 21483'); several entries lack full pagination.

Circularity Check

1 steps flagged

High-energy minimum-length prediction reduces to the chosen special-relativistic momentum input by construction; β_m is a relabeling of the SR coefficient and the minimum length is the Compton wavelength.

specific steps
  1. renaming known result [Eqs. (17)–(21) and SM6]
    "κ_{n−1} = κ^{re}_{n−1} = 0 for even n, κ_0 = κ^{re}_0 = 1, κ_2 = κ^{re}_2 = 1/(2m^2c^2), κ_4 = κ^{re}_4 = 3/(8m^4c^4) and so on. ... for the quadratic GUP-parameter β_m = 3ℏc/(2Gm^2)."

    The high-energy momentum operator is defined by setting its expansion coefficients equal to the special-relativistic series (Eqs. 17–19). The commutator then yields τ2 = 3/(2m^2c^2) in the GUP (Eq. 20). Writing this as β_m G/(ℏc^3) with β_m = 3ℏc/(2Gm^2) is a definition, not a derivation: G cancels from the resulting minimum length δx_min = √(3/2)ℏ/(mc), which is just the reduced Compton wavelength. The 'quantum gravity' parameter and the minimum length are the SR input rewritten, so the headline prediction is equivalent to the choice of input by construction.

full rationale

The low-energy GUP (Eqs. 4–11) is self-contained: it follows directly from the central-difference operators in Eqs. (2)–(3), and the numerical checks in SM3 test the same analytic expression. The circularity is in the high-energy minimum-length claim. Equations (17)–(19) set p̂_x^H to be exactly the special-relativistic momentum series; Eq. (20) then gives τ2 = 3/(2m^2c^2), which is relabeled in Eq. (21) as β_m G/(ℏc^3) by defining β_m = 3ℏc/(2Gm^2). The resulting δx_min = √(3/2)ℏ/(mc) is independent of G and is just the Compton wavelength, so the gravitational constant enters only through a redefinition of the SR coefficient. This is a renaming of a known relativistic result rather than an independent first-principles prediction. I also note a non-circular but load-bearing correctness issue: SM4 proves Eqs. (13)–(14) only under the assumption ÂÂᵀ = ÂᵀÂ = Î, whereas for the open-boundary shift operator in Eq. (2), ÂÂᵀ = I − |2^L⟩⟨2^L| and ÂᵀÂ = I − |1⟩⟨1|, so the assumption is false and Eq. (14) is not established for this lattice. This invalidates the high-energy GUP as a derivation, but it is an error rather than a circularity. The HUP-preserving alternative in Eq. (22) is also constructed by setting τ_t = 0, confirming that the existence of a minimum length is tied to the input choice rather than to a robust, model-independent result.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The central result rests on a grid-discretization ansatz (ΔL = 1/2^L), the choice of Laplacian momentum, a false operator identity (AA^T = I) in SM4, the point-momentum conjecture for matching special relativity, and the choice of the SR-matched high-energy momentum over the HUP-preserving one.

