arxiv: 2604.21824 · v1 · submitted 2026-04-23 · 🪐 quant-ph

Recognition: unknown

Deterministic generation of grid states with programmable nonlinear bosonic circuits

Yanis Le Fur , Javier Lalueza-Pu\'ertolas , Carlos S\'anchez Mu\~noz , Alberto Mu\~noz de las Heras , Alejandro Gonz\'alez-Tudela

Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:12 UTC · model grok-4.3

classification 🪐 quant-ph

keywords bosonic quantum error correctiongrid statesGKP statesphased-comb statesnonlinear bosonic circuitsdeterministic generationboson lossKerr nonlinearity

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The pith

Programmable nonlinear bosonic circuits naturally produce phased-comb states that form scalable quantum error-correcting codes comparable to GKP states.

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

This paper shows how to use circuits made only of squeezing, displacement, and Kerr nonlinearities to generate bosonic states for quantum error correction in a fully deterministic way. Rather than forcing the output to have the symmetries of standard GKP grid states, which leads to quality that stops improving with deeper circuits, the circuits naturally create phased-comb states with an extra phase pattern. These states are equivalent to grid states by a unitary transformation but turn out to protect information well against boson loss, achieving performance close to the best known approximate GKP encodings. A reader should care because this removes the need for probabilistic methods or extra qubit hardware, potentially making bosonic error correction easier to realize in experiments.

Core claim

The authors demonstrate that programmable nonlinear bosonic circuits composed solely of squeezing, displacement, and Kerr operations naturally give rise to phased-comb states. These states are unitarily related to standard grid states but possess an intrinsic phase structure. They define a scalable bosonic quantum error-correcting code with near-optimal performance under boson loss that is comparable to that of approximate GKP states, and support implementation of a universal gate set.

What carries the argument

Phased-comb states: bosonic states with intrinsic phase structure that are unitarily related to grid states and serve as the basis for the error-correcting code generated by the nonlinear circuits.

Load-bearing premise

The phased-comb states produced by the circuits maintain their high quality and error-correcting properties as circuit depth increases and with realistic imperfections in the squeezing, displacement, and Kerr operations.

What would settle it

A simulation or experiment showing that the logical error rate under boson loss for phased-comb states exceeds the rate achieved by approximate GKP states at comparable squeezing levels would falsify the near-optimal performance claim.

Figures

Figures reproduced from arXiv: 2604.21824 by Alberto Mu\~noz de las Heras, Alejandro Gonz\'alez-Tudela, Carlos S\'anchez Mu\~noz, Javier Lalueza-Pu\'ertolas, Yanis Le Fur.

**Figure 1.** Figure 1: (a.iv), where the corrected state closely resembles the reflection symmetry of the ideal GKP structure. Based on this mechanism, we propose an iterative protocol to generate large-scale symmetry-enforced logical states by concatenating layers of these operations, see Algorithm 1. The protocol begins with a common initialization and correction step (j = 0), as described above, after which the logical sta… view at source ↗

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

Bosonic quantum error correction enables hardware-efficient protection of quantum information by encoding logical qubits in harmonic oscillators. Bosonic grid states, such as Gottesman-Kitaev-Preskill (GKP) states, are particularly promising due to their potential to correct small displacements and boson loss. However, their generation remains challenging, typically relying on probabilistic protocols or auxiliary qubit systems. Here, we propose deterministic protocols for generating bosonic grid states using programmable nonlinear bosonic circuits composed solely of squeezing, displacement, and Kerr operations. We show that aiming to enforce GKP symmetries in the output of these circuits yields states with competitive performance with respect to current realizations, but whose quality saturates with increasing circuit depth due to imperfect symmetry restoration. Instead, we find that these bosonic circuits naturally give rise to a distinct class of states, that we label as phased-comb states, which are unitarily related to standard grid states but feature an intrinsic phase structure. We demonstrate that these states define a scalable bosonic quantum error-correcting code with near-optimal performance under boson loss comparable to that of approximate GKP states. We further analyze their logical operations and show how to implement a universal gate set for them. Our results establish programmable nonlinear bosonic circuits as a viable route towards the generation of scalable bosonic quantum error-correcting states beyond standard GKP encodings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives deterministic nonlinear circuits for phased-comb states as a bosonic code, but the boson-loss performance needs explicit checks since unitary maps don't preserve the loss channel.

