# 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 topological code — Local deformations of the surface code preserve its \(\mathbb{Z}_2\) topological order.

## Children

- Kitaev surface code — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
- Rotated surface code — Rotated surface codes can be obtained from surface codes via a constant-depth Clifford circuit.
- 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 \(XZXZ\).

## Cousins

- Dynamically-generated quantum error-correcting code — 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]
- Arpit Dua et al., “Clifford-deformed Surface Codes”. 2201.07802

## 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)

## Zoo code information

## 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