free parameters (2)
  • Grid spacing ΔL = 1/2^L (examples: 2^-10, 2^-40, 2^-41, 2^-42)
    Chosen by hand through the qubit count L; it defines the lattice momentum cutoff ℏ/ΔL and controls the low-energy GUP. The high-energy minimum-length claim is taken in the limit ΔL→0, so this parameter defines the simulation framework but does not set the final β_m.
  • Series truncation order n = 15 (Algorithm 2)
    Number of κ terms used in the QuTiP high-energy simulation; a numerical truncation of the infinite series in Eq. (15), not a physical parameter.
axioms (6)
  • standard math Robertson–Schrödinger uncertainty relation δx δp ≥ |⟨i[x,p]⟩|/2
    Used to convert the commutators (4) and (16) into the GUP inequalities (6), (16), (20), (21).
  • domain assumption p_x² is defined by the central-difference Laplacian (Eq. (3)), not as the square of the first-order momentum (Eq. (2))
    Needed so that (p_x)² = p_x² − (ΔL)²/(4ℏ²)p_x⁴ and the commutator identities (13), (14) can be written in the paper's form.
  • ad hoc to paper AA^T = A^T A = I for the open-boundary shift operators
    Asserted in SM4 to derive the generalised commutator identities, but false on the finite non-periodic grid: AA^T = I − |2^L⟩⟨2^L| and A^T A = I − |1⟩⟨1|. This invalidates Eq. (14) for q≥2.
  • ad hoc to paper Point-momentum conjecture: δp^(n)_x = ⟨p_x^n⟩ − ⟨p_x⟩^n = 0 for all n
    Used after Eq. (18) and in SM5 to set κ_{n−1} = κ^re_{n−1} so ⟨p_H⟩ matches the special-relativistic momentum. Without it the SR-matched high-energy momentum expectation does not match and the β_m formula in Eq. (21) lacks support.
  • domain assumption Operator expansion p_re = p_x(1 − p_x²/(m²c²))^{-1/2} is valid on the grid; requires |p_x| ≤ ℏ/ΔL < mc
    For an electron, ℏ/ΔL > m_e c for L>41, so the expanded relativistic operator is singular on some grid states; the paper restricts the minimum-length discussion to L≤41 (see the L=42 marker in Fig. 2 and the surrounding text).
  • ad hoc to paper Choice of SR-matched momentum (Eq. (18)) rather than HUP-preserving momentum (Eq. (22)) defines the 'prediction' of a minimum length
    The minimum length follows only for this choice; the paper explicitly presents a second choice with no minimum length, so the headline result is one fork of a trade-off rather than a unique consequence of the framework.

pith-pipeline@v1.3.0-alltime-deepseek · 17894 in / 34451 out tokens · 258398 ms · 2026-08-01T12:17:34.010079+00:00 · methodology

0 comments
read the original abstract

We present a recipe for simulating one-dimensional quantum systems for low and high energies with $L$ qubits within the framework of first quantisation. Assuming a minimum grid spacing $\Delta L$ in the finite-difference method, the generalised uncertainty principle (GUP) is derived analytically and shows distinct properties for low- and high-energy quantum systems. In the low-energy regime, the GUP approaches to the standard Heisenberg uncertainty principle (HUP) for $ \Delta L \ll \hbar$. However, for finite $\Delta L \neq 0$, the GUP mathematically provides three different regions dependent on the average momentum and suggests that a wavefunction can exhibit a single-point localisation at the finite momentum uncertainty due to lack of grid resolution. We then derive a new expression for the high-energy momentum and focus on two specific cases relevant to relativistic aspects. First, if the HUP is relaxed and the high-energy momentum is matched to the special-relativistic one, the resulting GUP predicts the existence of a minimum length with non-zero mass for high energy despite the continuous limit $\Delta L = 0$. Second, enforcing consistency with the HUP allows the recovery of a canonical high-energy representation with no minimum length. But this requirement brings incompatibility with the special-relativistic momentum and suggests a modified energy-momentum equation. Therefore, we believe that the proposed momentum formulation provides a novel pathway to investigate quantum gravity phenomena for high energy using quantum simulation tools.

Figures

Figures reproduced from arXiv: 2607.19602 by Andrew Burbanks, David J. Bacon, Jack Keable-Elliott, Jaewoo Joo.

Figure 1
Figure 1. Figure 1: FIG. 1. Low-energy GUP curves from Eqs. (8) to (10) with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. High-energy GUP curves for an electron using differ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Low-energy GUP curves using 10 qubits with ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. High-energy GUP curve with ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

38 extracted references · 3 linked inside Pith

  1. [1]

    P. Bossoet. al, Class. Quantum Grav.40, 195014 (2023)

    2023

  2. [2]

    L. N. Changet. al, Adv. High Energy Phys.2011, 493514 (2011)

    2011

  3. [3]

    R. J. Adler,Am. J. of Phys.,78, (2010)

    2010

  4. [4]

    A. Kempfet. al, Phys. Rev. D52, 1108 (1995)

    1995

  5. [5]

    P. Bossoet. al, Phys. Rev. D107, 126009 (2023)

    2023

  6. [6]

    Y. S. Myunget. al, Phys. Lett. B645, 393 (2006)

    2006

  7. [7]

    D. Amatiet. al, Phys. Lett. B216, 41 (1989)

    1989

  8. [8]

    Maggiore,Phys

    M. Maggiore,Phys. Lett. B304, 65 (1993)

    1993

  9. [9]

    L. J. Garay,Int. J. Mod. Phys. A10, 145 (1995)

    1995

  10. [10]