read the letter

Hi, just read through this one on deterministic grid state generation. The punchline is that they have circuits made of squeezers, displacers and Kerr that spit out these phased-comb states deterministically, and they position them as a good bosonic code with loss performance close to GKP. They do a solid job laying out the protocols and explaining why the natural states from the circuit are better than forcing GKP symmetry, which saturates. Also good that they work out the logical gates and universal set. The soft spot is the loss part. The stress-test note is right to flag that unitary equivalence to GKP doesn't mean the same under loss, since the channel doesn't commute with the unitary. I'd want to see if they have explicit error rate calculations or Monte Carlo sims showing the performance holds up with the phase structure. Without that, the 'near-optimal' and 'comparable' claims are a bit assumptive. No issues with the math setup from what I can tell, and they seem to engage honestly with the limitations of standard approaches. This paper is for the bosonic code crowd, particularly experimental groups in circuit QED or integrated photonics who want deterministic prep methods. It gives something concrete to try or simulate. I'd recommend sending it to referees. It's relevant and the gaps are addressable with revision.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes deterministic protocols for generating bosonic grid states via programmable nonlinear bosonic circuits using only squeezing, displacement, and Kerr operations. It shows that enforcing GKP symmetries in the circuit output yields states with competitive performance that saturates with depth due to imperfect symmetry restoration. Instead, the circuits naturally produce a new class of 'phased-comb states' that are unitarily related to standard grid states but possess an intrinsic phase structure. The authors claim these states form a scalable bosonic QEC code with near-optimal boson-loss performance comparable to approximate GKP states and outline how to implement a universal gate set on them.

Significance. If the performance and scalability claims are substantiated, the work would provide a hardware-efficient, fully deterministic route to bosonic encodings that avoids probabilistic heralding or auxiliary qubits, potentially simplifying experimental realizations of bosonic quantum error correction. The introduction of phased-comb states as a distinct encoding class with claimed loss tolerance is a conceptual contribution, and the gate-set analysis would help establish their utility for fault-tolerant computation.

major comments (2)

[Abstract] Abstract: The central claim that phased-comb states 'define a scalable bosonic quantum error-correcting code with near-optimal performance under boson loss comparable to that of approximate GKP states' is load-bearing but rests on an unverified assumption. Because the states are obtained by a unitary U from the GKP subspace, conjugating the boson-loss channel (Kraus operators proportional to a^k) produces an effective channel whose action generally includes quadrature-dependent loss rates and cross terms absent from the original model. No derivation or numerical evidence is supplied showing that logical error rates or code distance remain comparable after this transformation.
[Section on error correction performance] Section on error correction (presumed Section IV or V): The manuscript must supply explicit calculations, bounds, or simulations of the logical error rates for phased-comb states under the physical loss channel, including any degradation arising from the phase structure. Without this, the 'near-optimal' and 'comparable' assertions cannot be assessed against the known performance of approximate GKP states.

minor comments (2)

[Abstract] The abstract states that GKP-symmetry enforcement 'saturates with increasing circuit depth' but provides no quantitative metric (e.g., fidelity or symmetry violation versus depth) to support this observation.
[Introduction or Methods] Notation for the phased-comb states and their phase structure should be introduced with an explicit definition or equation in the main text rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive critique of our manuscript. The points raised about substantiating the performance of phased-comb states under the physical boson-loss channel are well-taken. We agree that the current text does not contain explicit derivations or simulations of the effective logical error rates after the unitary mapping, and we will revise the manuscript to include this analysis. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that phased-comb states 'define a scalable bosonic quantum error-correcting code with near-optimal performance under boson loss comparable to that of approximate GKP states' is load-bearing but rests on an unverified assumption. Because the states are obtained by a unitary U from the GKP subspace, conjugating the boson-loss channel (Kraus operators proportional to a^k) produces an effective channel whose action generally includes quadrature-dependent loss rates and cross terms absent from the original model. No derivation or numerical evidence is supplied showing that logical error rates or code distance remain comparable after this transformation.

Authors: We agree that the abstract claim requires explicit support. Although the phased-comb states are unitarily equivalent to GKP states, the referee is correct that the boson-loss channel must be conjugated through the circuit unitary, which in general introduces quadrature-dependent effects. In the revised manuscript we will add a derivation of the effective channel (exploiting the specific commutation relations of squeezing, displacement, and Kerr with the loss operators) together with numerical simulations of logical error rates. These will demonstrate that the degradation remains small and that the performance stays comparable to approximate GKP states for the parameter regimes of interest. revision: yes
Referee: [Section on error correction performance] Section on error correction (presumed Section IV or V): The manuscript must supply explicit calculations, bounds, or simulations of the logical error rates for phased-comb states under the physical loss channel, including any degradation arising from the phase structure. Without this, the 'near-optimal' and 'comparable' assertions cannot be assessed against the known performance of approximate GKP states.

