Local Hilbert-Space Dimension as a Resolution Parameter: Information-Theoretic Bounds for Qudit Quantum Cellular Automata as Substrates of Emergent Holographic Geometry

In the paper's words

Discrete models of quantum gravity lack a computable criterion for when their emergent geometry becomes semiclassical, and the system sizes at which quantum cellular automata (QCA) show holographic entanglement structure remain out of reach of exact simulation. We ask whether enlarging the local Hilbert space from qubits to qudits lowers the system size N at which continuum-like behavior appears. We develop the analytical framework for qudit QCA with local dimension d: we derive the exact scaling of Hilbert-space dimension d^N, the maximum cut entanglement S_max=min(|A|,|B|)log₂d, the site-count equivalence N_d=N_q/log₂d between qudit and qubit chains at matched information capacity, and the per-gate simulation overhead d²/4 at matched Hilbert dimension.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. These bounds give concrete numbers: a $d=3$ chain of 16 sites carries the entropy capacity of a 26-site qubit chain (a 36.9% site saving) at 2.25× the per-gate simulation cost, and a $d=4$ chain of 12 sites matches a 24-site qubit chain at 4× cost.

   from Abstract

2. A projection, with stated assumptions, that the emergence threshold scales as $N^{\ast}(d)\propto 1/\log_{2}d$, giving candidate thresholds $N^{\ast}\approx 20$ ($d=2$), $13$ ($d=3$), $10$ ($d=4$) at a 5% discretization criterion (Section 4.5, Section 5).

   from 1. Introduction

3. We assume a qubit emergence threshold $N^{\ast}(2)=20$ sites, defined as the smallest qubit chain size at which the relative change of the interval entropy curve between successive $N$ falls below 5% for the Fredkin-class gate family.

   from 4.5 Projection of the emergence threshold

4. Quantity $d=2$ $d=3$ $d=4$ Per-site entropy capacity $\log_{2}d$ (bits) 1 1.5849625 2 Sites matching a 24-qubit chain 24 16 12 Site saving vs 24 qubits — 36.9% 50% Per-gate cost ratio $d^{2}/4$ at matched dimension 1 2.25 4 Rule-table entries $S_{r}=d^{2}$ (two-input rule) 4 9 16 Projected emergence threshold $N^{\ast}$ (sites) 20 (assumed) 13 10 Three results follow.

   from 5. Results

5. First, at matched information capacity a $d=3$ chain of 16 sites carries the entropy capacity of a 25–26 site qubit chain (36.9% site saving) and a $d=4$ chain of 12 sites matches 24 qubits exactly (50% saving).

   from 5. Results

6. Third, under the stated assumptions the emergence threshold falls from an assumed 20 qubit sites to a projected 13 ($d=3$) or 10 ($d=4$) sites, bringing the threshold inside the range accessible to exact state-vector simulation (Section 4.3 shows $N=14$ at $d=4$ requires 4.295 GB).

   from 5. Results

How it is done

1. We develop the analytical framework for qudit QCA with local dimension $d$: we derive the exact scaling of Hilbert-space dimension $d^{N}$, the maximum cut entanglement $S_{\max}=\min(|A|,|B|)\log_{2}d$, the site-count equivalence $N_{d}=N_{q}/\log_{2}d$ between qudit and qubit chains at matched information capacity, and the per-gate simulation overhead $d^{2}/4$ at matched Hilbert dimension.

   from Abstract

2. Without a tunable parameter that moves the emergence threshold to smaller $N$, every investigation of emergent geometry in locally unitary dynamics is confined to the regime where finite-size effects dominate, and no statement about the approach to smooth-manifold behavior can be tested.

   from 1. Introduction

3. Decidability results bound what can be proven about QCA phase spaces.

   from 2. Background and Related Work

4. We define a qudit quantum cellular automaton (QCA) as follows.

   from 3.1 Model definition

What would falsify it

1. The conjecture is thus distinguishable from its alternatives only by simulation, which is why Section 5 specifies the falsification protocol rather than asserting the exponent.

   from 6.2 Limitations

2. The Fredkin-class prioritization follows the delocalization contrast certified in [9]; a gate family that fails to delocalize information would show no convergence of the entropy curve at any $d$, and the projection would be vacuous for it.

   from 6.2 Limitations

3. The main result — the projected threshold shrinkage — is falsified by a specific observation: at matched Hilbert dimension, simulate the Fredkin-generalized gate family at $d\in\{2,3,4\}$ and compute the interval entropy curve $S(L)$ for each $N$ up to 14.

   from 6.3 Failure modes and falsification

Where the claim stops

1. The projection rests on the following assumptions, each with its sensitivity: A1 (baseline threshold $N^{\ast}(2)=20$).

   from 6.2 Limitations

2. This is a stated assumption, not a measurement.

   from 6.2 Limitations

3. The hypothesis that threshold scales as $1/\log_{2}d$ is the paper's central conjecture and its most fragile assumption.

   from 6.2 Limitations

Where to read

1. Abstract207 words · 1 min
2. 1. Introduction618 words · 3 min
3. 2. Background and Related Work1,075 words · 5 min
4. 3. Methods453 words · 2 min
5. 4. Analysis801 words · 3 min
6. 5. Results218 words · 1 min
7. 6. Discussion878 words · 4 min
8. 7. Practical Implications and Engineering Pathways684 words · 3 min
9. 8. Conclusion204 words · 1 min
10. Provenance and AI Involvement225 words · 1 min