Multi-Mode Fock State Codes Overview

• Multi-mode Fock state codes are bosonic encodings that store logical information in fixed-total-particle-number sectors, exploiting correlations between bosonic modes.
• They employ distinct constructions such as constant-excitation, pair-cat, and GKP lattice codes to mitigate amplitude damping and loss errors via structured stabilizer frameworks.
• Experimental implementations span platforms like trapped ions, superconducting circuits, and optical systems, demonstrating practical resource generation, state preparation, and syndrome extraction.

Searching arXiv for recent and foundational papers on multi-mode Fock state codes. Multi-mode Fock state codes are bosonic encodings in which logical information is stored in selected populations and coherences of joint number states $\lvert n_1,\dots,n_K\rangle$ of several bosonic modes. In the superselection-rule-compliant formulation, a general $K$-mode pure state is written in a fixed-total-particle-number sector,

$$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$

so a multi-mode Fock state code is a logical subspace inside a fixed-$N$ manifold, or, if a number-nonconserving continuous-variable description is used, an equivalent fixed-$N$ code on a larger multimode Hilbert space with explicit reference modes (Descamps et al., 7 Jan 2025). In current bosonic-QEC practice, the topic spans constant-excitation amplitude-damping codes, driven-dissipative pair-cat and $M$-mode generalizations, multimode GKP lattice encodings, and group-orbit constructions such as the two-mode $2T$-qutrit (Elimelech et al., 16 Mar 2026, Albert et al., 2018, Shaw et al., 2022, Denys et al., 2022).

1. Formal setting and code space

A “mode” is a physical degree of freedom—spatial, spectral, polarization, temporal, motional, or cavity-defined—associated with bosonic annihilation and creation operators obeying $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$ (Descamps et al., 7 Jan 2025). For a $q$-mode bosonic system, the joint Fock space is

$$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$

and the constant-excitation sector of total photon number $K$0 is

$K$1

A multi-mode Fock state code of dimension $K$2 is then a subspace

$K$3

with orthonormal basis $K$4 (Elimelech et al., 16 Mar 2026).

This fixed-$K$5 viewpoint includes familiar encodings such as dual-rail $K$6, $K$7, higher-$K$8 occupancy encodings such as $K$9 and $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$0, and more general multi-rail states $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$1, all of which are fully compatible with the particle-number SSR (Descamps et al., 7 Jan 2025). A recurrent theme in the literature is that “multi-mode” refers not merely to using several oscillators, but to exploiting correlations between their occupation numbers, number-difference sectors, or lattice coordinates to define logical support, syndrome structure, and control.

2. Principal code families

The present literature contains several distinct but related multi-mode Fock-state code constructions.

Family	Defining structure	Distinctive property
Constant-excitation Fock codes	$$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$2 built from $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$3	Positive-rate, linear-distance asymptotic families with bounded per-mode occupancy
Pair-cat and $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$4-mode generalizations	Fixed number-difference sectors and stabilizers $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$5	Protection against large classes of loss, and for $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$6, loss/gain in one mode
Two-mode $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$7-qutrit	Span of 24 coherent states indexed by the binary tetrahedral group $$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$8	Small-$$\ket{\Psi}=\sum_{\{n_i\}: \sum_{i=1}^K n_i = N} c_{n_1,\dots,n_K}\ket{n_1}_1\cdots\ket{n_K}_K,$$9 limit approaches a two-mode Fock code
Multi-mode GKP	Lattice codewords with infinite Fock support	Stabilizer-subsystem decomposition with ideal decoding as partial trace

Constant-excitation codes have recently acquired an asymptotic theory. Exact and approximate error correction for amplitude damping can be formulated on $N$0 by restricting to a truncated amplitude-damping channel and by constructing codes from classical codes on the simplex $N$1 with large $N$2-distance. This yields asymptotically good exact and approximate Fock state codes with positive rate, linear distance, and bounded per-mode occupancy, including deterministic $N$3-local excitation and probabilistic $N$4 or $N$5 occupancy bounds (Elimelech et al., 16 Mar 2026).

Pair-cat codes begin from two-mode pair-coherent states in a fixed photon-number-difference sector $N$6. Logical states $N$7 are superpositions of pair-coherent states selected by a $N$8 projection, and the construction extends to $N$9 modes with one non-Hermitian stabilizer $N$0 and $N$1 number-difference stabilizers $N$2. In these codes, photon-number correlations across modes carry both the logical information and the syndrome information (Albert et al., 2018).

The two-mode $N$3-qutrit is a group-orbit code in $N$4. Its ambient space is a 24-dimensional span of coherent states indexed by the binary tetrahedral group $N$5, and its logical qutrit is the span of three coset superpositions over $N$6, $N$7, and $N$8. In the $N$9 limit, the orthonormal basis approaches $M$0 and two superpositions of $M$1, $M$2, and $M$3, matching a known two-mode Fock code (Denys et al., 2022).

