# Transverse-field Ising model (TFIM) code[1]

## Description

A 1D translationally invariant stabilizer code whose encoding is a constant-depth circuit of nearest-neighbor gates on alternating even and odd bonds that consist of transverse-field Ising Hamiltonian interactions. The code allows for perfect state transfer of arbitrary distance using local operations and classical communications (LOCC).

## Protection

Code distance is 1 for open boundary conditions similar to a repetition code, and 3 for periodic boundary conditions with an encoding circuit depth of 4.

## Encoding

1D geometrically local constant-depth brickwork circuit of nearest-neighbor gates on alternating even and odd bonds. Gates are generated by interaction terms of the transverse-field Ising Hamiltonian.

## Parents

## Cousins

- Majorana stabilizer code — The TFIM code stabilizers can be expressed in terms of Majorana operators.
- Quantum repetition code — When written in the computational basis, the phase-flip and TFIM codewords are superpositions of qubit states of fixed total parity. The superposition is equal for the phase-flip code, whereas some states appear with a \(-1\) coefficient for TFIM code. However, the TFIM code can be encoded in constant depth.

## References

- [1]
- Y. Hong et al., “Quantum error correction in a time-dependent transverse-field Ising model”, Physical Review A 106, (2022) arXiv:2205.12998 DOI

## Page edit log

- Victor V. Albert (2022-05-20) — most recent

## Cite as:

“Transverse-field Ising model (TFIM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/tfim