Encoding a qubit in an oscillator

Introduces the GKP (Gottesman-Kitaev-Preskill) code: encoding a qubit in the continuous Hilbert space of a harmonic oscillator using grid states in phase space. GKP states are simultaneous +1 eigenstates of displaced position and momentum operators, forming a lattice in phase space that can correct small displacement errors.

Key Results

• Qubit encoded in position/momentum grid states of an oscillator
• Corrects arbitrary small displacement errors in phase space
• Stabilizers: $S_q=e^{2i\sqrt{\pi }\hat{q}}$ and $S_p=e^{-2i\sqrt{\pi }\hat{p}}$
• Concatenation with surface code yields very high thresholds (~10% displacement error)

Links

• Journal: Physical Review A
• arXiv: quant-ph/0008040

Zoo Links

• gkp-codes
• cat-codes
• binomial-codes