Qunaught State: Ancilla GKP Resource

• Qunaught state is a non-Gaussian continuous-variable bosonic state defined by a one-dimensional grid that carries no logical qubit information.
• It functions as an ancilla or resource in error correction, Bell-pair generation, and cluster-state construction in quantum computing architectures.
• Preparation methods include Gaussian conversion and photonic circuit optimization, achieving high fidelity and effective squeezing near fault-tolerant thresholds.

Searching arXiv for papers on qunaught states and related GKP resources. A qunaught state is a special non-Gaussian continuous-variable bosonic state, also described as a grid state, canonical GKP state, or sensor state, whose phase-space lattice is chosen so that it spans a one-dimensional subspace and therefore carries no logical qubit information. In the Gottesman–Kitaev–Preskill framework, it is a grid state stabilized by commuting displacements in position and momentum, and it functions primarily as an ancilla or resource state for bosonic error correction, Bell-pair generation, cluster-state construction, and all-Gaussian implementations of encoded computation rather than as a logical data state itself (Zheng et al., 2023).

1. Definition and code-theoretic status

The defining distinction between a qunaught and a logical GKP qubit state is code dimension. A qunaught has the grid structure of a GKP codeword but does not encode logical information; this is the origin of the name “qu-naught.” In one formulation, the spacing of the grid in phase space is chosen so that the state is a simultaneous eigenstate of two commuting displacement operators and spans a one-dimensional subspace, in contrast to the two-dimensional code space of a logical GKP qubit (Zheng et al., 2023).

For the $d=1$ case emphasized in trapped-ion and error-correction analyses, the qunaught is stabilized by

$$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$

with $l=\sqrt{2\pi}$, giving a denser lattice than the $d=2$ GKP qubit code (Fontboté-Schmidt et al., 8 May 2026). In the rectangular encoding used for cluster-state constructions, the GKP qunaught is written as

$\ket{\varnothing}=\ket{0_{\sqrt{\pi/2}_{\mathrm{rec}}},$

and satisfies the Fourier-invariance property

$\hat F\ket{\varnothing}=\ket{\varnothing}.$

This fixed-point property under $\hat F$ is one reason the state is natural in Gaussian interferometric constructions (Østergaard et al., 26 Feb 2025).

A common misconception is that the qunaught is merely the logical zero of the standard GKP qubit. The data indicate a sharper distinction: the qunaught lattice is denser, the code space is one-dimensional, and expectation values of logical operators do not carry qubit information (Marqversen et al., 20 May 2025).

Feature	GKP logical qubit state	Qunaught state
Code dimension	Two-dimensional	One-dimensional
Encodes logical information	Yes	No
Typical role	Data / computation	Ancilla / resource / intermediate

2. Phase-space structure, variants, and finite-energy approximations

In idealized form, the qunaught is a Dirac-comb state. A single-mode position-space representation is

$\ket{\varnothing}=\sum_k \ket{q=k\sqrt{2\pi}},$

equivalently described as a superposition of infinitely squeezed position eigenstates spaced by $\sqrt{2\pi}$ (Fontboté-Schmidt et al., 8 May 2026). Another equivalent representation used in GKP error-correction analysis is

$$\ket{\,}_L=\sum_{n\in\mathbb{Z}}\ket{n\sqrt{2\pi}},$$

again emphasizing the half-spacing relative to standard logical GKP codewords (Marqversen et al., 20 May 2025).

The qunaught admits displaced variants obtained by half-grid shifts in $$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$0, $$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$1, or both. In the trapped-ion formulation, these include momentum-displaced, position-displaced, and doubly displaced qunaughts; the four possibilities correspond to different sign choices for stabilizer eigenvalues while still carrying no logical qubit information (Fontboté-Schmidt et al., 8 May 2026). This suggests that the physically relevant object is not a single isolated state but a small orbit of grid states with the same non-encoding character.

