Protected Grid States in Bosonic Codes

• Protected grid states are bosonic code states characterized by comb-like lattice wavefunctions stabilized by commuting displacement operators.
• In superconducting circuits, the gridium qubit combines Cooper-quartet and quantum phase-slip elements to implement an extended GKP Hamiltonian with a protected degenerate ground manifold.
• Deterministic unitary protocols and measurement-based techniques, such as phased-comb constructions, facilitate robust generation and near-optimal boson-loss performance of these grid states.

Searching arXiv for the cited papers and related work on protected grid states. Protected grid states are bosonic code states whose structure is organized by commuting translational symmetries in conjugate variables and whose protection derives either from Hamiltonians that enforce those symmetries or from preparation protocols that approximate the corresponding lattice structure. In the superconducting setting, the most explicit recent realization is the “gridium” qubit, whose eigenstates form protected grid states by combining an effective Cooper-quartet tunnel junction with a quantum phase-slip element in a high-impedance loop, thereby implementing an extended GKP Hamiltonian with a protected two-fold degenerate ground manifold (Nguyen et al., 18 Sep 2025). More broadly, protected grid states encompass approximate Gottesman–Kitaev–Preskill (GKP) codewords, hardware-encoded variants in non-reciprocal superconducting circuits, measurement-free and measurement-based generation schemes, and newer phased-comb constructions that are unitarily related to standard grid states while retaining near-optimal boson-loss performance (Rymarz et al., 2020, Hastrup et al., 2019, Eaton et al., 2021, Fur et al., 23 Apr 2026).

1. Conceptual and formal definition

Protected grid states are states whose wavefunctions form a comb-like lattice in conjugate representations and are stabilized by commuting displacement-like operators. In the single-mode GKP formulation, the stabilizers are

$$S_\phi=e^{i\,2\,\hat\phi},\qquad S_n=e^{i\,2\pi\,\hat n},$$

or, in quadrature notation,

$$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$

The defining feature is that these operators commute and determine a lattice of allowed peaks in phase space. The corresponding ideal Hamiltonian may be written as

$$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$

or, in a quadrature-rescaled form,

$$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$

with the confinement term vanishing in the ideal limit (Nguyen et al., 18 Sep 2025, Rymarz et al., 2020).

In the idealized limit, the wavefunctions become infinitely squeezed combs. For the superconducting gridium circuit, this appears as

$$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$

while in the standard GKP quadrature language the codewords are

$$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$

With finite confinement, charging, inductive energy, or finite squeezing, the delta-combs acquire Gaussian envelopes and become approximate grid states rather than exact stabilizer eigenstates (Nguyen et al., 18 Sep 2025, Fur et al., 23 Apr 2026).

A common misconception is that all protected grid states are identical to ideal GKP states. The literature instead distinguishes several families: Gaussian-truncated GKP states, hard-cutoff comb states, hardware-encoded approximate codewords in superconducting circuits, and phased-comb states that are related to standard grid states by a unitary phase operator $U_{\rm phase}=e^{i\Phi(\hat x)}$ (Fur et al., 23 Apr 2026). This suggests that “protected grid state” is best understood as a broader architectural category rather than a single canonical wavefunction.

2. Superconducting gridium and the extended GKP Hamiltonian

The gridium qubit is a single superconducting loop containing two complementary nonlinear elements in parallel: a Cooper-quartet tunnel junction and a quantum phase-slip element. The Cooper-quartet junction is realized by a four-junction “KITE” or rhombus at applied flux $\theta_{\rm ext}=\pi$, so that destructive interference removes single-Cooper-pair tunneling and leaves a leading second harmonic $I\propto \sin(2\phi_J)$, yielding an effective $\cos(2\phi)$ term. The quantum phase-slip element is a small Josephson junction with $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$0 embedded in a superinductance, producing a nonlinear capacitor term $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$1. The high-impedance environment satisfies $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$2, suppressing stray inductive and capacitive dispersion and confining the dynamics to a single mode (Nguyen et al., 18 Sep 2025).

Up to a constant, the circuit Hamiltonian is

$$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$3

Here $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$4 is the normalized Cooper-pair number, $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$5 is its conjugate phase, and $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$6. The potential in $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$7-space,

$$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$8

is a $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$9-periodic “egg-carton” modulated by a shallow parabola, while the $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$0-basis exhibits a $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$1-periodic $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$2 modulation (Nguyen et al., 18 Sep 2025).

