Exploiting Translational Symmetry for Quantum Computing with Squeezed Cat Qubits

TL;DR

The utility of the unexplored translational symmetry of the squeezed cat codes is revealed, with applications to autonomous QEC, reliable logical operations, and readout in a non-orthogonal basis.

Abstract

Squeezed cat quantum error correction (QEC) codes have garnered attention because of their robustness against photon-loss and excitation errors while maintaining the biased-noise property of cat codes. In this work, we reveal the utility of the unexplored translational symmetry of the squeezed cat codes, with applications to autonomous QEC, reliable logical operations, and readout in a non-orthogonal basis. Using the basis under subsystem decomposition spanned by squeezed displaced Fock states, we analytically show that our autonomous QEC protocol allows for correcting logical errors due to photon loss, although the translational symmetry in one direction does not uniquely specify the code space. We also introduce the implementation methods of reliable logical operations by repeated alternation of a small-step unitary operation with a subsequent step of QEC onto the code space. Finally, by appropriately treating the non-Hermitian nature of the logical $Z$ operator, we also propose a circuit for precisely reading out the squeezed cat code in a non-orthogonal basis.

Figures (14)

• Figure 1: (a) Wigner function of the SC state $\ket{{\rm sq}^+_{\alpha, r}}$ (Eq. \ref{['eq:sq']}) and its translational symmetry $\hat{T}_0$. (b) Subsystem decomposition of the bosonic Hilbert space. The photon loss process (red arrows) changes the parity and partially generates an excitation in the gauge space. Our proposed QEC protocol (blue arrow) approximately dissipates back to the SC code space (shaded area), along with the parity change, and thus appropriately corrects the correctable part of the photon-loss error.
• Figure 2: The circuits for stabilizing the squeezed cat code. They consist of two conditional displacements on the composite system with an $X$-rotation on the ancillary qubit between them.
• Figure 3: Entanglement fidelity $F_e$ against squeezing parameter $r$ after photon-loss noise with $\kappa t = 0.01$ followed by 5 (red), 10 (blue), 15 (green) applications of the sharpen-trim protocol. $\bar{n} = 5$.
• Figure 4: The population of $\ket -_L\ket 0_G$ after applying one cycle of ST to $\ket +_L\ket 1_G$, for different values of $\alpha(=1.0,1.4,1.8,2.2,2.6,3.0)$ and $r(=1.3,1.5,1.7,2.0,2.3)$. The subsystem decomposition analysis expects it to be $2\pi^2 /\alpha'^2$, which is accurate for larger value of rescaled displacement $\alpha'=\alpha e^r$.
• Figure 5: An improved measurement circuit of $\hat{Z}_L$ for the squeezed cat code, corresponding to the trim-like Trotterization \ref{['eq:MeasTrim']}.