Holstein Primakoff spin codes for local and collective noise

TL;DR

This work introduces Holstein-Primakoff spin codes (HP spin codes), a framework that imports bosonic codes into permutation-invariant spin ensembles under the Holstein-Primakoff approximation. HP spin codes inherit the error-correcting properties of their bosonic counterparts for collective spin errors and automatically exhibit robustness to local spin decoherence, with local noise effectively mapping within or between nearby total-spin irreps in a self-similar manner. A key contribution is a measurement-free local error recovery protocol (MFLER) using a collective SWAP gadget, which coherently converts local errors into correctable collective ones without syndrome measurements. The paper exemplifies the construction with spin-GKP, spin-cat, and spin-binomial codes, analyzes their performance under symmetric and asymmetric local noise, and discusses practical avenues for fault-tolerant quantum information processing in platforms with dominant collective interactions.

Abstract

Quantum error correction is essential for fault-tolerant quantum computation, yet most existing codes rely on local control and stabilizer measurements that are difficult to implement in systems dominated by collective interactions. Inspired by spin-GKP codes in PhysRevA.108.022428, we develop a general framework for Holstein-Primakoff spin codes, which maps continuous-variable bosonic codes onto permutation-symmetric spin ensembles via the Holstein-Primakoff approximation. We show that HP codes are robust to both collective and local-spin noise and propose an explicit measurement-free local error recovery procedure to map local noise into correctable collective-spin errors.

Figures (6)

• Figure 1: Collective-spin probability distributions for spin-GKP codewords, Eq. \ref{['eqn:spingkpdef']}, with $N=160$, $\delta=0.4$, and $\mathbf{T}=(5,5)$. (a) $J_x$-basis. (b) $J_y$-basis; note here that the distributions are nearly identical. (c) $J_z$-basis highlighting concentration near $\frac{N}{2}$, indicative of the HP regime. For the states shown, $\bra{{0}_L} \hat{J}_z \ket{{0}_L} = \tfrac{N}{2}-2.65$ and $\bra{{1}_L} \hat{J}_z \ket{{1}_L} = \tfrac{N}{2}-3$.
• Figure 2: Effects of local-symmetric depolarizing noise on states of $N=100$ spins. For a spin-GKP $\ket{{0}_L}$ state, (a) shows the total population in each spin-$J$ irrep after time $\gamma t$. Probability distributions for collective measurements in the $\hat{J}_y$ basis given (b) this spin-GKP state, (c) a SCS along $z$, and (d) the GHZ state in Eq. \ref{['eq:GHZ']}, respectively. Shown are initial (grey) and partially depolarized (red) distributions for $\gamma t = 0.025$, which is marked on subplot (a) by a dashed line.
• Figure 3: Self-similarity of HP spin-code states across spin-irreps under local-symmetric depolarizing noise. Probability distributions in each $J$-irrep for transverse collective spin observables $\hat{J}_x$ (a--c) and $\hat{J}_y$ (d--f) for three representative HP spin codes for $N=100$ spins: a spin-GKP code, a spin-cat code, and the smallest spin-binomial code. Each code is prepared in $\ket{{0}_L}$ with parameters tuned such that $\langle \hat{J}_z \rangle = J-2 = 48$. The state evolves under the local symmetric depolarizing channel, Eq. \ref{['eq:localsymmetricnoise']}, with noise strength $\gamma t = 0.025$, after which it is projected into various spin-irreps, and the corresponding marginal distributions are found. Note that at $\gamma t = 0.025$, the total probability is highest in the $J=49$ irrep, see Fig. \ref{['fig:irreps+dist']}(a), which is why the orange curves are above blue. For comparison, we also show in (g) a SCS along $z$ and in (h) a GHZ state, Eq. \ref{['eq:GHZ']}. While the SCS displays self-similarity as it is the canonical HPA state, the GHZ (a non-HP state) does not---the features arising from interference fringes that are present in the $J=50$ irrep are entirely washed out in lower irreps.
• Figure 4: Effects of a spin-dependent depolarizing map, Eq. \ref{['eq:asym_noise_profile']} with $\zeta = 1$ and varying $p_0$ over $N=60$ spins. The distribution of $p_n$ is shown in the inset plot for $p_0 = 1$. $\hat{J}_y$ probability distributions are shown in (a) for a spin-GKP $\ket{{0}_L}$ and in (b) for a GHZ state.
• Figure 5: Idealized MFLER for spin-GKP codes. Fidelity with the initial spin-GKP state $\ket{ {0}_L }$ for $N = 60$ after it undergoes symmetric local depolarizing noise of strength $\gamma t$. The red curve shows fidelity to the initial state without MFLER, and the blue curve shows the fidelity after MFLER that transfers population in the damaged state from lower irreps back to the symmetric subspace using the SWAP gadget in \ref{['eq:SWAPgadget']}.