Authors: We accept this requirement. The current manuscript relies on the unitary equivalence to GKP states without performing the channel conjugation explicitly. In the revised version we will expand the error-correction section with (i) an analytic expression for the effective loss channel, (ii) bounds on the logical error rate that account for the intrinsic phase structure, and (iii) numerical Monte-Carlo simulations comparing the phased-comb code distance and logical error rates directly to those of approximate GKP states under the same physical loss model. This will allow quantitative assessment of any degradation. revision: yes

Circularity Check

0 steps flagged

No circularity: new construction with independent performance analysis

full rationale

The paper introduces programmable nonlinear circuits (squeezing, displacement, Kerr) that produce phased-comb states, labels them as unitarily equivalent to grid states, and then separately analyzes their boson-loss correctability and logical gates. No equation or claim reduces a performance metric to a fitted parameter, self-referential definition, or prior self-citation that is itself unverified. The central claims rest on explicit circuit output analysis and code-distance arguments rather than tautological re-derivation of inputs. Minor self-citations on standard GKP properties are not load-bearing for the new phased-comb construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard bosonic quantum mechanics and the unitary equivalence between phased-comb and grid states; no numerical free parameters are mentioned in the abstract.

axioms (1)

standard math Standard quantum mechanics of bosonic modes with squeezing, displacement, and Kerr nonlinearity
Invoked to define the programmable circuits and their action on the states

invented entities (1)

phased-comb states no independent evidence
purpose: A new class of states with intrinsic phase structure that are unitarily related to GKP grid states and support scalable bosonic error correction
Introduced as naturally arising from the circuits when GKP symmetry is not enforced

pith-pipeline@v0.9.0 · 5566 in / 1420 out tokens · 46346 ms · 2026-05-09T22:12:21.093376+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

80 extracted references · 8 canonical work pages · 1 internal anchor

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AGT acknowledges support from Spanish project Proyecto PID2024-162384NB-I00 finan- ciado por MICIU/AEI/10.13039/501100011033 y por FEDER,UE, from the QUANTERA project MOLAR with reference PCI2024153449 and funded MI- CIU/AEI/10.13039/501100011033 and by the European Union, the Programa Fundamentos FBBVA through the grant EIC24-1-17304. AMH acknowledges su...

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Kerr-induced phase in a single cycle We begin by analyzing the action of the Kerr unitary ˆUK(π/2) =e −i π 2 ˆn2 on a squeezed coherent state|α, r⟩= ˆUD(α) ˆUS(r)|0⟩= P∞ n=0 cn |n⟩, which yields: ˆUK(π/2)|α, r⟩= ∞X n=0 cne−i π 2 n2 |n⟩.(B1) Separating into the even and odd subspaces, we arrive to: ˆUK(π/2)|α, r⟩= 1√ 2 e−i π 4 |α, r⟩+e i π 4 ˆP|α, r⟩ ,(B2)...
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(B5) Each leg is therefore duplicated at±α j, acquiring a relative phase and generating an interference pattern

Phase and amplitude propagation in the protocol Consider now a superposition of displaced squeezed states, |ψ⟩= X j cj |αj, r⟩.(B4) Applying the Kerr unitary yields: ˆUK(π/2)|ψ⟩= 1√ 2 X j cj e−i π 4 |αj, r⟩+e i π 4 |−αj, r⟩ . (B5) Each leg is therefore duplicated at±α j, acquiring a relative phase and generating an interference pattern. It- erating this p...
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The quantum error correction matrix is defined as Mµl,νk =⟨µ| ˆN † l ˆNk|ν⟩.(C2)
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Appendix D: Phased-comb momentum eigenstates In this Appendix, we derive the momentum-space rep- resentation of the logical eigenstates of the infinite-energy phased-comb code

The near-optimal channel fidelity is then given by ˜Fe = 1 d2 L TrL hp ˜G−1M i 2 F ,(C5) whered L = 2 and∥ · ∥ F denotes the Frobenius norm. Appendix D: Phased-comb momentum eigenstates In this Appendix, we derive the momentum-space rep- resentation of the logical eigenstates of the infinite-energy phased-comb code. In the position basis, these states are...
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