Because the ideal Dirac comb is unphysical, practical work uses finite-energy approximations. One representation is

$$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$2

with $$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$3 setting the effective width of the peaks (Solodovnikova et al., 8 Aug 2025). In numerical and experimental studies, quality is then quantified by grid-sensitive observables rather than exact code-space membership. A representative metric is the effective squeezing

$$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$4

computed from the characteristic function at a grid translation (Solodovnikova et al., 8 Aug 2025).

3. Resource state for error correction, Bell pairs, and universality

Two qunaught states interfered on a $$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$5 beam splitter produce an entangled GKP Bell resource. In the Octo-Rail Lattice analysis, the transformation is

$$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$6

and this Bell-pair generation is used as a primitive for macronode cluster-state construction (Østergaard et al., 26 Feb 2025). The same mechanism appears in trapped-ion experiments and in non-Hermitian bosonic-dimer theory, where the lossless limit recovers the standard beam-splitter generation of a square GKP$$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$7 Bell pair from a two-mode qunaught product state (Fontboté-Schmidt et al., 8 May 2026).

In teleportation-based GKP error correction, qunaught-based Bell ancillas are especially significant. Performance analysis of GKP error correction shows that when the Bell state is prepared from two qunaughts and a beam splitter, the post-correction peak width satisfies

$$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$8

for both quadratures, so the output GKP squeezing matches the squeezing of the prepared qunaught states regardless of the input’s initial width (Marqversen et al., 20 May 2025). By contrast, standard Bell-state preparation with logical GKP ancillas and a controlled-$$\hat S_q = e^{i l \hat p}, \qquad \hat S_p = e^{i l \hat q},$$9 gate yields quadrature-asymmetric output widths worse than the ancilla squeezing (Marqversen et al., 20 May 2025). The same study states that, with recently introduced qunaught states, the Knill approach achieves superior GKP squeezing and is the simplest to realize experimentally in the optical domain (Marqversen et al., 20 May 2025).

At the architecture level, the Octo-Rail Lattice uses eight input GKP qunaught states per macronode and combines them with static optical components and homodyne-basis changes. Analysis there reports compatibility with a fault-tolerant threshold of $l=\sqrt{2\pi}$0 squeezing when combined with the surface code, while preserving universality and fault tolerance without additional non-Gaussian states or feed-forward operations (Østergaard et al., 26 Feb 2025). In that formulation, qunaughts are the non-Gaussian enabling resource for Clifford operations via Gaussian measurements and for non-Clifford computation through magic-state protocols.

4. Preparation protocols and numerical optimization

A direct route to qunaught generation is the Gaussian conversion protocol from rotationally symmetric binomial code states. The reported protocol is iterative, heralded, and uses only Gaussian operations—beam splitters and alternating homodyne detection in $l=\sqrt{2\pi}$1 and $l=\sqrt{2\pi}$2 (Zheng et al., 2023). Starting from four-fold-symmetric binomial states corresponding to a zero-logical encoded qubit, the protocol progressively converts rotational symmetry into translational symmetry. Numerical simulation yields GKP qunaught states with fidelity over $l=\sqrt{2\pi}$3 and probability approximately $l=\sqrt{2\pi}$4 after only two steps, while additional iterations can increase fidelity at the cost of lower success probabilities (Zheng et al., 2023).

A complementary line of work uses continuous-variable photonic circuit optimization. The $l=\sqrt{2\pi}$5 framework simulates Gaussian Boson Sampling circuits with generaldyne and photon-number-resolving detection by combining linear combinations of Gaussians with coherent-state decomposition (Solodovnikova et al., 8 Aug 2025). In that setting, the qunaught is the heralded target state of multimode squeezed-vacuum circuits followed by interferometers, loss channels, and photon-number measurements on all but one mode. The reported optimization computes overlaps, characteristic functions, effective squeezing, and analytical gradients, and identifies circuit configurations with symmetric effective squeezing $l=\sqrt{2\pi}$6, described as close to the threshold for fault tolerance (Solodovnikova et al., 8 Aug 2025).

