Fault-tolerant bosonic quantum error correction with the surface-GKP code

Christopher Chamberland; Kyungjoo Noh

REVIEW 2 major objections 5 minor 82 references

Fault-tolerant bosonic quantum error correction with the surface-GKP code

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Fault-tolerant quantum error correction is possible with the surface-GKP code if GKP squeezing exceeds 11.2 dB.

desk verdict First full circuit-level simulation of surface-GKP code with concrete thresholds; the thresholds are solid within the stated twirled Gaussian noise model, but the 'conservative' mapping to that model is asserted, not proven. read the letter →

arxiv 1908.03579 v2 pith:BWYTSOCY submitted 2019-08-09 quant-ph

Kyungjoo Noh , Christopher Chamberland This is my paper · ORCID

classification quant-ph MSC 81P7081P68 PACS 03.67.Pp

keywords surface-GKPcodeGKPfault-tolerantquantumerrorcorrectionsurfacecontinuous-variablecomputingbosoniccodesminimum-weightperfectmatchingthresholds

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 a concrete noise budget for building a fault-tolerant quantum computer out of bosonic modes: it concatenates the GKP code, which stores a qubit in the continuous position and momentum variables of an oscillator, with the surface code, and simulates the full error-correction circuit under realistic loss and finite-squeezing noise. The result is a set of thresholds: fault tolerance is possible when GKP squeezing exceeds 11.2 dB if GKP states are the only noisy elements, when each circuit element fails below 0.81% if GKP states are ideal, and when both are comparably noisy at 18.6 dB squeezing with 0.69% component failure. The mechanism that carries the argument is a decoding scheme that uses the analog information from GKP-stabilizer measurements to renormalize edge weights in the 3D matching graph fed to a minimum-weight perfect matching decoder. These thresholds matter because they tell experimental groups how good GKP states and two-mode gates have to be for a scalable continuous-variable architecture.

What carries the argument

The central object is the concatenated code: each data mode, ancilla mode, and syndrome mode of a rotated surface code is itself a GKP-encoded qubit, with GKP stabilizers measured by SUM and inverse-SUM gates followed by homodyne detection. The load-bearing simplification is the noise model: finite-squeezing GKP states and lossy gates are represented as incoherent Gaussian random displacement channels with variances $\sigma_{\mathrm{gkp}}^2$ and $\sigma^2$, with covariance matrices derived from a Lindblad evolution that adds heating to convert photon loss into displacement noise. The decoding machinery is a 3D space-time graph whose horizontal and vertical edge weights are renormalized using the analog position and momentum outcomes of the GKP-stabilizer measurements, so that a minimum-weight perfect matching decoder can exploit soft information about how close a measured shift was to the decision boundary where a Pauli error becomes likely.

What would settle it

Run the distance-3 and distance-5 surface-GKP error-correction circuits under the paper's own noise model at $\sigma_{\mathrm{gkp}} = 0.19$ with $\sigma = 0$, and check whether the logical X-error rate drops as the code distance increases; the paper predicts a clear drop, so observing a flat or rising rate would falsify the Case I threshold claim. An experiment with GKP squeezing near 12 dB and otherwise ideal components that shows no error suppression with code distance would contradict the claim directly.

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Extended reading notes

Core claim

The central claim is that the concatenated surface-GKP code is fault tolerant below the reported thresholds: GKP squeezing above 11.2 dB when only the GKP states are noisy, component failure probability below 0.81% when GKP states are noiseless, and 18.6 dB squeezing with 0.69% component failure when both are comparably noisy. The paper further claims that, below threshold, the logical X or Z error rate decreases as the code distance increases, and that the decoder's use of continuous GKP-stabilizer information is essential, since ignoring it lowers the thresholds and raises logical error rates by one to several orders of magnitude. The paper also derives simple effective noise variances, such as $\sqrt{5\sigma_{\mathrm{gkp}}^2 + (59/3)\sigma^2}$ for the probability of a Pauli error after a GKP-stabilizer measurement, showing that these probabilities decay exponentially as the noise parameters approach zero.

Load-bearing premise

Everything rests on modelling realistic finite-squeezing GKP states, photon loss, and gate noise as incoherent Gaussian random displacement channels with known variances; if actual devices produce non-Gaussian or correlated noise, the numerical thresholds may not transfer.

