Squeezed-vacuum bosonic codes

TL;DR

This work introduces a new family of rotation-symmetric bosonic quantum error-correcting codes formed from rotated squeezed-vacuum states, with logical states supported on photon numbers $n \equiv 2k \pmod{2m}$ and a code distance of $d=m$. The authors show how to prepare these codes using probabilistic 2-legged schemes or deterministic multi-legged schemes built from conditional rotations and squeezing, and they quantify performance with Knill–Laflamme violation, revealing a trade-off: larger $m$ improves loss tolerance but increases dephasing sensitivity. Numerical analysis confirms the expected loss-vs-dephasing trade-off and benchmarks against cat codes, demonstrating the practical advantages of the squeezed-vacuum family for hardware-ready bosonic error correction. They also discuss realizations in circuit QED and trapped-ion platforms, arguing that the requisite Gaussian operations and conditional controls are increasingly available and that the squeezed-vacuum codes offer a scalable, finite-energy route toward robust CV quantum information processing.

Abstract

We introduce a family of bosonic quantum error-correcting codes built as a rotation-symmetric superposition of squeezed vacuum states, which promise protection against both loss and dephasing noise channels. The robustness of these "squeezed-vacuum codes" arises from being arranged at evenly spaced angles in phase-space, and simultaneously in evenly spaced photon-number support $n \equiv {2k} \! \pmod {2m}$. We present simple preparation circuits: a two-legged code using a Hadamard-conditional-squeezing-Hadamard sequence on an ancilla qubit, and for general "$m$-legged" codewords using sequences of conditional rotations. The performance of these codes is evaluated against loss and dephasing noises using the Knill-Laflamme violation function and benchmarked against cat codes. As the number $m$ of squeezed-vacuum states in a code increases, the code exhibits improved loss tolerance at the cost of higher dephasing sensitivity. We outline implementations in circuit QED and trapped-ion platforms, where high-fidelity Gaussian operations and conditional controls are available or under active development. These results help establish squeezed-vacuum codes as practical, hardware-ready, members of the bosonic codes class.

Figures (4)

• Figure 1: Generation of multi-legged squeezed states: We present a new family of quantum error-correcting bosonic codes built from squeezed-vacuum states. Each codeword can be created by a sequence of single-qubit gates, conditional-rotations and post-selection. Each resulted bosonic state can be fed back to generate a codeword with a larger superposition of squeezed-vacuum states. Selecting different measurement results, generates different members of the family. In this example, the $8$-legged logical-$1$ codeword is generated.
• Figure 2: The family of multi-legged squeezed bosonic codes: Wigner functions wignerFunction of $m$-legged squeezed codes for $m\!=\!2\ldots8$ at fixed squeezing strength $r=1.5$. For each $m$, we show $2$ logical codewords. Increasing the number of legs increases $m$-fold rotation symmetry and protection against particle loss. The upper-left Wigner function was previously shown NICACIO20104385 but had not been analyzed there as a bosonic code.
• Figure 3: Generating scheme of multi-legged squeezed codes vs cat codes: Comparing a conventional circuit for generating the $2$-legged Schrödinger's cat state using a conditional-displacement gate $CD(\alpha)$ ( a1 ) and our $2$-legged squeezed code using a conditional-squeezing gate $CS(r;0,\frac{\pi}{2})$\ref{['eq:conditional-squeezing']} ( b1 ) . Wigner functions wignerFunction of the resultant states are shown in ( a2,b2 ) . The support of the codes in Fock space is also compared ( a3,b3 ) : While the 2-legged cat code has a number distance of $d{=}1$ ( a3 ) with respect to photon loss events, our code has a number distance of $d{=}2$, meaning that single-photon loss events map to disjoint subspaces ( b3 ) . The number distance increases linearly for both codes as the number of legs $m$ increases. Circuits drawn with quantikz2018tutorial.
• Figure 4: Analysis of error correction properties, comparing the squeezed- and cat- bosonic codes: Plotting $V_\text{KL}( \mathcal{N} )_{0,1}$ (\ref{['eq:KL_violation_formula']}) for loss noise and $V_\text{KL}( \mathcal{N} )_{+,-}$ for dephasing noise, over strengths $\gamma$. Exact KL conditions satisfactions occurs at $V_\text{KL}( \mathcal{N} ) = 0$ hence smaller values indicate better code performance. We compare our squeezed-vacuum codes to the established cat codes. Here the squeezing strength and the displacement magnitude are chosen so that both codes are compared with the same mean photons number $\bar{n}=2$ for $\ket{0_L}$ (See \ref{['eq:nk-mean']} in \ref{['sec:supp:code-derivation']}). One can observe that increasing the number of legs protects against loss better at the cost of worse protection against dephasing.