    Wagner,Phys

    F. Wagner,Phys. Rev. D104, 126010 (2021)

    2021

  11. [11]

    M. H. Al Ghifariet. al, Gen. Relativ. Gravit.57, 121 (2025)

    2025

  12. [12]

    Y. C. Ong,J. Cosmol. Astropart. Phys.09, 015 (2018)

    2018

  13. [13]

    A. F. Ali and A. N. Tawfik,Int. J. Mod. Phys. D22, 1350020 (2013)

    2013

  14. [14]

    Pikovskiet

    I. Pikovskiet. al, Nat. Phys.8, 393 (2012)

    2012

  15. [15]

    S. Boseet. al, Phys. Rev. Lett.119, 240401 (2017)

    2017

  16. [16]

    S. Boseet. al,arXiv:2509.01586

    Pith/arXiv arXiv

  17. [17]

    Petruzziello and F

    L. Petruzziello and F. Illuminati,Nat. Comm.12, 4449 (2021)

    2021

  18. [18]

    S. P. Kumar and M. B. Plenio,Nat. Comm.11, 3900 (2020)

    2020

  19. [19]

    Yoshiokaet

    N. Yoshiokaet. al, Nat. Comm.16, 5014 (2025)

    2025

  20. [20]

    Yamamotoet

    K. Yamamotoet. al, PRX Quantum7, 020319 (2026)

    2026

  21. [21]

    L. E. Fischeret. al, Nat. Phys.22, 302 (2026)

    2026

  22. [22]

    Di Meglioet

    A. Di Meglioet. al, PRX Quantum5, 037001 (2024)

    2024

  23. [23]

    Joo and T

    J. Joo and T. Spiller,New J. Phys.25, 083041 (2023)

    2023

  24. [24]

    Lubaschet

    M. Lubaschet. al, Phys. Rev. A101, 010301(R) (2020)

    2020

  25. [25]

    Joo and H

    J. Joo and H. Moon, arXiv:2109.09216

    Pith/arXiv arXiv

  26. [26]

    J. Jooet. al,arXiv:2509.22579

    Pith/arXiv arXiv

  27. [27]

    P. Jizbaet. al, Phys. Rev. D81, 084030 (2010)

    2010

  28. [28]

    Heisenberg,Z

    W. Heisenberg,Z. Phys.43, 172 (1927)

    1927

  29. [29]

    A. H. Compton,Physical Review,21483, (1923)

    1923

  30. [30]

    Balasubramanianet

    V. Balasubramanianet. al, Ann. Phys.360, 1 (2015)

    2015

  31. [31]

    M. F. Gussonet. al, Eur. Phys. J. C81336 (2021)

    2021

  32. [32]

    Pedram,Phys

    P. Pedram,Phys. Lett. B714, 317 (2012)

    2012

  33. [33]

    Todorinovet

    V. Todorinovet. al, Ann. Phys.405, 92 (2019)

    2019

  34. [34]

    Das and E

    S. Das and E. C. Vagenas,Phys. Rev. Lett.101, 221301 (2008)

    2008

  35. [35]

    Petruzziello,Class

    L. Petruzziello,Class. Quantum Grav.38, 135005 (2021)

    2021

  36. [36]

    Amelino-Camelia and V

    G. Amelino-Camelia and V. Astuti,Mod. Phys. Lett. A, 2640001 (2026)

    2026

  37. [37]

    G. G. Luciano and L. Petruzziello,Eur. Phys. J. C79, 283 (2019)

    2019

  38. [38]

    ∆L xj + ∆L−x 0 + 2(∆L)2 (xj −x 0)2 −(∆L) 2 # ,(44) 9 we can simplify our expression to ⟨ˆpx⟩s =− iℏ 2∆L N 2 s 2L−1X j=1

    G. Amelino-Camelia,Living Reviews in Relativity16, 1 (2013). 6 SUPPLEMENT AR Y MA TERIAL SM1: POSITIVITY OF⟨ˆp2 x⟩ Let us start by defining a general form for the first-quantised localised wavefunction |ψ⟩= 2L X j=1 cj|j⟩,(23) with P2L j=1 |cj|2 = 1. As such, we say that our wavefunction is well localised as we are confining it to a relatively narrow regi...

    2013