Hybrid Oscillator-Qudit Stabilizer States

• Hybrid oscillator–qudit stabilizer states are composite quantum codes that integrate continuous-variable oscillators with finite-dimensional qudits to achieve fault-tolerant quantum information encoding.
• They employ a unified phase space formalism and lattice embedding that factorizes Gaussian and Clifford operations, enabling robust error detection and correction.
• Preparation protocols using controlled displacement with multi-qubit ancillas realize approximate GKP comb states, achieving exponential suppression of errors with minimal overhead.

A hybrid oscillator–qudit stabilizer state is a quantum state or codeword defined on a composite system comprising both continuous-variable (CV) modes (typically oscillators) and finite-dimensional discrete subsystems (qudits). These states generalize the stabilizer formalism, originally formulated for qubits and later extended by the Gottesman-Kitaev-Preskill (GKP) code to oscillators, to heterogeneous systems. Hybrid stabilizer codes enable fault-tolerant quantum information encoding employing both types of subsystems simultaneously, resulting in unique error-correcting capabilities and entanglement features inaccessible to tensor products of pure CV or qudit codes (Chakraborty et al., 6 Aug 2025, Bjerrum et al., 2024).

1. Hybrid Phase Space Formalism

Hybrid oscillator–qudit systems are described by a phase space formalism that unifies continuous and discrete variables. Define the configuration group as $$M = \mathbb{R}^p \times \mathbb{Z}_{c_1} \times \dots \times \mathbb{Z}_{c_k}$$, representing $p$ oscillators and $k$ qudits of dimensions $c_j$. The full phase space is

$$G = M \times \widehat{M} = (\mathbb{R}^p \times \mathbb{R}^p) \times (\mathbb{Z}_{c_1} \times \mathbb{Z}_{c_1}) \times \cdots$$

The skew-symmetric symplectic form splits into CV and DV sectors:

$J_{cv} = \begin{pmatrix}0&I_p\-I_p&0\end{pmatrix},\quad J_{dv} = \begin{pmatrix}0&\mathrm{diag}(1/c_j)\-\mathrm{diag}(1/c_j)&0\end{pmatrix}$

Canonical displacement operators act as

$D(x) = D(m_{cv}, s_{cv}) \otimes D(m_{dv}, s_{dv})$

with

$$D(m_{cv}, s_{cv}) = \exp(-i \sqrt{2\pi} m_{cv} \cdot \hat{p}) \exp(i \sqrt{2\pi} s_{cv} \cdot \hat{x}),\quad D(m_{dv}, s_{dv}) = X^{m_{dv}} Z^{s_{dv}}$$

Their commutation is governed by

$$D(x) D(y) = e^{-2\pi i \langle x, J y \rangle} D(y) D(x)$$

A stabilizer code arises from a lattice embedding $T: \mathbb{Z}^{2n} \to G$, with Gram matrix $\Theta = T^T J T$ antisymmetric integer, and cell volume

$$\mathrm{Vol}(\mathscr{D}) = |\det T| = |\mathrm{Pf}(\Theta)|$$

For a single oscillator and qudit ($n = 2$), the minimal cell has area $2\pi c$, growing linearly with the qudit dimension $c$ (Chakraborty et al., 6 Aug 2025).

2. Construction and Properties of Hybrid Stabilizer States

Hybrid stabilizer states are defined as +1 eigenstates of a commuting subgroup of hybrid displacement operators. For a single oscillator and single qudit ($p=1, c_1=c$), one sets

$$S_X = \exp(-i\sqrt{2\pi/c}\,\hat{p}) \otimes X^\dagger,\quad S_Z = \exp(i\sqrt{2\pi/c} \,\hat{x}) \otimes Z$$

These generators commute since the oscillator's phase-space cell area ($2\pi c$) is precisely canceled by the qudit commutator ($X Z = e^{-2\pi i/c} Z X$). The unique stabilizer state is

$$|\mathrm{LCA}\rangle = \sum_{\ell \in \mathbb{Z}} \Bigl| \ell \sqrt{\tfrac{2\pi}{c}} \Bigr\rangle_x \otimes \bigl| -\ell \bigr\rangle_Z$$

This state can also be expressed as a superposition of $c$ GKP comb states in the oscillator, each entangled with a different qudit computational basis state. The resulting entanglement between the CV and DV subsystems cannot be generated by local Gaussian (CV Clifford) and qudit Clifford gates alone; thus, these states constitute a non-Gaussian entanglement resource (Chakraborty et al., 6 Aug 2025).