Multi-mode GKP codes occupy a different corner of the subject. Physically they are bosonic/Fock-state codes because each ideal logical basis state is a highly structured superposition over multi-mode Fock states, but their native organization is a lattice in phase space. The stabilizer-subsystem decomposition shows that a general multi-mode GKP code can be written as $M$4 logical GKP qudits plus qunaughts, possibly coupled by a Gaussian unitary, all implemented inside the multi-mode Fock space (Shaw et al., 2022).

3. Error models and correction mechanisms

The central noise model for constant-excitation Fock codes is amplitude damping. For a $M$5-mode system with independent loss, an error pattern is labeled by $M$6, with Kraus operator

$M$7

where $M$8 removes $M$9 photons from mode $2T$0. A code with distance $2T$1 corrects any combination of up to $2T$2 photon losses across modes, namely the set $2T$3 (Elimelech et al., 16 Mar 2026). A key formal advance is the normalized truncated amplitude-damping channel $2T$4, obtained by conditioning on losing at most $2T$5 photons. Within this formulation, exact and approximate asymptotic goodness become operationally equivalent: asymptotically good exact codes for the truncated channel are equivalent in usefulness to asymptotically good approximate codes for the full amplitude-damping channel, and correcting only sublinear losses $2T$6 is insufficient at fixed loss parameter $2T$7 (Elimelech et al., 16 Mar 2026).

Pair-cat codes emphasize a different protection mechanism. The number-difference syndrome $2T$8 changes in opposite directions under losses in the two modes,

$2T$9

so losses in different modes are distinguishable by monitoring $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$0. In the code sector $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$1, all single-mode loss errors $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$2 or $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$3 are detectable, whereas the leading uncorrectable process is the correlated two-mode loss $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$4, which acts approximately as a logical bit flip. In the $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$5-mode extension, the lowest-weight undetectable error is $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$6 (Albert et al., 2018). A common misconception is therefore that distributing information over many modes automatically protects against arbitrary simultaneous local loss; the pair-cat analysis shows instead that multimode structure distinguishes many local loss patterns, but correlated losses across all participating modes remain decisive.

The same pair-cat framework also yields exponential suppression of logical dephasing. Projected powers of the number operators act as the identity plus exponentially small logical-$[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$7 corrections of order $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$8, and for the special $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$9 code there is a dephasing “sweet spot” at $q$0 photons per mode (Albert et al., 2018). This illustrates a general design principle: multi-mode Fock-state codes often trade exact finite-dimensional syndrome structure against approximate bosonic protection whose quality depends on excitation scale.

4. Stabilizers, logical subsystems, and symmetry structure

Multi-mode Fock-state codes are not tied to one stabilizer language. In pair-cat theory, the code space is the joint $q$1 eigenspace of $q$2 and the number-difference stabilizers $q$3, and some stabilizers are explicitly non-Hermitian and not diagonalizable in the usual sense (Albert et al., 2018). In the $q$4-qutrit, the qutrit subspace is selected inside a 24-dimensional dark-state manifold by a stabilizer set

$q$5

which implies the congruence constraints $q$6, $q$7, and $q$8 (Denys et al., 2022).

For GKP codes, the stabilizer viewpoint is sharpened by a subsystem decomposition

$q$9

with $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$0 the logical subsystem and $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$1 an infinite-dimensional stabilizer subsystem. Its defining property is that tracing out $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$2 is exactly ideal GKP decoding for the chosen primitive cell. This converts physical bosonic noise into a finite-dimensional logical channel without Fock truncation, a particularly strong result for multi-mode lattice codes (Shaw et al., 2022).

A broader SSR-compliant control framework is provided by Schwinger-boson generators. For a pair of modes, Gaussian SSRC unitaries are $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$3, while non-Gaussian SSRC unitaries are $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$4; together they form a universal set on fixed-$$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$5 sectors. Gaussian operations preserve total photon number and implement mode rotations, but they do not change the intrinsic number of modes and cannot create genuine mode entanglement; non-Gaussian operations are therefore necessary for universality and for nontrivial multi-mode code manipulations (Descamps et al., 7 Jan 2025).

A complementary structural tool is the multi-mode Jordan–Schwinger map. Fixed-$$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$6 $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$7-mode bosonic subspaces carry reducible $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$8 representations generated by collective operators $$\mathcal H_q=\bigotimes_{i=0}^{q-1}\mathcal H_{\text{Fock}} =\mathrm{Span}\bigl\{\lvert \mathbf n\rangle=\lvert n_0\rangle\otimes\cdots\otimes \lvert n_{q-1}\rangle\bigr\},$$9. For general $K$00, the labels $K$01 must be augmented by a counting number $K$02, yielding basis states $K$03; in the bosonic case, the multiplicities are linked to Gaussian polynomials (Dubus et al., 2024). This provides a symmetry-adapted indexing of fixed-excitation code spaces.

