Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier

TL;DR

The paper constructs a family of non-Abelian self-correcting quantum memories in $D\geq 5+1$ spacetime dimensions by gauging twisted $\ ext{Z}_2^3$ higher-form SPT phases, yielding magic stabilizer codes with non-Pauli stabilizers. It shows thermal stability and self-correction via a probabilistic local cellular-automaton decoder, and demonstrates a fault-tolerant non-Clifford CCZ logical gate implemented through higher cup products in the 5D Cubic Code. A key technical breakthrough is achieving a code distance $d=O(n^{2/5})$, surpassing the Bravyi-König $O(n^{1/3})$ barrier for transversal CCZ in conventional topological codes. The Cubic Theory generalizes to non-Abelian TQFTs without particles, with a rich spectrum of loop and membrane excitations, including Borromean-type membrane braiding, and can be connected to twisted compactifications of color codes. The work lays groundwork for universality via code switching between Abelian 4D and non-Abelian 5D codes and suggests potential experimental paths, including ion-trap platforms, while highlighting future directions in deterministic decoders and fracton-like non-Abelian memories.

Abstract

We construct a family of infinitely many new candidate non-Abelian self-correcting topological quantum memories in $D\geq 5+1$ spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of $\mathbb{Z}_2^3$ higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group $\mathbb{D}_8$ gauge theory in (2+1)D. The simplest example includes a new non-Abelian self-correcting memory in (5+1)D with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability, and devise a probabilistic local cellular-automaton decoder. We also construct fault-tolerant non-Clifford CCZ logical gate using constant depth circuit from higher cup products in the 5D non-Abelian code. The use of higher-cup products and non-Pauli stabilizers allows us to get an $O(n^{2/5})$ distance overcoming the $O(n^{1/3})$ distance barrier in conventional topological stabilizer codes, including the 3D color code and the 6D self-correcting color code.

Figures (9)

• Figure 1: Braiding between $k$ dimensional and $k'$ dimensional excitations in $D$ spacetime dimension requires $k+k'+2=D-1$, where there are 2 dimensions from the radial and orbital directions. The two excitations are extended in extra orthogonal $k,k'$ directions not shown in the figure, respectively, so the two excitations appear as two points that braid with each other on the 2D plane. The blue excitation moves around the red one and braids with each other.
• Figure 2: (a): The cup product $\alpha_1\cup \beta_2$ evaluated on a 3d cube. It sums over the possible sequence of a 1d edge and a 2d square sharing a single vertex, starting at $(0,0,0)$ and ending at $(1,1,1)$. (b): Coboundary $d\alpha$ of a $\mathbb{Z}_2$ 1-cochain $\alpha$ on a 2d square is given by summing over $\alpha$ on edges bounding a square.
• Figure 3: The SPT Hamiltonian is obtained by conjugating the trivial SPT Hamiltonian $H^0$ by the SPT entangler $U$. The figure shows the $\mathbb{Z}_2^3$ SPT phase in (2+1)D, with $l=m=n=1$. The local Hamiltonian term involves the six CZ gates support at a pair of sites at the boundaries of a green dashed curve. The arrow in the dashed curve for $\text{CZ}_{j,k}$ specifies that the outgoing site has the qubit $Z^j$ and the ingoing one has the qubit $Z^k$. This SPT Hamiltonian corresponds to the expression in \ref{['eq:SPT_Hamiltonian']} with $l=m=n=1$ in terms of cup product.
• Figure 4: The process of gauging $\mathbb{Z}_2$ 0-form symmetry in (2+1)D (i.e., $l=m=n=1$). (a): We initially have a model with $\mathbb{Z}_2$ 0-form symmetry with charged matter fields $\lambda=(1-\tilde{Z})/2$ on each $(l-1)$-hypercubes, which are vertices when $l=1$. The first step is to add the $\mathbb{Z}_2$ gauge fields (blue qubits $\{X',Z'\}$) on $l$-hypercubes $s_l$, which are edges when $l=1$. The $\mathbb{Z}_2$ gauge field is expressed as $a'=(1-Z')/2$. (b): The next step is to impose the Gauss law to define the gauge transformation, and the physical Hilbert space $\mathcal{H}_{\text{phys}}$ is characterized by the gauge invariant states with $G=1$. (c): The gauge invariant operators are the ones commuting with the Gauss law. We define the gauge invariant Pauli operators $\{X,Z\}$, and new $\mathbb{Z}_2$ gauge field $a=(1-Z)/2$, which is related to $a'$ by a gauge transformation. (d): After minimally coupling the original $\mathbb{Z}_2$ symmetric Pauli Hamiltonian to gauge fields in a gauge invariant manner, the Hamiltonian is expressed in terms of these gauge invariant Pauli operators $\{X,Z\}$. This gauged Hamiltonian can be obtained by the map shown in the figure; a single Pauli $\tilde{X}$ is mapped to a product of $X$ due to the Gauss law, and the symmetric product of $\tilde{Z}$ (which is associated with $d\lambda$) is mapped to a single $Z$ (which is associated with the gauge field $a$), due to minimal gauge invariant coupling.
• Figure 5: Mapping of operators under the gauging operation in table \ref{['tab:gauging']} for $l=2$ on 3D cubic lattice. Here $s_1$ and $s_2$ represent the 1- and 2-dimensional hypercubes, i.e., an edge and a face respectively.