3. Symplectic and Clifford Operations in Hybrid Systems

The group of physical symplectic transformations factoring into hybrid systems is $$\mathrm{Sp}(2p, \mathbb{R}) \times \mathrm{Sp}(2k, \mathbb{Z}_{c_1} \times \cdots)$$, acting as Gaussian (CV) and Clifford (DV) gates respectively. Critically, no mixed continuous–discrete symplectics exist: symplectic transformations on $\mathbb{R} \times \mathbb{Z}_c$ always factor, and cannot generate hybrid entanglement.

Logical Clifford operations on codewords correspond to automorphisms of the stabilizer lattice, formalized via Morita equivalence on the torus parameterized by $\Theta$. Specifically, given a transformation $g \in \mathrm{SO}(2n,2n | \mathbb{Z})$, one defines

$$\Theta \mapsto g \cdot \Theta = (A \Theta + B)(C \Theta + D)^{-1}$$

This transformation is physically realized by a metaplectic lift, implementing logical Clifford gates on codewords while conjugating stabilizer generators according to the induced lattice action (Chakraborty et al., 6 Aug 2025).

4. Hybrid Error-Correcting Codes and Logical Structures

Hybrid stabilizer codes are in one-to-one correspondence with noncommutative tori defined by their commutation structure. A code is determined by a Gram matrix $\Theta$ and its "dual" (logical) data arises through Morita equivalence. For example, for a single oscillator and qudit code (dimension $c$), the commutation relations for logical Pauli operators take the form

$$\bar{X} \bar{Z} = e^{2\pi i (a\theta + b)/(c\theta + d)} \bar{Z} \bar{X}$$

where integers $a, b$ satisfy $b c - a d = 1$, and the code's size is $K = |c \theta + d|$.

Logical Clifford gates correspond to specific integer symplectic matrices. In simple single-mode, single-qudit $(c, d)$ codes, the logical Hadamard, for instance, is implemented by the metaplectic lift of matrices of the form $$\begin{pmatrix}-a & -b \ c & d\end{pmatrix}$$ (Chakraborty et al., 6 Aug 2025).

5. Preparation Protocols: Many-Qubit Mapping and Stabilizer Realization

Controlled displacement protocols allow the preparation of hybrid oscillator–qudit stabilizer states using multi-qubit ancillas and harmonic oscillators. The multi-qubit qudit (“logical qudit”) is defined using generalized Pauli operators: $$\tilde{X} = e^{-i\frac{2\pi}{M} Y_N}, \quad \tilde{Z} = e^{+i\frac{2\pi}{M} X_N}, \quad \tilde{Z} \tilde{X} = e^{i 2\pi/M} \tilde{X} \tilde{Z}$$ where $X_N$ and $Y_N$ play the roles of discrete position and its Fourier conjugate.

The key interaction Hamiltonian is

$H_{\mathrm{cd}} / \hbar = \chi q X_N$

with evolution

$U_I = e^{+i\frac{2\pi}{P_q} q X_N}$

and disentanglement by

$U_D = e^{-i(P_q/M) p X_N}$

The protocol consists of: 1) preparing the oscillator in a squeezed vacuum, 2) initializing the multi-qubit register, 3) applying inverse and forward QFTs, 4) applying $U_I$ and $U_D$.

After disentanglement, the oscillator wave function realizes an approximate GKP comb, while the logical qubit remains encoded in the multi-qubit system. The constructed stabilizer group combines the oscillator’s GKP shifts and the qudit’s $\tilde{Z}$, with logical operators jointly realized as (for code dimension $M$):

$$\bar{X} = D(\sqrt{\pi}) \otimes \tilde{X}, \quad \bar{Z} = D(i\sqrt{\pi}) \otimes \tilde{Z}$$

The protocol requires only two time-independent interactions and achieves exponential suppression of the peak width ($\Delta \propto 2^{-N}$), corresponding to approximately 6 dB of squeezing per qubit (Bjerrum et al., 2024).

6. Physical Significance and Applications

Hybrid oscillator–qudit stabilizer states enable the simultaneous encoding and fault-tolerant protection of quantum information in systems combining CV and DV subsystems. The unique feature is the ability to measure arbitrarily large sets of non-commuting displacement errors, with the phase space unit cell volume scaling as $2\pi c$. Entanglement between oscillators and qudits in these codes cannot be generated by Gaussian ⊗ Clifford operations, so these states constitute distinct non-Gaussian resources.

Practical applications include hybrid hardware platforms (superconducting circuits, trapped ions) that support both qudit and cavity-oscillator modes. Error correction in hybrid codes leverages both the grid structure of the oscillator and the syndrome structure of the qudits, allowing robust correction of displacement and discrete errors. The explicit realization using controlled displacement enables quantum error-correction with minimal interaction overhead and high efficiency, making these codes promising for scalable hybrid quantum processors (Chakraborty et al., 6 Aug 2025, Bjerrum et al., 2024).