In fault-tolerant quantum computing with the surface code, non-Clifford gates are essential for achieving universal computation. However, existing approaches—such as magic state distillation and code switching—are resource-intensive. In this work, we propose a new protocol that combines magic state preparation with code switching to implement logical non-Clifford operations.

Our approach starts with a specially prepared logical state in the Z4 surface code. Through a sequence of transformations involving gauging and anyon condensation, the system transitions through various topological codes, notably including the non-Abelian D4 quantum double model. This process ultimately produces a magic state in a condensed Z2 surface code, which enables the implementation of a logical T gate in the standard Z2 surface code through gate teleportation.

Related publications:

1. "Generating logical magic states with the aid of non-Abelian topological order",
   Sheng-Jie Huang, Yanzhu Chen,
   arXiv:2502.00998

The essential idea of the topological holography is to encode the generalized symmetry and symmetry charges into a topological order in one higher dimension. This topological order have many different names such as symmetry topological field theory (SymTFT) or symmetry topological order (SymTO). 

Together with Meng Cheng, we have applied the topological holography picture to study a variety of exotic quantum critical points and gapless phases in (1+1)d. These includes

• Phase transitions between SPT phases

• Symmetry enriched quantum critical points (transitions between a SPT and a symmetry breaking phase) 

• Deconfined quantum critical points

• Intrinsically gapless SPT phases

Many of these exotic quantum critical points and gapless phases can be characterized by the non-trivial responses in the symmetry defect operators and twisted partition functions.

We also developed a topological holographic picture for conformal boundary states of (1+1)d RCFTs. This picture allows us to uncover the nature of the gapped phases that corresponding to the conformal boundary states.

I have further generalized the topological holography picture to fermionic systems and studied exotic fermionic quantum critical points from this veiwpoint. 

Recently, my collaborators and I have generalized this picture to studied (2+1)d gapped and gapless phases with non-invertible symmetries. We further developed a general construction of the lattice models of (2+1)d gapped phases with non-invertible symmetries.  

What I like about this picture is that we can use the intuitions and techniques in the theory of topological orders to derive general statements about the critical points and gapless phases.

Related publications:

1. “(2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry”,
   Kansei Inamura, Sheng-Jie Huang, Apoorv Tiwari, Sakura Schafer-Nameki,
   arXiv:2506.09177

2. "Gapless Phases in (2+1)d with Non-Invertible Symmetries",
   Lakshya Bhardwaj, Yuhan Gai, Sheng-Jie Huang, Kansei Inamura, Sakura Schafer-Nameki, Apoorv Tiwari, Alison Warman,
   arXiv:2503.12699

3. "Fermionic quantum criticality through the lens of topological holography",
   Sheng-Jie Huang,
   Phys. Rev. B 111, 155130 (2025)

4. "Topological holography, quantum criticality, and boundary states",
   Sheng-Jie Huang and Meng Cheng,
   arXiv:2310.16878

Crystalline symmetry can protect non-trivial topological phases and leads to interesting higher-order boundary phenomana such as hinge modes or corner modes. I have developed a general theoretical framework, which we called a topological crystal approach, to systematically classify and characterize these phases. The essential idea is that any topological crystalline phase have a corresponding real-space crystalline stacking pattern of lower-dimensional topological states, which we systematically classified.

I have applied this approach to classify bosonic crystalline SPT phases and fermoinic topological crystalline insulators (TCIs). Based on this picture, I have also studied various different surface field theories for bosonic SPT phases with point group symmetry, and studied exotic crystalline SPT phases and surface theories in 4d space. Recently, I have further generalized the topological crystal approach to classify interacting Dirac semimetals. This framework has the advantage of having a clear physical picture, being applicable to bosonic and fermionic systems, easily allowing one to consider the effect of strong interactions, and study the boundary signertures and topological responses. 

