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Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata

This paper establishes a microscopic realization of non-invertible symmetries and SPT stacking for $\mathbb{Z}_p$ one-form symmetries in 3+1d as quantum cellular automata acting on a local operator algebra, revealing that the resulting fusion rules are refined by non-trivial QCAs that correspond to the central elements of the underlying twisted Witt group extensions.

Original authors: Kansei Inamura, Oskar Wojdel, Lukasz Fidkowski, Sakura Schafer-Nameki

Published 2026-07-27

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CC BY 4.0

Original authors: Kansei Inamura, Oskar Wojdel, Lukasz Fidkowski, Sakura Schafer-Nameki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic game of LEGO. For decades, physicists have been trying to figure out the rules of this game, specifically how the tiny building blocks (particles) stick together and how they can be rearranged without breaking the structure. One of the most exciting recent discoveries in this field is the idea of "non-invertible symmetries." Think of a normal symmetry like a mirror: if you look in it, you see a reflection, and if you look in the mirror again, you get back to yourself. You can reverse the process. But a non-invertible symmetry is like a magical blender. If you throw a strawberry and a banana into the blender, you get a smoothie. You can't un-blend the smoothie back into a whole strawberry and a whole banana. In the quantum world, these "blenders" are special operations that change the state of a system in a way that can't simply be undone.

To understand how these blenders work, scientists often look at "Quantum Cellular Automata" (QCAs). Imagine a giant grid of light switches, where every switch can be on or off, but they are connected by invisible rules. A QCA is a rulebook that tells you how to flip these switches in a pattern that spreads across the grid, like a wave of dominoes falling, but in a way that respects the speed limit of the universe (nothing moves faster than light). The big question this paper tackles is: what happens when you try to build these "un-undoable" blenders on a real, physical grid of quantum switches, rather than just in a smooth, continuous mathematical world?

This paper, written by Kansei Inamura, Oskar Wojdel, Lukasz Fidkowski, and Sakura Schäfer-Nameki, dives deep into this question for a specific type of symmetry called a "1-form symmetry" in a 3D world (plus time). They discover that when you try to build these quantum blenders on a lattice (a grid), the rules get a little messy. The smooth, perfect mathematical rules they expect from the "continuum" (the idealized, smooth universe) don't quite match up with the rules on the grid. Instead, the grid introduces a new ingredient: a specific type of quantum "entanglement" machine that acts like a hidden twist in the system.

The authors show that for a simple case (using a symmetry called $Z_2$ , which is like a switch that can be on or off), the group of these quantum operations forms a structure known as the "single-qubit Clifford group." In the language of quantum computing, this is the set of operations you can do on a single quantum bit (qubit) using standard gates like the Hadamard and Phase gates. They prove that the "blender" operation (called the Kramers-Wannier-Wegner duality) and the "entanglement" operation (called the Tsui-Wen entangler) mix together on the grid in a very specific way. When you combine them, they don't just return to the start; they create a "framing" effect, which is a non-trivial quantum machine that twists the system in a way that can't be undone by simple moves.

For more complex cases (using a symmetry called $Z_p$ where $p$ is an odd prime number), the story is slightly different. The authors find that the mixing between the blender and the entangler is less dramatic; the group structure splits apart more cleanly. However, they suggest that even here, there is a subtle "mixing" if you look closely at how these operations extend to the full system. They argue that the lattice version of these symmetries is refined by these QCAs, meaning the grid adds a layer of complexity that the smooth mathematical theory misses.

The paper doesn't just guess; they build these operations explicitly using local rules on a cubic lattice (a 3D grid of cubes). They show that the "blender" operator, which swaps electric and magnetic properties, and the "entanglement" operator, which stacks a special quantum phase, can be written down as concrete algorithms. They calculate exactly what happens when you combine them. For the $Z_2$ case, they find that combining the blender and the entangler in a specific sequence results in a "3-fermion" quantum machine, a known type of non-trivial QCA. For the odd prime cases, they find a similar but distinct mixing involving "Clifford" QCAs.