Method	Operations	Reported outcome
Gaussian conversion from binomial states	Beam splitters + alternating $l=\sqrt{2\pi}$7 homodyne + heralding	Fidelity over $l=\sqrt{2\pi}$8, probability $l=\sqrt{2\pi}$9 after two iterations
GBS-based heralding with $d=2$0	Squeezers + interferometer + loss + PNRDs + optimization	Symmetric effective squeezing $d=2$1

These two approaches illuminate different preparation regimes. The first emphasizes code-to-code Gaussian conversion between rotationally symmetric and translationally symmetric bosonic resources; the second emphasizes realistic photonic compilation and optimization under inefficient components.

5. Experimental realization and dynamical deformation

Experimental generation of entangled GKP states from qunaughts has been demonstrated with two motional modes of a trapped ion. By interfering two qunaught states on a beam splitter, all four Bell states were generated with an average fidelity of $d=2$2, and an extension of the entangled-state lifetime was subsequently demonstrated through quantum error correction (Fontboté-Schmidt et al., 8 May 2026). The same report describes qunaught product states as the pre-entanglement resource: before the beam splitter, each mode contains stabilizer information but no logical information; after interference, logical Bell structure emerges.

The dynamical behavior of qunaught-derived grids under monitored loss has also been analyzed in a no-jump non-Hermitian bosonic dimer. There, the effective quadratic non-Hermitian dynamics induces a complex symplectic flow that deforms both the primitive lattice vectors and the regularized origin seed (Rodriguez-Lara et al., 15 Jun 2026). The reduced sector exhibits elliptic, parabolic, and hyperbolic regimes, associated respectively with oscillatory, linear, and exponential lattice deformations. Although projected lattice areas can change, the full four-dimensional phase-space deformation has determinant one, and the postselection cost is set by the no-jump probability $d=2$3 (Rodriguez-Lara et al., 15 Jun 2026).

This body of work indicates that qunaught states are not only static code resources. They also provide a controlled probe of how Gaussian and non-Hermitian evolutions reshape grid structure, entanglement, and postselected survivability.

6. Stabilization, metrological use, and terminological ambiguity

Reservoir engineering provides a dissipative route to finite-energy qunaught stabilization. A two-dissipator Lindblad master equation has been proposed for a quantum harmonic oscillator,

$d=2$4

with

$d=2$5

where $d=2$6 targets a qunaught and $d=2$7 regularizes the finite energy (Robin et al., 15 Apr 2026). The study gives explicit energy bounds, establishes exponential convergence of periodic observables to the codespace with a spectral gap, and numerically shows that for the qunaught case the fine phase-space grid structure is maintained even with moderate photon loss, although the contrast fades (Robin et al., 15 Apr 2026). The same framework presents steady-state preparation of metrological states, reinforcing the use of qunaughts as sensor states rather than encoded logical qubits.

There is also a terminological ambiguity outside the GKP literature. In a geometric-quantization treatment of symplectic vector spaces with inner product, the label “qunaught state” is used for the Sorkin–Johnston state, a distinguished pure quasi-free state on the Weyl algebra with

$d=2$8

That construction is shown to coincide with the state obtained by geometric quantization followed by Berezin–Toeplitz dequantization and evaluation at the origin (Hawkins et al., 2022). This is a distinct usage: it concerns a distinguished state in algebraic quantum field theory and geometric quantization, not the GKP grid-state resource used in bosonic error correction.

Taken together, these results define the qunaught state primarily as a one-dimensional GKP grid resource with no logical payload, but with unusually broad operational reach: heralded state preparation, loss-aware simulation, linear-optical Bell-pair generation, teleportation-based GKP error correction, cluster-state fault tolerance, dissipative stabilization, and metrological steady-state engineering all use the same underlying grid structure, while a separate mathematical literature employs the same term for an unrelated distinguished quasi-free state (Zheng et al., 2023).