# Very small logical qubit (VSLQ) code[1]

## Description

The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)|0\rangle|0\rangle\), where the total Hilbert space is the tensor product of two qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords) and two oscillators. In the original proposal for implementation, the single logical qubit is given by the two lowest energy states of a circuit composed of two transmons coupled to two lossy resonators, but the resonators can also be thought of as qubits since only a few low-lying Fock states are used by the code.

## Protection

Passively protects against single photon loss.

## Encoding

Engineering a circuit made of two transmons and two oscillators coupled through three driven superconducting quantum interference devices (SQUIDs) results in passive stabilization of the logical states.

## Gates

Single logical qubit operations implemented by resonant physical qubit driving and phase shifting the SQUID drives.A CZ gate between two logical qubits implemented by coupling devices through another driven SQUID and applying a pulse to the coupling squid simultaneously with a single qubit operation on one of the logical qubits.

## Decoding

Logical qubit can be measured with physical qubit measurements along \(X\). Can be implemented by engineering a coupling of one of the qubits to a readout cavity via the interaction \(\sigma_x (a+a^\dagger)\) [2]. This results in an \(X\)-dependent shift of the readout cavity resonance which can be measured.

## Parent

- Hybrid qudit-oscillator code — VSLQ code yields a logical qubit out of two physical qubits and two oscillators.

## Cousin

- 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.

## Zoo code information

## References

- [1]
- E. Kapit, “Hardware-Efficient and Fully Autonomous Quantum Error Correction in Superconducting Circuits”, Physical Review Letters 116, (2016). DOI
- [2]
- 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

## Cite as:

“Very small logical qubit (VSLQ) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/very-small-logical-qubit