Bosonic quantum error correction using squeezed Fock states

TL;DR

The paper addresses quantum error correction for bosonic systems subject to particle loss and dephasing by introducing a protocol based on squeezed Fock (SF) states and benchmarking it against squeezed Schrödinger cat (SSC) states. The authors formalize two evaluation measures—the Knill–Laflamme (KL) cost function and the Petz map fidelity—along with channel fidelity and an explicit optimal recovery via semidefinite programming, using Choi matrices to quantify performance. They demonstrate that the second SF state, with $|0_L;2\rangle=\hat S(0.57)|2\rangle$ and $|1_L;2\rangle=\hat S(-0.57)|2\rangle$, achieves competitive KL cost and Petz-bounded channel fidelity relative to SSC encodings for low noise rates $\gamma_1,\gamma_2$, while highlighting practical generation advantages of SF states. The work suggests SF-based QEC is a viable, experimentally favorable alternative to SSC—and potentially to GKP approaches—in mitigating particle loss and dephasing in bosonic channels, with future directions including entangled logical operations and multi-mode scaling.

Abstract

In the paper, we develop a bosonic quantum error correction code based on squeezed Fock states. We compare our proposed code with one based on squeezed Schrodinger's cat states using the Knill-Laflamme cost function and the Petz map fidelity. We demonstrate that squeezed Fock states are competitive in protecting information in a channel with particle loss and dephasing.

Figures (4)

• Figure 1: Schematic illustration of the effect of errors $\hat{K}_{j}$ on codewords $|0_{L}\rangle$ and $|1_{L}\rangle$. The encoding of quantum information in the two-dimensional subspace $\mathcal{H}_{d}$ of the Hilbert space $\mathcal{H}$ is shown as a black solid Bloch sphere. Codewords $|0_{L}\rangle$ and $|1_{L}\rangle$ lie respectively at the north and south poles of the Bloch sphere of the logical qubit. The action of the noise map (marked as red arrows) is shown as a red ellipsoid. In this case, codewords are no longer poles of the ellipsoid. The action of the recovery map (marked as blue arrows) is demonstrated as two blue spheres, the radii of which correspond to the long and short axes of the red ellipsoid. Between the two spherical surfaces, there is a figure that corresponds to the optimal recovery procedure.
• Figure 2: Schematic representation of phase portraits of codewords $|0_{L}\rangle$ and $|1_{L}\rangle$, respectively: a) the second squeezed Fock state with squeezing parameters $r$ and $-r$; b) even and odd squeezed Schrödinger’s cat states with squeezing parameters $r$, when coherent states $\ket{\alpha}$ and $\ket{-\alpha}$ are spaced along the axis and squeezed along the orthogonal one; c) even and odd squeezed Schrödinger’s cat states with squeezing parameters $r$, when coherent states $\ket{\alpha}$ and $\ket{-\alpha}$ are spaced and squeezed along the axis.
• Figure 3: Dependence of the KL cost function on: a) the rate of particle loss error $\gamma_{1}$ for the set of errors $\lbrace \hat{I}, \hat{a}, \hat{a}^\dagger \hat{a}\rbrace$; b) the rate of dephasing error $\gamma_{2}$ for the set of errors $\lbrace \hat{I},\left(\hat{a}^\dagger \hat{a}\right)^2\rbrace$. The following quantum states were investigated (see Table \ref{['tab:state']}): 1) squeezed Schrödinger's cat states: $\alpha_{0.5}^{\|}$ (marked in pink color), $\alpha_{0.5}^{\perp}$ (marked in orange color), $\alpha_{1.0}^{\|}$ (marked in purple color), $\alpha_{1.0}^{\perp}$ (marked in brown color); 2) the second squeezed Fock state with the squeezing $r=0.57$ (marked in green color). The mean number of photons in the considered states is $\langle n \rangle=3.83$.
• Figure 4: Dependence of the Petz map infidelity of a channel on: a) the rate of particle loss error $\gamma_{1}$ for the set of errors $\lbrace \hat{I}, \hat{a}, \hat{a}^\dagger \hat{a}\rbrace$; b) the rate of dephasing error $\gamma_{2}$ for the set of errors $\lbrace \hat{I},\left(\hat{a}^\dagger \hat{a}\right)^2\rbrace$. The following quantum states were investigated (see Table \ref{['tab:state']}): 1) squeezed Schrödinger's cat states: $\alpha_{0.5}^{\|}$ (marked in pink color), $\alpha_{0.5}^{\perp}$ (marked in orange color), $\alpha_{1.0}^{\|}$ (marked in purple color), $\alpha_{1.0}^{\perp}$ (marked in brown color); 2) the second squeezed Fock state with the squeezing $r=0.57$ (marked in green color). The mean number of photons in the considered states is $\langle n \rangle=3.83$.