Editorial extensions

If this is right

Below the thresholds, larger surface-GKP codes suppress logical error rates with increasing code distance, so the architecture is scalable without post-selection.
The decoder works without 3D space-time correlated edges, simplifying the matching-graph construction relative to optimized bare-qubit surface-code decoders.
Circuit-element failure thresholds around 0.7 to 0.8 percent are comparable to the roughly one-percent threshold of the rotated surface code with depolarizing noise, suggesting that bosonic encoding does not inherently require an order-of-magnitude worse noise tolerance.
Because the GKP inner code converts small shift errors into exponentially suppressed Pauli-error probabilities, a small-distance surface code may reach a target logical error rate with fewer total modes than a bare surface code in the low-noise regime.

Reading between the lines

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

The twirling step that justifies replacing coherent finite-squeezing noise with an incoherent displacement channel is acknowledged to be impractical because it raises the average photon number; a natural follow-up is to repeat the threshold simulation with coherent, non-twirled GKP states and check how much the 11.2 dB and 18.6 dB values move.
If SUM-gate implementations introduce Kerr nonlinearities, the assumed Gaussian displacement noise model breaks down, so the thresholds should be re-evaluated for any specific gate hardware that is not well approximated by the loss-plus-heating Lindblad model.
The renormalized-edge-weight technique is not obviously limited to GKP codes and could plausibly transfer to other analog bosonic codes, such as cat or binomial codes, when concatenated with a topological code, provided their stabilizer measurements supply similar soft error information.
The comparison with measurement-based GKP schemes suggests a systems-level trade-off: deterministic operation costs higher squeezing thresholds than post-selection-based approaches, which is relevant when choosing between gate-based and measurement-based architectures.

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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

Two things you should know about this paper. First, it is the first full circuit-level simulation of the surface-GKP code, and it gives concrete thresholds: 11.2 dB squeezing when GKP states are the only noise, 0.81% circuit-element failure when GKP states are ideal, and 18.6 dB / 0.69% when both are comparable. Second, those thresholds are computed for a twirled, incoherent Gaussian displacement noise model, and the paper itself admits the twirling operation is not practically implementable. That does not kill the result, but it does mean the thresholds are conditional on a model, not on a fully realistic device.

What is genuinely new: the detailed circuit-level treatment. Prior work on concatenated GKP codes used code-capacity or phenomenological noise models, manually injecting shift errors. Here, Appendix B spells out every noise update step from GKP state preparation through SUM gates, idling, homodyne measurement, and surface-code stabilizer extraction. The renormalized edge-weight method, which uses the analog GKP measurement information, is clearly explained and shown to improve the threshold substantially over ignoring it. The circuit scheduling with inverse-SUM gates to cancel propagated shift errors is also a nice, concrete contribution. The paper is careful about comparisons with earlier measurement-based schemes that rely on post-selection, and it is honest about the practical obstacles, such as Kerr nonlinearities in the SUM gate implementation.

Soft spots, in proportion. The biggest one is the claim that replacing coherent finite-squeezing states and lossy gates by incoherent Gaussian displacements is "conservative." For the GKP state, the twirling argument in Appendix A averages over stabilizer shifts; that is a standard and plausible way to make the analysis tractable, but it is not a proven monotonicity result. A coherent superposition of displacements can behave differently under repeated correction than the twirled mixture, especially with a MWPM decoder that uses soft information. Similarly, adding heating to the loss channel may be pessimistic, but the paper does not show that the logical error rate is monotonic under that change. The stress-test note captures this well, and I think it is the right concern to raise. But it is a limitation, not a fatal flaw: the thresholds are explicitly thresholds for the stated noise model, and the model is stated very clearly. The paper does not overclaim beyond that, aside from the word "conservative."

Other issues are minor by comparison. No code or data are released, and the figures have no error bars, though they report 10,000–100,000 samples per point. The simulation seems reproducible in principle given the level of detail in Appendix B, but a release would be better.

Who this is for: anyone working on GKP codes, bosonic QEC, or concatenation with topological codes. It is a useful reference point for experimental groups chasing squeezing thresholds and for theorists studying decoder design. It deserves a serious referee: I would send it to peer review and lean toward accepting after asking the authors to (1) release code/data, and (2) make the conservative-mapping caveat more prominent, ideally with a proof or at least a cleaner statement of what would be needed to prove it.

