# 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

Depolarizing noise: the threshold under ML decoding corresponds to the value of a critical point of the weight-two (two-body) two-dimensional random-bond Ising model (RBIM) on the Nishimori line [1–3]. Utilizing this statistical mechanical mapping yields a phase diagram for a CDSC.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 [4]. Local Clifford deformations preserve this topological order.

## Children

- Kitaev surface code — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
- 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 — The XZZX surface 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 QECC — To create CDSCs, a dynamical process is applied on top of the surface code [1].
- Random stabilizer code — Many useful CDSCs are constructed using random Clifford circuits.
- Asymmetric quantum code — Random Clifford deformation can improve performance of surface codes agaisnt biased noise [1,5].
- 3D lattice stabilizer code — Applying Clifford deformations to various 3D stabilizer codes, including the 3D surface code, 3D color code, X-cube model code, the SFSL code, yields a \(50\%\) code capacity threshold under infinitely biased Pauli noise [5].

## References

- [1]
- A. Dua et al., “Clifford-Deformed Surface Codes”, PRX Quantum 5, (2024) arXiv:2201.07802 DOI
- [2]
- H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
- [3]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [4]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971) DOI
- [5]
- E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 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