# Crystalline-circuit code[1]

## Description

Code dynamically generated by unitary Clifford circuits defined on a lattice with some crystalline symmetry. A notable example is the circuit defined on a rotated square lattice with vertices corresponding to iSWAP gates and edges decorated by \(R_X[\pi/2]\), a single-qubit rotation by \(\pi/2\) around the \(X\)-axis. This circuit is invariant under space-time translations by a unit cell \((T, a)\) and all transformations of the square lattice point group \(D_4\).

The input state to the circuit is taken to be a product stabilizer state with finite entropy density. If the input is translation-invariant, then this periodicity is preserved by the circuit at all later times, so the code is a quantum quasi-cyclic code with unit cell \(a\). The initial product state recurs after a time \(\tau(n)\) that is linear in \(n\) for \(n=a 2^k\), but is thought to be exponential for generic \(n\).

## Protection

The code protects against Pauli errors. The circuit composed of iSWAP and \(R_X[\pi/2]\) gates on the square lattice is a “good scrambler” with non-fractal operator spreading and thus behaves like a random circuit in that regard, motivating the use of contiguous code distance as a proxy for code distance.

For the \(D_4\) example above, the unit cell \(a=2\), and the initial product group is chosen to have code rate \(1/2\). The parameters of the code are \([[n, n/2, d(t)]]\), and the contiguous code distance [2] grows linearly before saturating at \(O(n)\).

Selecting the code defined by the stabilizer group at the time when the contiguous distance is maximized and subjecting it to random erasures, an optimal threshold of \(1/4\) is achieved for a subset of system sizes [3][4]. The subthreshold scaling is competitive with random codes, which obey the random matrix theory ansatz [4].

## Encoding

## Parents

## Child

- Floquet code — Floquet codes are crystalline-circuit codes with non-commuting measurements in 2D.

## Cousins

- Random-circuit code — Crystalline-circuit codes can be thought of as random-circuit codes with symmetries.
- Monitored random-circuit code — Projective measurements can be included in crystalline-circuit codes in a spacetime translation-invariant fashion, turning such codes into monitored crystalline-circuit codes. However, the unit cell of measurements must be large enough to avoid purification.

## References

- [1]
- Grace M. Sommers, David A. Huse, and Michael J. Gullans, “Crystalline Quantum Circuits”. 2210.10808
- [2]
- S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009). DOI; 0810.1983
- [3]
- N. Delfosse and G. Zémor, “Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel”, Physical Review Research 2, (2020). DOI; 1703.01517
- [4]
- M. J. Gullans et al., “Quantum Coding with Low-Depth Random Circuits”, Physical Review X 11, (2021). DOI; 2010.09775

## Page edit log

- Victor V. Albert (2022-10-19) — most recent
- Grace Sommers (2022-10-19)

## Zoo code information

## Cite as:

“Crystalline-circuit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/crystalline_dynamic_gen