Referee Report

2 major / 5 minor

Summary. The manuscript studies fault-tolerant quantum error correction with a concatenated surface-GKP code under a circuit-level noise model. Finite-squeezing GKP states are modeled as ideal GKP states subject to the Gaussian random displacement channel N[σ_gkp] (Eq. (16)), and noisy SUM/inverse-SUM gates are modeled, via a Lindblad equation with loss and added heating (Eqs. (17)-(18)), as ideal gates followed by correlated Gaussian displacements with variance σ² = κ/g (Eq. (19)). The full syndrome extraction circuit, alternating GKP stabilizer measurements with four-step surface-code stabilizer measurements, is simulated for increasing code distances and decoded by minimum-weight perfect matching on 3D space-time graphs whose edge weights are renormalized using the conditional error probabilities p[σ](z) derived from GKP stabilizer measurements (Appendix B). The paper reports three thresholds: σ*_gkp = 0.194, corresponding to 11.2 dB squeezing, when GKP states are the only noisy components; σ* = 0.09, corresponding to κ/g = 0.81%, when GKP states are ideal; and σ* = σ*_gkp = 0.083, corresponding to 18.6 dB and κ/g = 0.69%, when both are comparably noisy. Below these thresholds the logical X/Z error rate is suppressed with increasing code distance d. The results are compared with earlier toric-GKP and measurement-based GKP computations, and the authors argue that their scheme is deterministic (no post-selection) and scalable.

Significance. If the thresholds are taken at face value, the paper is a significant advance in the practical assessment of bosonic quantum error correction: it provides, for this concatenated setting, a full circuit-level simulation in which noise propagation through every gate is tracked, a decoder that exploits the analog syndrome information from GKP stabilizer measurements, and concrete falsifiable predictions (the three threshold values and the exponential scaling of perr(σ) in Eqs. (20)-(21)) that connect state-preparation squeezing to a fault-tolerance threshold. Strengths of the manuscript are the exceptionally detailed simulation protocol (Appendix B gives explicit stochastic update equations for every noise step and the complete MWPM decoding procedure), the careful scheduling of SUM and inverse-SUM gates to avoid correlated-noise amplification (Fig. 5), and the transparent comparison with prior toric-GKP and measurement-based results. The central claims are conditional on the noise model, and the 'conservative' reduction of physical finite-squeezing and loss noise to Gaussian displacement channels is asserted rather than proven; this is the main risk.

major comments (2)

[II C and Appendix A] The thresholds reported in Section III are computed for the model of Eq. (16), in which every finite-squeezing GKP state is replaced by an ideal state subject to the incoherent Gaussian displacement channel N[σ_gkp], and Section II C asserts that this replacement is 'conservative' compared to the coherent superposition of displacements in Eq. (15). The twirling derivation in Eqs. (A2)-(A4) shows that the incoherent model is obtained from the coherent state by averaging over GKP stabilizer shifts and dropping the (k1,k2)≠(0,0) terms, but it does not establish that the decoded logical error rate is larger for the incoherent model: the logical error probability after repeated rounds of measurement and feedback is not a convex function of the input state, and coherences between distinct displacement components within a peak, which are destroyed by the twirl, could in principle affect subsequent rounds. Because the abstract's headline claim ('fault-tolerant quantum error correction is possible ... if the squeezing ... is higher than 11.2 dB') is phrased in terms of physical squeezing, the identification of the physical state with N[σ_gkp] is load-bearing. Please either prove the monotonicity, provide numerical evidence by simulating the coherent states of Eq. (15) directly for small d and comparing with the N[σ_gkp] model, or explicitly qualify the abstract and Section III as applying to the twirled noise model.
[II C, Eqs. (17)-(19)] The gate error model in Eq. (18) replaces the loss-only Lindbladian L± of Eq. (17) by L'± = L± + κ(D[a1†] + D[a2†]), which doubles the added displacement variance relative to pure photon loss and removes the attenuation; the resulting correlated Gaussian displacement model (Eq. (19)) is directionally pessimistic, but the paper gives no monotonicity argument for the final logical error rate under this substitution. Since the MWPM edge weights in Eqs. (B22)-(B33) are themselves computed from the same noise parameters σ and σ_gkp, changing the noise model also changes the decoder, so a monotone relation between the physical loss rate and the simulated logical error rate cannot simply be assumed. The authors should either provide such an argument, or state explicitly that the thresholds (κ/g)* = 0.81% and 0.69% are for the diffusion-type gate model and may not match a pure-loss implementation.

minor comments (5)