The key structural point is that the Hamiltonian commutes with both $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$3 and $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$4. As stated in the source description, $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$5 is simultaneously diagonalizable with the two stabilizers, and logical code states $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$6 and $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$7 lie in the joint $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$8 eigenspace of $$H_{\rm GKP}=-E_S\cos(2\pi \hat n)+E_{2J}\cos(2\hat\phi),$$9, forming a two-fold degenerate ground manifold (Nguyen et al., 18 Sep 2025). Passive protection is therefore built into the hardware Hamiltonian rather than relying solely on active correction cycles.

The associated spectroscopy reveals pairs of degenerate states separated by large energy gaps, in excellent agreement with theoretical predictions, and the observations indicate that the circuit tolerates small disorders and gains robustness against environmental noise as its parameters approach the ideal regime (Nguyen et al., 18 Sep 2025). A plausible implication is that gridium realizes a superconducting embodiment of the long-envisioned idea of enforcing GKP stabilizers directly at the Hamiltonian level.

3. Degeneracy, wavefunctions, and spectral protection

At $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$0 or $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$1, with $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$2, the two lowest eigenstates $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$3 and $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$4 become exactly degenerate, forming a protected doublet. The source description attributes the absence of first-order splitting to the symmetry-enforced vanishing of the linear matrix elements

$$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$5

(Nguyen et al., 18 Sep 2025). This is central to the notion of “protected” in this context: the encoded manifold is not merely low in energy, but symmetry suppresses the dominant local couplings that would otherwise split or mix it.

Above the ground manifold lie excited doublets split by an energy $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$6, separated from the ground doublet by a gap

$$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$7

Changing the ratios $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$8 or $$H_{\rm LLL} = \frac{\hbar\omega_{LC}^2}{\omega_c}\,\frac{P^2+X^2}{2} - V_0\bigl[\cos(2\sqrt\pi\,X)+\cos(2\sqrt\pi\,P)\bigr],$$9 modifies both the support of the grid wavefunction and the gap sizes. In experiment, spectroscopic transitions to higher doublets appear as pairs of nearly parallel lines, with large avoided crossings when they couple to the readout resonator (Nguyen et al., 18 Sep 2025).

The approximate wavefunctions retain the lattice structure of the ideal code. In the $$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$0 basis, the source gives

$$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$1

and each computational eigenstate is described as a coherent superposition of alternating peaks in $$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$2 and $$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$3, differing only by an overall parity (Nguyen et al., 18 Sep 2025). In the non-reciprocal superconducting architecture, the analogous approximate codewords take the form

$$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$4

$$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$5

with $$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$6 for strong protection (Rymarz et al., 2020).

The non-reciprocal circuit also identifies the actual ground and first excited states as Hadamard-diagonal combinations,

$$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$7

which approximate the two GKP codewords with disjoint comb structure in $$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$8 and $$\psi(\phi)\sim\sum_k \delta(\phi-k\pi),\qquad \tilde\psi(n)\sim\sum_\ell \delta(n-\ell),$$9 (Rymarz et al., 2020). This makes clear that protected grid states can arise in different canonical coordinates while preserving the same underlying lattice logic.

4. Protection mechanisms against noise and disorder

In gridium, passive protection arises from three mechanisms explicitly identified in the source: commuting stabilizers, symmetry-enforced vanishing of dipole matrix elements, and exponential suppression of dispersion as wavefunctions delocalize (Nguyen et al., 18 Sep 2025). Small asymmetries in Josephson energies or inductances shift the location of the degeneracy points but do not lift the doublet degeneracy nor open first-order splitting. Charge-noise signatures observed in less-protected devices vanish to below Hz level as $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$0 becomes large, while flux-tuning experiments show vanishing dispersion of $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$1 versus $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$2 in the protected regime (Nguyen et al., 18 Sep 2025).

The same source reports that $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$3 grows from $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$4 in the weak regime to milliseconds in the protected regime, demonstrating exponential enhancement as grid support increases (Nguyen et al., 18 Sep 2025). Because the data explicitly link this growth to increased grid support, the relevant interpretation is not only improved coherence in a generic sense, but specifically improved logical bit-flip suppression as the encoded wavefunction becomes more delocalized over the lattice.

The non-reciprocal superconducting proposal identifies an overlapping but not identical protection mechanism. Charge noise appears as random gate charges $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$5 on the capacitors, but static offsets gauge away and dynamical shifts couple only to the cyclotron variables $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$6, whereas the code lives in guiding-center space $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$7, leading to exponentially suppressed matrix elements (Rymarz et al., 2020). Flux noise through the gyrator loops enters the cosine terms and, upon expansion, all noise terms either commute with the GKP stabilizers or act trivially within the code subspace. Flux noise on the outer loops is minimized at sweet spots $$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$8, and the source states that

$$\lvert\mu_{\rm GKP}\rangle \propto \sum_{s\in\mathbb Z} \bigl|(2s+\mu)\sqrt\pi\bigr\rangle_x,\qquad \mu=0,1.$$9

for local flux or quasiparticle operators (Rymarz et al., 2020).

