## Description

The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), where the total Hilbert space is the tensor product of two transmon qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords). Since the code is intended to protect against losses, the qutrits can equivalently be thought of as oscillator Fock-state subspaces.

In the original proposal for autonomous stabilization [1], the single logical qubit is given by the two lowest energy states of a time-dependent Hamiltonian acting on two transmon qutrits and two lossy oscillators.

## Protection

## Encoding

## Gates

## Decoding

## Realizations

## Parents

## Cousins

- Hybrid qudit-oscillator code — VSLQ decoder utilizes two ancillary oscillators.
- Quantum repetition code — Parts of the VSLQ codewords resemble the two-qubit phase-flip repetition code, though the code cannot correct phase errors. Unlike the phase-flip code, the VSLQ code can correct for single photon loss because it uses the second excited state in the construction, which remains distinct from the vacuum even after photon loss.

## References

- [1]
- E. Kapit, “Hardware-Efficient and Fully Autonomous Quantum Error Correction in Superconducting Circuits”, Physical Review Letters 116, (2016) arXiv:1510.06117 DOI
- [2]
- Z. Li et al., “Autonomous error correction of a single logical qubit using two transmons”, (2023) arXiv:2302.06707
- [3]
- N. Didier, J. Bourassa, and A. Blais, “Fast Quantum Nondemolition Readout by Parametric Modulation of Longitudinal Qubit-Oscillator Interaction”, Physical Review Letters 115, (2015) DOI

## Page edit log

- Victor V. Albert (2021-12-09) — most recent
- Jonathan Kunjummen (2021-12-07)

## Cite as:

“Very small logical qubit (VSLQ) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/very-small-logical-qubit