[III, Fig. 6] The caption does not state which symbol shape corresponds to which code distance d, nor the set of distances used in each panel; listing the distances (apparently d = 3, 5, 7, and possibly 9) would make the threshold crossings checkable.
[III] The threshold values (σ*_gkp = 0.194, σ* = 0.09, and σ* = 0.083) are quoted to three significant figures, but Fig. 6 shows no error bars or confidence intervals; Appendix B only reports that 10,000-100,000 samples were used, so please quantify the statistical precision of the extracted thresholds.
[Appendix B, MWPM decoding step 3] The decoding procedure says to 'find the path with the minimum total weight' for all pairs of highlighted vertices but does not state which shortest-path algorithm is used; a one-sentence statement (e.g., Dijkstra on the spatio-temporal graph) would aid reproducibility.
[II A, after Eq. (10)] There is a typo in the sentence 'for some intergern' which should read 'for some integer n.'
[II C] The paper defines separate noise parameters σ_p, σ_c, σ_m for preparation, gates, and measurement but simulates only the case σ_p = σ_c = σ_m ≡ σ; a sentence on how unequal parameters would be handled, or why equal-time noise is the relevant regime, would clarify the sense in which the noise model is general.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fault-tolerance thresholds are outputs of a Monte Carlo simulation over a stated circuit-level noise model, with no fitted parameter renamed as a prediction and no load-bearing self-citation.

full rationale

The derivation chain is self-contained. The approximate GKP state is expanded in Eq. (15), then converted to the incoherent displacement channel N[sigma_gkp] in Eq. (16) via the twirling calculation in Appendix A; the gate noise model in Eqs. (17)-(19) is derived from the Lindbladian with added heating. These are model assumptions, not circular definitions. The thresholds in Section III are obtained by simulating the full noisy syndrome-extraction circuit and applying MWPM to 3D space-time graphs; the edge weights in Appendix B use the same noise variances that generate the data, which is standard matched decoding and does not fit any parameter to the logical error rates being predicted. The analytical error probabilities in Eqs. (20)-(21) are derived from noise propagation and are used as decoder weights, not as the logical error rates that determine the threshold. The only self-citations (Refs. [5, 50, 67]) are background or technique citations; the loss-to-displacement conversion is re-derived in Appendix A, so the central claim does not reduce to a self-citation. The paper itself flags the twirling operation as impractical and notes possible Kerr limitations (Section IV), but those are caveats about model realism, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, or physical entities. It relies on the existing GKP code and surface code. The only new elements are the noise model choices and the edge-weight computation method.

assumptions (5)

domain assumption Finite-squeezing GKP states can be modeled by an incoherent Gaussian displacement channel N[σ_gkp] with σ_gkp^2 = (1 - e^-Δ)/(1 + e^-Δ).
This relies on the twirling argument in Appendix A, which replaces coherent superpositions of displacements by an incoherent mixture. The paper explicitly notes this is not practical to implement and that the incoherent model is more noisy, hence conservative.
domain assumption Photon loss and heating during SUM/inverse-SUM gates can be converted into a correlated Gaussian displacement error after an ideal gate, via the Lindblad model in Eqs. (17)-(19).
The heating term D[a^†] is added to make the error Gaussian; this is conservative but not a physically exact model of a real two-mode gate.
domain assumption The displacement operations used for error correction are noiseless and can be tracked in a Pauli frame.
Section II C states displacement operations need not be physically implemented; this assumes perfect software tracking.
domain assumption The decoder has perfect knowledge of the noise parameters σ_gkp and σ and uses them in the renormalized edge weight formulas.
The edge weights in Appendix B are functions of the known noise variances; in practice these would need to be calibrated.
standard math Standard quantum mechanics and Gaussian quantum information tools (CPTP maps, Trotter's formula, Poisson summation) are used.
Background mathematical facts invoked in Appendix A and the main text.