A broader misconception is that protected grid states are necessarily immune to all local noise. The cited works instead describe channel-specific suppression mechanisms: exponential collapse of charge and flux dispersion in gridium, decoherence-free behavior for certain gyrator-loop fluctuations in the non-reciprocal circuit, second-order protection at sweet spots, and continuous energetic penalties for small displacements (Nguyen et al., 18 Sep 2025, Rymarz et al., 2020). This suggests that protection is structured and model-dependent rather than absolute.

5. Generation protocols beyond static Hamiltonian protection

Protected grid states can also be created deterministically by unitary protocols rather than by direct Hamiltonian engineering. In the measurement-free protocol of Hastrup et al., the goal is to start from a squeezed vacuum and end in an approximate GKP logical $U_{\rm phase}=e^{i\Phi(\hat x)}$0 on a rectangular or hexagonal lattice using only unitary Rabi-type interactions between a bosonic mode and a two-level ancilla (Hastrup et al., 2019). The building blocks are

$U_{\rm phase}=e^{i\Phi(\hat x)}$1

implemented through gate layers

$U_{\rm phase}=e^{i\Phi(\hat x)}$2

After $U_{\rm phase}=e^{i\Phi(\hat x)}$3 rounds, the protocol yields a superposition of $U_{\rm phase}=e^{i\Phi(\hat x)}$4 squeezed-Gaussian peaks along $U_{\rm phase}=e^{i\Phi(\hat x)}$5, reproducing the 1-GKP grid on a square lattice when the parameters are appropriately chosen (Hastrup et al., 2019).

That work also treats rectangular and hexagonal lattice geometries. For rectangular lattices one may take $U_{\rm phase}=e^{i\Phi(\hat x)}$6 and $U_{\rm phase}=e^{i\Phi(\hat x)}$7, while for hexagonal lattices the generators satisfy $U_{\rm phase}=e^{i\Phi(\hat x)}$8 with a distinct complex choice of $U_{\rm phase}=e^{i\Phi(\hat x)}$9 and $\theta_{\rm ext}=\pi$0 (Hastrup et al., 2019). Effective squeezing is quantified by

$\theta_{\rm ext}=\pi$1

These formulas make explicit that approximate protection is operationally tied to the sharpness of stabilizer expectation values.

Measurement-based generation has also been proposed within continuous-variable cluster states via PhANTM, the Photon-counting-Assisted Node-Teleportation Method. In that framework, Gaussian cluster teleportation is augmented by photon subtraction and photon-number-resolving detection to realize Kraus operators of the form

$\theta_{\rm ext}=\pi$2

and repeated application of these gadgets builds polynomials in $\theta_{\rm ext}=\pi$3 that turn squeezed vacuum into large-amplitude cat states. Those cats are then “bred” into GKP grid states through cluster-state CZ-based operations (Eaton et al., 2021). In this setting, finite-squeezing teleportation noise continually tends to wash out non-Gaussian features, while PhANTM both injects new non-Gaussian structure and teleports through fresh squeezing, thereby preserving or replenishing the ingredients needed for grid-state formation (Eaton et al., 2021).

6. Phased-comb states and generalized protected-grid encodings

A significant recent development is the introduction of phased-comb states, generated deterministically using programmable nonlinear bosonic circuits built only from squeezing, displacement, and Kerr operations (Fur et al., 23 Apr 2026). The resources are

$\theta_{\rm ext}=\pi$4

with all protocols starting from $\theta_{\rm ext}=\pi$5. One variant attempts symmetry-enforced GKP preparation by concatenating Kerr and displacement layers together with correction displacements $\theta_{\rm ext}=\pi$6 chosen to minimize a GKP squeezing operator $\theta_{\rm ext}=\pi$7. The other, defining phased-comb states, omits the correction displacements and retains the deterministic Kerr-induced phase structure (Fur et al., 23 Apr 2026).

The ideal GKP states are zero-eigenvectors of

$\theta_{\rm ext}=\pi$8

The source states that for genuine comb states $\theta_{\rm ext}=\pi$9 as the number of legs increases, whereas the symmetry-enforced circuit saturates due to residual Kerr phases (Fur et al., 23 Apr 2026). This is an important correction to the assumption that deeper circuits automatically converge to better GKP symmetry: the data instead state that quality saturates with increasing circuit depth because of imperfect symmetry restoration.

