# Clifford-deformed surface code (CDSC)[1]

## Description

A generally non-CSS derivative of the surface code defined by applying a constant-depth Clifford circuit to the original (CSS) surface code. Unlike the surface code, CDSCs include codes whose thresholds and subthreshold performance are enhanced under noise biased towards dephasing. Examples of CDSCs include the XY code, XZZX code, and random CDSCs.

## Protection

As a stabilizer code, \([[n=O(d^2), k=O(1), d]]\).

## Fault Tolerance

In order to leverage the benefits of CDSCs into practical universal computation, we have to implement syndrome measurement circuits and fault-tolerant logical gates in a bias-preserving way.

## Code Capacity Threshold

A class of random CDSCs, parametrized by the probabilities \(\Pi_{XZ},~ \Pi_{YZ}\) of \(X\leftrightarrow Z\) and \(Y\leftrightarrow Z\) Pauli permutations, respectively, has \(50\%\) code capacity threshold at infinite \(Z\) bias.Certain translation-invariant CDSCs such as the XY code and the XZZX code also have \(50\%\) code capacity threshold at infinite \(Z\) bias.XZZX code and the \((0.5,\Pi_{YZ})\) random CDSCs have a \(50\%\) code capacity threshold for noise infinitely biased towards either Pauli-\(X\), \(Y\), or \(Z\) errors.

## Parents

- Qubit stabilizer code
- Abelian quantum-double stabilizer code — When treated as ground states of the code Hamiltonian, surface codewords realize \(\mathbb{Z}_2\) topological order, a topological phase of matter that also exists in \(\mathbb{Z}_2\) lattice gauge theory [2]. Local Clifford deformations preserve this topological order.

## Children

- XY surface code — XY code is obtained from the surface code by applying \(H\sqrt{Z}H\) to all qubits, thereby exchaning \(Z\leftrightarrow Y\).
- XZZX surface code — XZZX code is obtained from the rotated surface code by applying Hadamard gates on a subset of qubits such that \(XXXX\) and \(ZZZZ\) generators are both mapped to \(XZXZ\).
- Kitaev surface code — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.

## Cousins

- Dynamically-generated QECC — To create CDSCs, a dynamical process is applied on top of the surface code.
- Random quantum code — Many useful CDSCs are constructed using random Clifford circuits.

## References

- [1]
- A. Dua et al., “Clifford-deformed Surface Codes”, (2023) arXiv:2201.07802
- [2]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971) DOI

## Page edit log

- Victor V. Albert (2022-09-28) — most recent
- Aleksander Kubica (2022-09-28)
- Liang Jiang (2022-09-28)
- Steven T. Flammia (2022-09-28)
- Michael Gullans (2022-09-28)
- Victor V. Albert (2022-01-20)
- Arpit Dua (2022-01-19)

## Cite as:

“Clifford-deformed surface code (CDSC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/clifford-deformed_surface