Measurement Quantum Cellular Automata and Anomalies in Floquet Codes

TL;DR

This paper develops a rigorous framework for the dynamics of quantum information under periodic Pauli measurements, introducing locally reversible measurement cycles (LRMC) and a broad measurement-QCA (MQCA) formalism. It proves that one-dimensional MQCA flow is always integer-valued, while two-dimensional topological LR cycles carry a ${\mathbb{Z}}_2$ bulk invariant that obstructs trivial boundaries. Through concrete examples—the Wen plaquette translation and the Hastings–Haah honeycomb code—the authors reveal boundary anomalies realized as Majorana-chain-like edge algebras and show how period doubling can gap or neutralize these flows. The work connects measurement dynamics to topological phases and boundary phenomena, offering a robust language for classifying Floquet codes and guiding fault-tolerant edge engineering. It also lays groundwork for extending to qudits and fermionic systems, with potential implications for error thresholds and Floquet-code design.

Abstract

We investigate the evolution of quantum information under Pauli measurement circuits. We focus on the case of one- and two-dimensional systems, which are relevant to the recently introduced Floquet topological codes. We define local reversibility in context of measurement circuits, which allows us to treat finite depth measurement circuits on a similar footing to finite depth unitary circuits. In contrast to the unitary case, a finite depth locally reversible measurement circuit can implement a translation in one dimension. A locally reversible measurement circuit in two dimensions may also induce a flow of logical information along the boundary. We introduce "measurement quantum cellular automata" which unifies these ideas and define an index in one dimension to characterize the flow of logical operators. We find a $\mathbb{Z}_2$ bulk invariant for two-dimensional Floquet topological codes which indicates an obstruction to having a trivial boundary. We prove that the Hastings-Haah honeycomb code belongs to a class with such obstruction, which means that any boundary must have either nonlocal dynamics, period doubled, or admits anomalous boundary flow of quantum information.

Figures (16)

• Figure 1: (a) A schematic for quantum teleportion of a single qubit. Broken into steps (from bottom to top), we take an unknown qubit (left), create a Bell pair (cup), measure the unknown qubit with one of the qubit of the Bell pair (cap), and the remaining qubit (right) carries the original quantum information, up to Pauli corrections. (b) A schematic for a measurement circuit which implements a qubit-translation along an infinite chain. The circuit, acting on a chain of qubits along with ancilla between qubits, comprises of two (composite) steps. First the qubit on each site is teleported to its adjacent ancilla, followed by another teleportation of the ancilla to the next qubit. This results in a one-dimensional locally reversible measurement cycle which have a nontrivial MQCA index. In both subfigures, solid lines indicate flow of quantum information, the dashed lines are classical information which results from the measurements.
• Figure 2: Wen's plaquette model with a vertical boundary. (Left) A square lattice with a vertical boundary to its right. (Middle) Stabilizers of the model, which include both bulk terms and terms touching the boundary. (Right) Generators of the logical algebra, which are 2-body terms which lives on the boundary.
• Figure 3: Circuit picture for the teleportation protocol discussed in the main text. The first two measurements (read from the bottom) prepare bells states between parties $2n$ and $2n+1$. The second two measurements measures parties $2j-1$ and $2j$ in the Bell basis, which results in teleportation of the state on qubit $1$ to qubit $2n+1$. This circuit is reversible, but not locally reversible (as $n$ become large while fixing $\ell$).
• Figure 4: A locally reversible circuit which implements a translation on a chain of $n$ qubits with $n$ ancilla's. After one cycle, the logical operators at odd-numbered sites are transports two sites over to the right.
• Figure 5: Illustrative example of a blending in one dimension, between $({\mathcal{A}}_0, {\mathfrak{A}})$ (yellow) with $({\mathcal{B}}_0, {\mathfrak{B}})$ (blue). The lattice is divided into $A$ (left half) and $B$ (right half); the interface region $I$ has 4-site width. Stabilizers and circuit operators supported on $A \setminus I$ and $B \setminus I$ matches those of $({\mathcal{A}}_0, {\mathfrak{A}})$ and $({\mathcal{B}}_0, {\mathfrak{B}})$ respectively.

Theorems & Definitions (69)

• Proposition 2.1
• proof
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• proof
• Definition 3.1
• Definition 3.2