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REVIEW 3 major objections 5 minor 53 references

Holographic quantum error-correcting codes have been realized on a trapped-ion processor, with bulk qubits recovered from boundary regions and the Ryu-Takayanagi entanglement area law measured in the laboratory.

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2026-08-01 20:45 UTC pith:MAVCQVXM

load-bearing objection First real laboratory implementations of holographic pentagon and heptagon codes, with genuinely new decoding circuits and a quasi-transversal Hadamard gate; the experimental work deserves a serious referee, but the RT-law test is quantitatively unsupported as written. the 3 major comments →

arxiv 2607.16503 v1 pith:MAVCQVXM submitted 2026-07-17 quant-ph

Holographic quantum codes with trapped ions

Alex Steiner , Gerard Angl\`es Munn\'e , Robert Freund , Ivan Pogorelov , Michael Meth , Robert J. Harris , Gavin Brennen , Thomas M. Stace
show 4 more authors
Thomas Monz Rainer Blatt Felix Huber Martin Ringbauer
This is my paper
classification quant-ph MSC 81P7081P68 PACS 03.67.Pp03.67.Lx
keywords holographic quantum error correctionpentagon codeheptagon codetrapped-ion quantum processorRyu-Takayanagi formulapartial decodingquasi-transversal Hadamard gatestabilizer graph code
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 reports the experimental realization of holographic quantum error-correcting codes on a trapped-ion processor. For the holographic pentagon code, the authors encode four logical bulk qubits into twelve boundary qubits, then recover two, three, or four bulk qubits from only their nearby boundary regions. They also measure boundary entanglement entropies that track the cut length, matching the Ryu-Takayanagi area law. For the holographic heptagon code, they show that a physical Hadamard gate on one logical block acts as a logical Hadamard up to a single correctable error on the neighboring block, and they correct that error in post-processing. If correct, this establishes holographic codes as practical experimental tools for studying bulk-boundary correspondence and geometry-aware quantum error correction.

Core claim

The paper establishes, on its own terms, that holographic quantum error correction can be brought into the laboratory. A small pentagon code—built from four six-qubit absolutely maximally entangled states—successfully encodes four bulk qubits into twelve boundary qubits, and logical information can be recovered from boundary subregions alone: two bulk qubits from five boundary qubits, three from seven, and four from eight. The measured von Neumann entropies of the boundary regions are 3.96(2), 4.13(2), and 3.818(10), close to the ideal values 3, 4, and 4 predicted by the Ryu-Takayanagi formula. For the heptagon code, formed by contracting two seven-qubit CSS codes, the transversal Hadamard o

What carries the argument

The load-bearing object is the stabilizer graph-code representation of the tensor-network holographic code: the code state |G0000> is a graph state prepared by applying 28 controlled-Z gates to twelve |+> qubits, and the graph structure supplies the logical operators and the decoding circuits. The six-qubit absolutely maximally entangled states—states where every half is maximally entangled with its complement—act as the pentagon building blocks. A decoding cut is implemented by a unitary built from concatenated AME states, each expressed as a three-to-three-qubit unitary extracted from the graph code. The Ryu-Takayanagi formula enters as S(∂E) = |γ|, equating the entropy of a boundary regio

Load-bearing premise

Everything holographic rests on the claim that the graph-code representation used in the experiments reproduces the original tensor-network holographic code exactly, including which boundary regions can recover which bulk regions; if that equivalence fails for the chosen cuts, the observed partial-recovery regions and entropies would not demonstrate holography.

What would settle it

Simulate the original tensor-network pentagon code directly for cuts 1–3 and compare the partial-recovery maps and boundary entropies with the graph-code circuits used here; if any of the 2-, 3-, or 4-qubit recovery maps fail to act as the claimed isometries, or if the entropy departs from |γ| beyond depolarizing-noise expectations, the holographic interpretation is undermined. For the heptagon code, verify that the sign flip of the product g5g10 after the physical Hadamard is the only induced error; finding a second uncorrectable error would falsify the quasi-transversal claim.

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

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If this is right

  • The heptagon result provides a template for logical gates in holographic codes: a gate that is transversal in the constituent code may induce only a single correctable error in the glued code, so a short round of error correction restores fault-tolerant operation.
  • Partial decoding from a confined boundary region means holographic codes can protect information even when large parts of the quantum processor are inaccessible or noisy, a property relevant for burst errors.
  • The measured entropies support the Ryu-Takayanagi area law in a finite-size code, connecting laboratory data to a formula originally motivated by quantum gravity.
  • Unique syndrome tables and a code distance of three mean the implemented pentagon instance can serve as a small geometric quantum error-correcting memory.
  • The quasi-transversal Hadamard with Pauli-frame correction is a concrete fault-tolerance primitive; extending it to a full set of logical gates would be the next step toward holographic quantum computation.

