Implementation-independent sufficient condition of the Knill-Laflamme type for the autonomous protection of logical qudits by strong engineered dissipation

Autonomous quantum error correction utilizes the engineered coupling of a quantum system to a dissipative ancilla to protect quantum logical states from decoherence. We show that the Knill-Laflamme condition, stating that the environmental error operators should act trivially on a subspace, which then becomes the code subspace, is sufficient for logical qudits to be protected against Markovian noise. It is proven that the error caused by the total Lindbladian evolution in the code subspace can be suppressed up to very long times in the limit of large engineered dissipation, by explicitly deriving how the error scales with both time and engineered dissipation strength. To demonstrate the potential of our approach for applications, we implement our general theory with binomial codes, a class of bosonic error-correcting codes, and outline how they can be implemented in a fully autonomous manner to protect against photon loss in a microwave cavity.

Figure 1

Schematic of the density matrix, Lindblad jump operators, and noiseless subsystem symmetry. The total Hilbert space is divided into $d\times d$ arrays of $d^2$ large blocks. The logical code space $\mathcal{C}$ is formed by the cross-striped areas. The arrows represent dissipative jump operators and the gray-shaded areas represent the space of corrupted states, connected via Lindblad operator $f$ [green dashed arrow, Eq. (A1)] to the code states. Engineered dissipation $F_{\mathrm{eng}}$ [blue double-line arrow, Eq. (5)] pumps the corrupted states back to the code space and in consequence each large block supports a steady state $U_{\mu }{\rho }_{0,\mathrm{st}}{U}_{\nu }^{†}$, and ${\mathcal{S}}_{\mathrm{ccs}}$ forms a noiseless subsystem. In addition, $U_{\mu }$ intertwines between ${\mathcal{S}}_0$ and ${\mathcal{S}}_{\mu }$ and is defined in Eq. (A4). The remaining intrinsic dissipation $\tilde{f}$ [red-dotted arrow, cf. Eq. (A1), for clarity only one such process is shown] breaks this noiseless subsystem symmetry and induces an undesired error.

Figure 2

Averaged fidelity for various choices of $|\mathrm{\Phi }\rangle$ in (9). Fidelity is averaged over six initial input logical states $|{\psi }_{\mathrm{in}}\rangle =|W_0\rangle$, $|W_1\rangle$, $\left(\right|W_0\rangle \pm |W_1\rangle )/\sqrt{2}$, and $\left(\right|W_0\rangle \pm i|W_1\rangle )/\sqrt{2}$. The solid gray line represents the fidelity of the lowest-energy uncorrected physical qubit (i.e., $|W_0^{\mathrm{phys}}\rangle =|0\rangle$ and $|W_1^{\mathrm{phys}}\rangle =|1\rangle$) for single photon loss $D\left(a\right)$ with decay rate $\gamma$. The other data show fidelities of the autonomously corrected binomial code with engineered dissipation strength $0\le M\le 1000$ under the same loss rate $\gamma$. The engineered jump operators are given in Eq. (9), where $|\mathrm{\Phi }\rangle$ was chosen as shown next to the sublabels (a)–(c).