Quantum Error Correction with Superpositions of Squeezed Fock States

Abstract

Bosonic codes, leveraging infinite-dimensional Hilbert spaces for redundancy, offer great potential for encoding quantum information. However, the realization of a practical continuous-variable bosonic code that can simultaneously correct both single-photon loss and dephasing errors remains elusive, primarily due to the absence of exactly orthogonal codewords and the lack of an experiment-friendly state preparation scheme. Here, we propose a code based on the superposition of squeezed Fock states with an error-correcting capability that scales as $\propto\exp(-7r)$, where $r$ is the squeezing level. The codewords remain orthogonal at all squeezing levels. The Pauli-X operator acts as a rotation in phase space is an error-transparent gate, preventing correctable errors from propagating outside the code space during logical operations. In particular, this code achieves high-precision error correction for both single-photon loss and dephasing, even at moderate squeezing levels. Building on this code, we develop quantum error correction schemes that exceed the break-even threshold, supported by analytical derivations of all necessary quantum gates. Our code offers a competitive alternative to previous encodings for quantum computation using continuous bosonic qubits.

Figures (3)

• Figure 1: (a) Deviation from the KL condition versus the squeezing amplitude $r$ for different values of $n$. An exponential decay is observed, with $n=1$ exhibiting the best performance at large $r$. (b) Errors from the set $[\hat{I}, \hat{a}, \hat{n}, \hat{n}^2]$ acting on the logical subspace define the corresponding error spaces. The code and error spaces approximately satisfy the KL condition $K_{\text{er}}$, enabling recovery via QEC. (c) Wigner functions of the codewords (related by a $\pi/2$ rotation) and associated error states (also related by a $\pi/2$ rotation) at 8 dB squeezing ($r \approx 0.921$). As the $\hat{n}$ and $\hat{n}^2$ operators do not affect the parity of the codewords (as compared to $\hat{a}$), the resulting error states have nonzero overlap with the codewords and with each other. This, in turn, is the reason for the nonvanishing deviation of the KL condition [cf. Eq. \ref{['seri']}].
• Figure 2: The encoded bosonic mode resides in an infinite-dimensional Hilbert space, while the auxiliary system is a discrete-level system, such as a qutrit. The bosonic mode is initialized in an encoded state $|\psi_{\mathrm{L}}\rangle$, and the auxiliary system in its ground state $|g\rangle$. Noise operators $\hat{F}_i$, derived from the error set $E$, map the code space into approximately orthogonal error subspaces. In the autonomous QEC protocol, control operations $U_i$ conditionally restore the encoded state while exciting the auxiliary system. A reset operation $\hat{R}$ returns the auxiliary system to $|g\rangle$, completing a measurement-free QEC cycle. In the measurement-based protocol, a parity measurement identifies the error subspace: even parity invokes $\hat{U}_a$ in Eq. (S23) for direct recovery; odd parity triggers a two-step correction via $\hat{U}_1$ and $\hat{U}_2$. Iterating these cycles enables long-term protection of encoded quantum information.
• Figure 3: Mean fidelity of the encoded state—averaged over the entire error space—versus time under $8$ dB squeezing with a photon loss to dephasing rate ratio $\kappa / \kappa_{\phi} \approx 8.5$. $\tau_w$ represents the waiting time within a single QEC cycle and is much longer than the duration of the recovery unitary operations ($\tau_w \gg \tau_g$).