Where Pith is reading between the lines

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

  • Editorial inference: if the graph-code equivalence holds, the same extraction method should scale to larger hyperbolic tilings; a natural test is to verify that the measured boundary entropy remains linear in the cut length as the tiling depth increases.
  • Editorial inference: the burst-error discussion suggests holographic codes may outperform block codes on spatially correlated noise; a direct experiment injecting adjacent two- and three-qubit errors and comparing with the constituent seven-qubit code would test this advantage.
  • Editorial inference: the quasi-transversal mechanism may generalize to a design rule—transversal in the constituent block, correctable in the glued code—that could guide the construction of fault-tolerant gate sets for other holographic or concatenated codes.
  • Editorial inference: the corrections here are applied in post-processing via Pauli-frame updates; implementing them in real time would test whether holographic codes can support active, hardware-level error correction.

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 / 5 minor

Summary. The paper reports trapped-ion experiments on small instances of holographic pentagon and heptagon codes. The pentagon-code experiment prepares graph-code states, measures stabilizers and logical operators, detects single-qubit errors, performs partial decoding of two, three, and four bulk qubits from boundary regions, and measures tomographic entropies that are compared with the Ryu-Takayanagi formula. The heptagon-code experiment initializes a two-logical-qubit code, applies logical operations, and demonstrates a quasi-transversal Hadamard gate whose induced error is corrected by a Pauli-frame update. All data are compared with a depolarizing-noise simulation.

Significance. If the claims are substantiated, this would be one of the first laboratory implementations of holographic quantum codes, with partial bulk-to-boundary decoding, an RT-type entropy law, and a quasi-transversal logical Hadamard gate. The paper benefits from using an external trapped-ion hardware benchmark and from grounding the code construction in the published graph-code framework of Ref. [23]. The most valuable evidence is the state-preparation stabilizer data, which lie above the mixed-state threshold, and the syndrome look-up table. However, the two headline results—'recovering logical bulk qubits' and 'testing the RT area law'—are not yet quantitatively supported, as detailed below.

major comments (3)
  1. [Entropy measurement (Pentagon code), Eq. (4)] The RT-law claim is not quantitatively supported. The measured entropies are 3.96(2) vs ideal 3 (cut 1), 4.13(2) vs ideal 4 (cut 2), and 3.818(10) vs ideal 4 (cut 3). These are deviations of +0.96, +0.13, and -0.18 bits, many standard deviations from the prediction. The text explains cut 1 by a 'finite upper bound in entropy' and cut 3 by residual pure-state contributions, but no noisy simulation of the entropy is provided; Supp. SX compares tomography only to a noise-free simulation. Since Eq. (4) is the central prediction being tested, a quantitative error budget is needed before 'testing the RT area law' can be claimed.
  2. [Partial decoding, §II.A and Supp. SIX] The decoded-qubit expectation values are far below the ideal value of +1 and in several cases disagree with the depolarizing simulation at many standard deviations. For cut 1, A=0.616(15) vs sim 0.436(16) and B=0.296(14) vs sim 0.422(16); for cut 3, B=0.142(11) vs sim 0.103(9). A logical Z-expectation near 0.3 means the recovered state is nearly random in that basis. The statement that the decoding 'reliably recovers the encoded logical qubits' is therefore not supported by the data as presented. Please provide a fidelity or process metric and address the simulation mismatch.
  3. [Fig. 3 caption and Supp. SVIII] The holographic interpretation relies on the claim that the graph code of Fig. 2(b) is, up to local Clifford operations, equivalent to the directly contracted tensor network. The paper does not give the explicit Clifford transformation or show that it preserves the partial-recovery regions and the RT formula for the cuts used. Since the decoding circuits are built from the graph code, this equivalence is load-bearing for the holographic claims. Please provide a derivation or a precise reference to the proof.
minor comments (5)
  1. [Heptagon code, §II.B] Typo: 'sate preparation' should be 'state preparation'.
  2. [Fig. 5(c) caption] Typo: 'the the graph code' should be 'the graph code'.
  3. [Supplementary material] The numbering is inconsistent: 'Supp. Mat. SIX' is used for the decoding expectation values, while 'Supp. Mat. SX' is used for the tomography. Please renumber for clarity.
  4. [Fig. 6 caption] The caption labels the induced gate as 'RXX 6,11' while the text says the MS gate is applied between qubits 6 and 11. Please make the notation consistent.
  5. [Data availability] The data availability statement says data can be made available on request. Given the complexity of the tomography and decoding data, a permanent public repository would improve reproducibility.