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Cite this review

Pith. "Pith review of Fault-tolerant bosonic quantum error correction with the surface-GKP code." pith.science (2026). https://pith.science/paper/BWYTSOCY

@misc{pith2026190803579,
  author       = {Pith},
  title        = {Pith review of: Fault-tolerant bosonic quantum error correction with the surface-GKP code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWYTSOCY}},
  note         = {Machine review of arXiv:1908.03579}
}

read the original abstract

Bosonic quantum error correction is a viable option for realizing error-corrected quantum information processing in continuous-variable bosonic systems. Various single-mode bosonic quantum error-correcting codes such as cat, binomial, and GKP codes have been implemented experimentally in circuit QED and trapped ion systems. Moreover, there have been many theoretical proposals to scale up such single-mode bosonic codes to realize large-scale fault-tolerant quantum computation. Here, we consider the concatenation of the single-mode GKP code with the surface code, namely, the surface-GKP code. In particular, we thoroughly investigate the performance of the surface-GKP code by assuming realistic GKP states with a finite squeezing and noisy circuit elements due to photon losses. By using a minimum-weight perfect matching decoding algorithm on a 3D space-time graph, we show that fault-tolerant quantum error correction is possible with the surface-GKP code if the squeezing of the GKP states is higher than 11.2dB in the case where the GKP states are the only noisy elements. We also show that the squeezing threshold changes to 18:6dB when both the GKP states and circuit elements are comparably noisy. At this threshold, each circuit component fails with probability 0.69%. Finally, if the GKP states are noiseless, fault-tolerant quantum error correction with the surface-GKP code is possible if each circuit element fails with probability less than 0.81%. We stress that our decoding scheme uses the additional information from GKP-stabilizer measurements and we provide a simple method to compute renormalized edge weights of the matching graphs. Furthermore, our noise model is general as it includes full circuit-level noise.

Figures

Figures reproduced from arXiv: 1908.03579 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. The surface-GKP codes with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Circuits for surface code stabilizer measurements. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Noise propagation from the X4 qubit to the Z2 qubit during surface code stabilizer measurements. The red lightening [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. The logical [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Visualization of the function [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. Measurement of the GKP stabilizers for [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: FIG. 9. Measurement of the surface code stabilizers for [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

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Reference graph

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Constructing ﬁnite-dimensional codes with optical con- tinuous variables,

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Encoding a qubit into a cavity mode in circuit QED using phase estimation,

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Encoding qubits into oscillators with atomic ensembles and squeezed light,

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Generating grid states from Schr¨ odinger-cat states without postselec- tion,

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Machine learning method for state preparation and gate synthesis on pho- tonic quantum computers,

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yields exp[L′ ±∆t] = lim N→∞ [ exp [ V± ∆t N ] exp [ Lerr ∆t N ]]N . (A7) Note that both exp[ V±∆t/N] and exp[ Lerr∆t/N] are 14 Gaussian channels with the characterization matrices T± =   1 0 0 0 0 1 0 ∓1/N ±1/N 0 1 0 0 0 0 1  , N± = 0, Terr =   1 0 0 0 0 1 0 0 0 0 1 ...
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2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Simulate d rounds of noisy stabilizer measurements followed by one round of ideal stabilizer measure- ments and construct the Z-type and X-type 3D space-time graphs as described above

[2] [2]

If the number of highlighted vertices is odd, highlight a boundary vertex

Highlight all vertices whose assigned stabilizer value is changed from the previous round. If the number of highlighted vertices is odd, highlight a boundary vertex. Thus, the number of highlighted vertices is always even

work page

[3] [3]

Then, save the minimum total weight and all edges in the path

For all pairs of highlighted Z-type (X-type) ver- tices, ﬁnd the path with the minimum total weight. Then, save the minimum total weight and all edges in the path. Then, we are left with a Z-type (X- type) complete graph of highlighted vertices, where the weight of the edge ( v,w ) is given by the mini- mum total weight of the path that connects v and w. 21

work page

[4] [4]

For all matched pairs of Z-type (X-type) vertices, highlight all the Z- type (X-type) edges contained in the path that connects the matched vertices

Apply the minimum-weight perfect matching algo- rithm [34, 35] on the Z-type (X-type) complete graph of highlighted vertices. For all matched pairs of Z-type (X-type) vertices, highlight all the Z- type (X-type) edges contained in the path that connects the matched vertices

work page

[5] [5]

For each Z-type (X-type) horizontal edge, count how many times it was highlighted

Suppress all vertical edges and project the Z-type (X-type) 3D space-time graph onto the 2D plane. For each Z-type (X-type) horizontal edge, count how many times it was highlighted. If it is high- lighted even times, do nothing. Otherwise, apply the Pauli correction operator ˆXgkp ( ˆZgkp) to the corresponding data GKP qubit. Equivalently, up- date the qu...

work page

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Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping,

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work page 1999

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Quantum computation over continuous variables,

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work page 1999

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Eﬃcient quantum computa- tion using coherent states,

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work page 2002