After $I\propto \sin(2\phi_J)$0 cycles, phased-comb states take the form

$I\propto \sin(2\phi_J)$1

with $I\propto \sin(2\phi_J)$2. In the large-cycle limit,

$I\propto \sin(2\phi_J)$3

so phased-comb states are unitarily related to hard-cutoff comb states (Fur et al., 23 Apr 2026).

Under boson loss, the relevant benchmark is the near-optimal channel fidelity $I\propto \sin(2\phi_J)$4, computed from the QEC matrix $I\propto \sin(2\phi_J)$5 and the overlap matrix $I\propto \sin(2\phi_J)$6. The reported results are that for $I\propto \sin(2\phi_J)$7, comb and phased-comb codes slightly outperform Gaussian-GKP approximations; over the full $I\propto \sin(2\phi_J)$8 range, all three bosonic codes are comparable and far exceed the trivial $I\propto \sin(2\phi_J)$9 encoding; and as the number of legs grows, infidelity drops for all codes while phased-comb tracks comb states nearly identically (Fur et al., 23 Apr 2026). This supports the narrower conclusion stated in the source: exact translational symmetry is not strictly required for near-optimal boson-loss protection.

7. Experimental parameters, logical operations, and performance outlook

The non-reciprocal superconducting circuit proposal includes explicit design parameters: capacitance $\cos(2\phi)$0, inductance $\cos(2\phi)$1, charging energy $\cos(2\phi)$2, Josephson energy $\cos(2\phi)$3, inductive energy $\cos(2\phi)$4, and gyrator conductance $\cos(2\phi)$5, yielding $\cos(2\phi)$6, $\cos(2\phi)$7, and $\cos(2\phi)$8 (Rymarz et al., 2020). That architecture further describes logical $\cos(2\phi)$9 and $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$00 gates implemented by dc currents applied to gyrator ports, a phase gate $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$01 realized by temporarily detuning one inductive or Josephson parameter, and a two-qubit SUM gate produced by tunable inductive coupling (Rymarz et al., 2020).

The measurement-free preparation protocol gives platform-specific timing estimates. Typical interaction times are stated as $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$02 for trapped ions and $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$03 for circuit QED (Hastrup et al., 2019). Under realistic noise, the work reports that with $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$04 rounds and input squeezing $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$05 dB one can still achieve output effective squeezing $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$06 dB; noiseless end-to-end fidelities exceed $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$07 for $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$08 and remain above $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$09 for realistic noise; and the effective shift-error probability falls below $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$10 for $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$11 with infinite squeezing, remaining $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$12 for input squeezing at least $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$13 dB under realistic noise (Hastrup et al., 2019).

The PhANTM cluster-state approach gives complementary benchmarks. After $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$14 steps with node squeezing $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$15 dB, one obtains cat amplitude $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$16 with fidelity $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$17. For GKP synthesis, a single breeding round gives $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$18 with fidelity $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$19, while two rounds give $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$20 with fidelity $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$21 (Eaton et al., 2021). These results remain below the 10 dB line-quadrature variance threshold quoted in that source for fault-tolerant GKP encoding, but they demonstrate a route to deterministic, embedded generation and continued stabilization within a Gaussian cluster resource (Eaton et al., 2021).

The deterministic Kerr-based bosonic circuits likewise report quantitative operating points. Symmetry-enforced GKP states achieve fidelities $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$22 for $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$23 cycles and $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$24 dB squeezing, but the source emphasizes saturation with circuit depth; phased-comb codes, by contrast, preserve the grid, grow unboundedly in size, and require Kerr-angle control $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$25 and loss rate $$\hat S_x = e^{-2\,i\sqrt\pi\,\hat p},\qquad \hat S_p = e^{2\,i\sqrt\pi\,\hat x}.$$26, stated to be within reach in microwave-cavity or superconducting-nonlinearity platforms (Fur et al., 23 Apr 2026).

Across these architectures, a consistent pattern emerges. Static Hamiltonian protection, deterministic unitary synthesis, and measurement-based breeding all seek to realize lattice-structured bosonic codewords whose local error channels are suppressed by symmetry, energy penalties, or code geometry. The principal differences concern where the protection resides: in commuting stabilizers engineered into the physical Hamiltonian, in carefully timed ancilla-assisted unitaries, in cluster-state teleportation gadgets that continually replenish non-Gaussianity, or in alternative phase-framed encodings such as phased-comb states (Nguyen et al., 18 Sep 2025, Hastrup et al., 2019, Eaton et al., 2021, Fur et al., 23 Apr 2026).