5. State determination, syndrome extraction, and measurement primitives

Characterizing multi-mode Fock-state codes requires both population readout and coherence-sensitive tomography. In trapped ions, a direct route is simultaneous blue-sideband spectroscopy on multiple modes. For a $K$04-mode motional state with Fock populations $K$05, the probability that all addressed ions remain spin-down after a blue-sideband pulse of duration $K$06 is

$K$07

Least-squares fits to these basis functions yield the full multi-mode Fock distribution, while displaced measurements combined with a discrete Fourier transform over displacement phases reconstruct the density matrix in a chosen cutoff. This protocol was experimentally verified on two- and three-mode motional states and produced a reconstructed fidelity $K$08 for the Bell-like state $K$09 using 16 displacement pairs (Jia et al., 2022).

A second trapped-ion primitive uses far-detuned multimode Jaynes–Cummings interactions. In the dispersive regime, a single spin accumulates a phonon-number-dependent Ramsey phase, leading in the Lamb–Dicke limit to

$K$10

A selective-decoupling scheme cancels the carrier AC-Stark phase while preserving the phonon-number-dependent dispersive phase. Experimentally, this enabled extraction of two-mode Fock-state distributions, parity-based filtering of two-mode motional states, and a nondestructive single-shot measurement of a single-mode Fock state via repeated filtering steps (Choi et al., 8 Jan 2026).

These two approaches are complementary. The multi-ion sideband protocol provides informationally complete tomography for offline benchmarking of state preparation and noise channels, whereas the single-spin SDR primitive supplies parity and number-modulo filters that are closer to runtime syndrome extraction in multi-mode bosonic encodings (Jia et al., 2022, Choi et al., 8 Jan 2026).

6. Implementations, resource generation, and outlook

The experimental landscape is now broad enough that multi-mode Fock-state codes are no longer purely formal. Trapped ions provide naturally multimode bosonic registers and long-lived spin ancillas; the trapped-ion tomography protocol is explicitly stated to extend to any system with Jaynes–Cummings-type interactions, including circuit QED, optomechanical systems, and hybrid spin–mechanical architectures (Jia et al., 2022). In superconducting cavity QED, a weakly coupled transmon can be Rabi-driven to induce strong sideband interactions on demand. Using a flute cavity with two high-$K$11 modes, deterministic Fock-state preparation was demonstrated up to $K$12 at operation times of less than $K$13 per photon, together with a single-photon SWAP in approximately $K$14 and a dual-rail Bell state (Karaev et al., 8 Apr 2026). For multi-mode Fock codes, this directly addresses the usual trade-off between high cavity isolation and fast state transfer.

Optical resource generation remains relevant as well. Pulsed SPDC has produced multi-photon Fock states up to three photons in well-defined spatial-temporal modes synchronized with a classical clock, with the three-photon state generated at a rate of one per second and characterized by homodyne tomography over 12 hours (Cooper et al., 2012). More elaborate linear-optical and nonlinear constructions generate $K$15-mode $K$16-photon NOON states with arbitrary photon number, using either coherent-state inputs with Fock-state filtration or $K$17-photon Fock-state inputs with single-photon coincidence detection (Zhang et al., 2017). These states are not themselves robust bosonic memories, but they exemplify how highly structured multi-mode Fock superpositions can be synthesized.

A broader implication is that “multi-mode” need not imply an uncontrollably large effective support. For large-$K$18 waveguide-emitted multimode Fock states, orthogonal-mode decompositions can concentrate about $K$19 of the photons in one mode, $K$20 in two modes, and $K$21 in three modes at $K$22, while number-resolved measurements that do not distinguish those internal modes can still saturate the QFI in the relevant interferometric setting (Perarnau-Llobet et al., 2019). This suggests that practical code design may often proceed in an effective few-mode subspace even when the underlying bosonic state is nominally highly multimode.

The current subject therefore spans three scales simultaneously: fixed-$K$23 formalism and SSR-compliant encoding rules; concrete code families with explicit stabilizers and noise models; and platform-specific measurement and control primitives that realize state preparation, syndrome readout, and inter-mode transfer. The recent appearance of asymptotically good constant-excitation codes with bounded per-mode occupancy indicates that multi-mode Fock-state codes are not limited to small handcrafted examples, while pair-cat, GKP, and group-orbit constructions show that number-difference sectors, lattice subsystems, and finite-group symmetries supply distinct and technically mature routes to bosonic encoding (Elimelech et al., 16 Mar 2026, Albert et al., 2018, Shaw et al., 2022, Denys et al., 2022).