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arXiv:2610.34411

Bosonic Loss Codes with Bounded Occupation

Avraham Kreindel, Aryeh Lev Zabokritskiy

quant-phSubmitted Oct 25, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

We study Fock-state codes on $q$ bosonic modes, with total excitation $N$ and at most $B$ excitations per mode, that exactly correct every pattern of at most $δN$ photon losses, for $q$ proportional to $N$. For every fixed cap $B$ and every ratio $q/N>1/B$, we prove explicit achievable rates, positive for all sufficiently small $δ>0$, in logical qubits per excitation. At $B=2$ and $q=N$, the rate is positive for every $δ<0.1355$ and is about $0.281$ at $δ=0.1$, compared with the previously established exact-code qutrit rate of about $0.275$. We also give an explicit upper bound on the dimension of every such exact code, including degenerate codes. For the pure-loss channel truncated to at most $δN$ losses and renormalized, we obtain approximate codes with exponentially small worst-case error. When the loss probability is positive and at most $1/(B+1)$, they improve our positive exact-code rate bounds; at $B=2$, $q=N$, and loss probability $0.1$, they remain positive up to $δ\approx0.158$.

Comments
51 pages, 4 tables. Supplementary computational certificates: https://doi.org/10.5281/zenodo.22693414

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We study Fock-state codes on qqq bosonic modes, with total excitation NNN and at most BBB excitations per mode, that exactly correct every pattern of at most δNδNδN photon losses, for qqq proportional to NNN. For every fixed cap BBB and every ratio q/N>1/Bq/N>1/Bq/N>1/B, we prove explicit achievable rates, positive for all sufficiently small δ>0δ>0δ>0, in logical qubits per excitation. At B=2B=2B=2 and q=Nq=Nq=N, the rate is positive for every δ<0.1355δ<0.1355δ<0.1355 and is about 0.2810.2810.281 at δ=0.1δ=0.1δ=0.1, compared with the previously established exact-code qutrit rate of about 0.2750.2750.275. We also give an explicit upper bound on the dimension of every such exact code, including degenerate codes. For the pure-loss channel truncated to at most δNδNδN losses and renormalized, we obtain approximate codes with exponentially small worst-case error. When the loss probability is positive and at most 1/(B+1)1/(B+1)1/(B+1), they improve our positive exact-code rate bounds; at B=2B=2B=2, q=Nq=Nq=N, and loss probability 0.10.10.1, they remain positive up to δ≈0.158δ\approx0.158δ≈0.158.
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The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat-binomial-GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multiqubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a multiqudit code.
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