Crucially, the authors point out that on the grid, these operations don't just fuse together to give a simple "identity" (doing nothing) as the smooth theory might suggest. Instead, they fuse to give a "translation" (shifting the whole grid) plus a non-trivial QCA. This means the fusion rules on the lattice are "refined" by these QCAs. The paper suggests that this is the full picture for these symmetries on a lattice, matching a mathematical classification called the "graded Witt group." While they don't prove every single step with absolute mathematical rigor (some parts rely on reasonable physical assumptions about how quantum phases are classified), the evidence they provide is strong and consistent. They effectively show that the "un-undoable" symmetries of the quantum world, when built on a real grid, come with a hidden twist that only appears because the world is made of discrete blocks, not a smooth continuum.

Technical Summary: Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata

Problem Statement
The paper addresses the challenge of constructing a lattice framework for non-invertible symmetries, specifically those arising from duality operations and Symmetry-Protected Topological (SPT) stacking in 3+1 dimensions. While continuum field theory describes these symmetries via topological defects and fusion rules involving central extensions of modular groups (e.g., $SL(2, \mathbb{Z}_p)$ ), a rigorous microscopic realization on a lattice has been lacking. The central problem is to determine how the fusion rules of these non-invertible symmetries manifest on a spatial lattice, particularly whether they close strictly as in the continuum or if they mix with non-trivial Quantum Cellular Automata (QCAs) and lattice translations. The authors focus on systems with $\mathbb{Z}_p^{(1)}$ 1-form symmetries, where $p=2$ and $p$ is an odd prime.

Methodology
The authors employ a dual approach combining categorical field theory and explicit lattice constructions:

Categorical Framework (SymTFT): They utilize the Symmetry Topological Field Theory (SymTFT) perspective. For a $\mathbb{Z}_p^{(1)}$ symmetry in 3+1d, the relevant SymTFT is a 4+1d $\mathbb{Z}_p$ BF-theory. The non-invertible symmetries are realized as invertible topological interfaces in this bulk. The classification of these interfaces is mapped to the graded, syllepsis-twisted pointed Witt group, denoted $\text{Witt}_{\text{pt}}(\mathbb{Z}_p \oplus^c \mathbb{Z}_p, s)$ . This group captures the fusion of defects and the associated anyon theories living on codimension-one interfaces.
Lattice Construction: The authors construct these symmetries microscopically on a 3D cubic lattice.
- Setup: They place qubits (for $p=2$ ) or qudits (for odd $p$ ) on the faces of the lattice. The $\mathbb{Z}_p^{(1)}$ symmetry is generated by the product of Pauli $X$ operators around each cube.
- Operator Algebra: They define a local operator algebra $\mathcal{A}_{\mathbb{Z}_p^{(1)}}$ generated by operators commuting with the symmetry generators, modulo the ideal generated by the symmetry operators themselves ( $X_{\partial c} = 1$ ). This quotient algebra represents the constrained Hilbert space.
- QCA Implementation: They explicitly construct locality-preserving automorphisms (QCAs) on this quotient algebra. Key operators include:
  - The Kramers-Wannier-Wegner (KWW) duality ( $S$ or $S_p$ ), implemented as an automorphism $\alpha_{\text{KWW}}$ .
  - The Tsui-Wen SPT entangler ( $T$ or $T_p$ ), implemented as $\alpha_{\text{TW}}$ .
  - Non-trivial QCAs such as the framing QCA (related to the 3-fermion theory) and $\mathbb{Z}_p$ -Clifford QCAs.
Fusion Analysis: They compute the fusion rules (composition of QCAs) on the lattice and compare them directly with the continuum defect fusion rules derived in Section 2 and the Witt group calculations in Section 3.