Circularity Check

0 steps flagged

No significant circularity: the experimental claims are benchmarked against external hardware and prior published code constructions; no measured quantity is fitted and repackaged as a prediction.

full rationale

The paper's derivation chain is: (i) take the HaPPY pentagon code [9] and heptagon CSS code [12], (ii) use the graph-code representation of [23] (which the paper states is Clifford-equivalent to the tensor-network contraction) to construct preparation and decoding circuits, (iii) measure stabilizers, logical operators, decoded qubit expectation values, and reduced-state entropies on a trapped-ion processor, and (iv) compare to a depolarizing-noise simulation with error rates p1=0.005, p2=0.025 taken from prior hardware characterization [28]. No step fits a free parameter to the target observable and then reports it as a prediction. The partial-recovery circuits are derived from the stabilizer/graph structure, not from the measured data; the RT 'ideal' values |gamma| are counts of contracted indices, not fitted constants. The observed deviations from the RT prediction (e.g., 3.96(2) vs 3) are a quantitative verification issue, not circularity. The main caveat is that the holographic interpretation relies on the Clifford-equivalence assertion from [23], which is co-authored by current authors; however, that assertion is a mathematical tool with stated assumptions independent of the present measurements, so it is prior evidence rather than a circular premise. No equation reduces to its input by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

All load-bearing content is either standard stabilizer/graph-state theory, prior constructions from the same group ([12,23]), or the hardware-specific depolarizing model. No new physical entities are postulated. The only numerical inputs are the simulator error rates, which are hardware parameters measured previously rather than fit to the presented data.

free parameters (2)
  • single-qubit depolarizing error rate p1 = 0.005
    Used in the simulator to produce all 'sim' expectation values. Taken from prior hardware characterization [28], not fitted to this experiment's data, but the agreement between sim and experiment is central to the paper's validation strategy.
  • two-qubit depolarizing error rate p2 = 0.025
    Same as above; the two-qubit gate error rate in the depolarizing simulation.
axioms (5)
  • standard math Stabilizer formalism and graph-state code basis
    Section I: code subspace defined by stabilizer generators; Eq. (1) graph states; logical operators derived from graph.
  • domain assumption HaPPY pentagon/AME tensor-network code satisfies partial-recovery and RT formula for the chosen cuts
    Section I and Fig. 1: the RT formula Eq. (4) and the partial-decoding property Eq. (3) are taken from [9,27]; the experiment assumes these hold for the small 12-qubit instance.
  • domain assumption Graph code of [23] is equivalent up to local Clifford to the tensor-network holographic code
    Fig. 3 caption and Supp SVIII: the decoding circuits are derived from the graph code of [23] and assumed to reproduce the holographic partial-decoding and entropy properties of the direct tensor-network contraction.
  • ad hoc to paper Depolarizing noise model captures experimental errors
    Supp SIII: all simulations use depolarizing channels with p1,p2; the paper uses agreement with this model as evidence of correct implementation, but the model ignores correlated errors, leakage, and calibration drift.
  • domain assumption Pauli-frame update suffices to correct the stabilizer sign flip induced by the logical Hadamard
    Eq. (7) and Supp SXI: post-processing correction is applied assuming the induced error is a single Pauli on one logical qubit.

pith-pipeline@v1.3.0-alltime-deepseek · 27308 in / 12632 out tokens · 130761 ms · 2026-08-01T20:45:33.957524+00:00 · methodology

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Cite this review

Pith. "Pith review of Holographic quantum codes with trapped ions." pith.science (2026). https://pith.science/paper/MAVCQVXM

@misc{pith2026260716503,
  author       = {Pith},
  title        = {Pith review of: Holographic quantum codes with trapped ions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MAVCQVXM}},
  note         = {Machine review of arXiv:2607.16503}
}
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read the original abstract

Holography is a central concept at the intersection of gravity, condensed matter theory, and quantum information, linking the interior bulk of a system to its boundary. A model realizing key features of holographic systems is the holographic pentagon code by Pastawski et al. Here we experimentally implement instances of the holographic pentagon and heptagon codes with trapped ions and test their properties: For the pentagon code, we recover logical bulk qubits from their nearby boundary and test the Ryu-Takayanagi entanglement area law. For the heptagon code, we show that the transversal Hadamard gate native to the constituent Steane codes induces a single-qubit, correctable error in the holographic code. Our implementation paves the way towards the use of holographic quantum codes for quantum information processing.

Figures

Figures reproduced from arXiv: 2607.16503 by Alex Steiner, Felix Huber, Gavin Brennen, Gerard Angl\`es Munn\'e, Ivan Pogorelov, Martin Ringbauer, Michael Meth, Rainer Blatt, Robert Freund, Robert J. Harris, Thomas Monz, Thomas M. Stace.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

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