Given the progress in classifying crystalline topological phases, a natural question is whether topological phenomena can be protected by the structure of quasicrystals. Building on the topological crystal framework, My collaborators and I have further generalize it to study quasicrystalline topological phases and their topological responses. We also discovered that there are intrinsically quasicrystalline phases, which have no crystalline analogs. This framework provides the theoretical foundation to understand quasicrystalline topological phases.

Related publication:

1. "Classification of Interacting Dirac Semimetals",
   Sheng-Jie Huang, Jiabin Yu, Rui-Xing Zhang,
   Phys. Rev. B 110, 035134 (2024)

2. “Effective field theories of topological crystalline insulators and topological crystals”,
   Sheng-Jie Huang, Chang-Tse Hsieh, Jiabin Yu,
   Phys. Rev. B 105, 045112 (2022) 

3. “Quantum many-body topology of quasicrystals”,
   Dominic V. Else, Sheng-Jie Huang, Abhinav Prem, Andrey Gromov,
   Phys. Rev. X 11, 041051 (2021)

4. “4D beyond-cohomology topological phase protected by C2 symmetry and its boundary theories”,
   Sheng-Jie Huang,
   Phys. Rev. Research 2, 033236 (2020)

5. “Bosonic crystalline symmetry protected topological phases beyond the group cohomology proposal”,
   Hao Song, Zhaoxi Xiong, Sheng-Jie Huang,
   Phys. Rev. B 101, 165129 (2020)

6. “Topological states from topological crystals”,
   Zhida Song, Sheng-Jie Huang, Yang Qi, Chen Fang, and Michael Hermele,
   Sci. Adv. 5, eaax2007 (2019)

7. “Surface field theories of point group symmetry protected topological phases”,
   Sheng-Jie Huang, Michael Hermele,
   Phys. Rev. B 97, 075145 (2018)

8. “Building crystalline topological phases from lower-dimensional states”,
   Sheng-Jie Huang*, Hao Song*, Yi-Ping Huang, Michael Hermele,
   Phys. Rev. B 96, 205106 (2017) [Editor’s suggestions]

9. “Topological phases protected by point group symmetry”,
   Hao Song, Sheng-Jie Huang, Liang Fu, and Michael Hermele,
   Phys. Rev. X 7, 011020 (2017)

Non-Abelian quasiparticles are fascinating excitations of quantum many-body systems, which lead to quantum information spread nonlocally throughout space and to exotic “braiding” properties different from those of familiar particles. While non-Abelian quasiparticles do occur in theoretical models in two spatial dimensions, one factor limiting the possibilities for their realization in 3D is the fact that fully mobile quasiparticles in 3D systems can only be bosons or fermions. In this work, we show this limitation is circumvented in so-called fracton phases of matter.

We introduce a class of gapped 3D fracton models, dubbed “cage-net fracton models,” which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility. Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional “flux strings” built out of pointlike excitations. Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton-ordered (equivalently, flux-string-condensed) phase, such as its ground-state wave function and spectrum of excitations. We further show that the ground-state wave function of such phases can be understood as a fluctuating network of extended objects—cages—and strings, which we dub a cage-net condensate. 

The second class of models that we introduced can be viewed as gauging certain subsystem symmetries in the d = 3 Abelian Dijkgraaf-Witten models. These models can support non-Abelian fractons for certain choices of non-Abelian gauge groups. I have also used tools in quantum information science such as entanglement entropy and entanglement spectra to characterize fracton orders and subsystem SPT phases.

Related publication:

1. “Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter”,
   Albert T. Schmitz, Sheng-Jie Huang, Abhinav Prem,
   Phys. Rev. B 99, 205109 (2019)

2. “Cage-Net Fracton Models”,
   Abhinav Prem*, Sheng-Jie Huang*, Hao Song, Michael Hermele,
   Phys. Rev. X 9, 021010 (2019)

3. “Twisted Fracton Models in Three Dimensions”,
   Hao Song, Abhinav Prem, Sheng-Jie Huang, M.A. Martin-Delgado,
   Phys. Rev. B 99, 155118 (2019)