Key Contributions and Results

Lattice Realization of Non-Invertible Symmetries: The paper provides the first explicit microscopic realization of the full non-invertible symmetry structure for $\mathbb{Z}_2^{(1)}$ and $\mathbb{Z}_p^{(1)}$ ( $p$ odd) symmetries as automorphisms of a constrained local operator algebra.
Mixing with QCAs: A primary result is that lattice fusion rules differ from continuum fusion rules by non-trivial QCAs and lattice translations.
- For $p=2$ : The group of locality-preserving automorphisms modulo finite-depth circuits and translations is identified as a $\mathbb{Z}_8$ central extension of $PSL(2, \mathbb{Z}_4) \cong S_4$ . This group is isomorphic to the single-qubit Clifford group. The relation $(ST)^3 = Y$ in the continuum (where $Y$ is a purely gravitational TQFT) becomes an identity of $2 \times 2$ unitaries on the lattice, where the "central" element $Y$ corresponds to the semion QCA (order 8). Specifically, $(ST^2)^4$ on the lattice yields the 3-fermion QCA (framing QCA) rather than the identity.
- For odd primes $p$ : The group of automorphisms splits as $SL(2, \mathbb{Z}_p) \times \text{Witt}_{\text{pt}}$ (where $\text{Witt}_{\text{pt}}$ depends on $p \pmod 4$ ). However, the authors argue that the splitting is not canonical when requiring a unique assignment of Witt classes to QCAs extendable to the full tensor product algebra. The relation $(S_p T_p)^3 = Y_p$ in the continuum manifests on the lattice as a mixing with a $\mathbb{Z}_p$ -Clifford QCA ( $\tilde{\beta}_p$ ) and lattice translations.
Witt Group Correspondence: The authors demonstrate a precise match between the categorical calculation of the graded Witt group $\text{Witt}_{\text{pt}}(\mathbb{Z}_p \oplus^c \mathbb{Z}_p, s)$ $Witt_{pt} (Z_{p} \oplus^{c} Z_{p}, s)$ and the group of lattice QCAs.
- For $p=2$ , the graded Witt group is shown to be the single-qubit Clifford group (up to decoupled factors).
- For odd $p$ , the structure is $SL(2, \mathbb{Z}_p) \times W_p$ , where $W_p$ is the pointed Witt group of the relevant anyon theory.
Construction of Specific QCAs: The paper explicitly constructs the 3-fermion Kramers-Wannier-Wegner QCA and the $\mathbb{Z}_p$ -Clifford QCAs as compositions of the elementary KWW duality and SPT entangler operations. This shows that all non-trivial QCAs in the conjectured Witt classification for 3+1d can be generated from these elementary lattice operations.

Significance and Claims
The paper claims to establish a rigorous bridge between the continuum description of non-invertible symmetries (via topological defects and modular groups) and their microscopic lattice realization.

Refinement of Fusion Rules: The authors demonstrate that on the lattice, the fusion rules of non-invertible symmetries are "refined" by QCAs. While continuum defects may fuse to the identity (or a TQFT coefficient that acts trivially on local operators), the corresponding lattice automorphisms fuse to a non-trivial QCA (e.g., the semion or 3-fermion QCA) plus a translation.
Classification of Symmetric QCAs: The work supports the conjecture that the classification of symmetric QCAs is given by the graded pointed Witt group $\text{Witt}_{\text{pt}}(\mathbb{Z}_p \oplus^c \mathbb{Z}_p, s)$ . The lattice calculations provide evidence that this group correctly captures the mixing of duality symmetries with invertible phases (QCAs).
Modesty on Proofs: The authors note that some arguments rely on physically motivated assumptions, such as the completeness of the classification of $\mathbb{Z}_2^{(1)}$ SPT phases and the assumption that any two entanglers of the same phase are related by a symmetric finite-depth circuit. They state that while they provide strong evidence and precise assumptions under which their claims hold, some steps (like the rigorous proof that $(ST)^3$ corresponds exactly to the semion QCA) rely on these assumptions rather than a fully rigorous mathematical proof of the QCA's explicit form.

In summary, the paper identifies the group of locality-preserving automorphisms of $\mathbb{Z}_p^{(1)}$ -symmetric algebras as a central extension of the modular group by a Witt group of QCAs, explicitly realizing this structure on a cubic lattice and showing how continuum fusion relations are modified by non-trivial QCAs in the lattice setting.

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