Error stabilized logical qubits in qudit generalizations of the monitored Kitaev model

TL;DR

We study monitored dynamics in qudit ($d=4$) generalizations of Kitaev models on honeycomb and square lattices, mapping measurement-only evolution to multi-flavor loop models and diagnosing entanglement phases via tripartite mutual information. The approach reveals three main phases: a topological area-law that protects two (or three, in AL-II) logical qubits, a critical phase with $S_A \sim L\ln L$, and a volume-law phase induced by two-site measurements that introduce Majorana interactions; single-site measurements stabilize the area-law, while two-site interactions can collapse it or create new topological-area-law regimes depending on lattice geometry. These results illustrate how local monitoring and simple perturbations can engineer and protect quantum information in Majorana-based spin liquids, with implications for error-resilient quantum dynamics in higher-dimensional qudit systems.

Abstract

Monitored dynamics in quantum circuits provide tunable platforms for the realization of novel non-equilibrium phases. Motivated by recent advances in monitored Kitaev circuits, we investigate the monitored dynamics of the qudit ($d=4$) generalizations of the Kitaev model on the honeycomb and square lattices. In the absence of additional perturbations, the measurement-only dynamics of these models map onto multi-flavor loop models and display either critical or area-law entanglement scaling. Magnetic field terms couple different flavors and when measured with sufficiently large probability, they enhance the stability of the area-law phase that hosts the logical qubits. In a circuit picture, these terms correspond to single-qubit measurements and can be interpreted as errors. We also examine the impact of two-qubit measurements that commute with the plaquette operator, which induce effective non-quadratic interactions between Majorana fermions. These interactions can drive a transition to a volume-law-entangled phase and, for sufficiently strong coupling, stabilize a distinct area-law phase with an additional logical qubit for the square lattice model. Our results reveal a rich interplay between quantum spin liquids and monitored circuit dynamics, highlighting new mechanisms for engineering and controlling entanglement phases in multi-flavor Majorana systems.

Figures (9)

• Figure 1: (a),(b) Schematic of our circuit models for square and honeycomb lattices. Two-qudit measurement checks act along bonds; for an $x-$bond a representative operator is $P_i^\sigma P_j^\sigma X_i^\tau X_j^\tau$. Single-site $S_i^\sigma$ measurements are indicated by black squares. (c) Circuit schematic: at discrete time steps, random single or two qubit checks are performed. (d) Square lattice phase diagram for $(p_x,p_y,p_z,p_I)$: tuning $p_h$ shifts the boundary between the critical and the area-law phase; beyond a threshold $p_h^\star$ the critical region vanishes and the entire tetrahedral exhibits area-law scaling with two logical qubits.
• Figure 2: (a) Measurement of plaquettes leads to flux purification where the dynamics of the density matrix only depends on the dynamics of the Majorana fermions. There are two flavors of Majorana fermions for the square lattice model represented by purple and orange circles. (b) Example measurements of both species of Majorana fermions as well as onsite $\sigma_z$ term, marked as boxes, and the resulting string configuration. (c) Worldlines corresponding to measurements in (b) for three labeled sites.
• Figure 3: (a)Dynamical purification from a fully mixed state for selected points in each of the three phases. All phases begin with an exponential decrease in entropy as the plaquettes are measured. After that, the critical phase quickly removes all remaining entropy. The area-law phase plateaus then descends to a constant value. The volume-law phase is similar but with a system size dependent final value. (b), (c) Bipartite and tripartite mutual information establishes a critical to area-law transition where curves cross as a function of $p_z$. (c) inset shows partitioning of the torus in four equal cylinders for mutual information calculations. Transition is marked by dashed line. (d, e, f) Subsystem entanglement entropy with cylinders of varying width: (d) In area-law (e) critical and (f) volume-law phases. Inset (e) data collapse of scaled entropy. Inset (f) shows data collapse of half system entanglement entropy showing $L^2$ scaling. All the data presented in the above panels is obtained for the square lattice model. The appropriate choice of parameters is made to display the representative properties of the respective phases.
• Figure 4: Stabilizers shared between regions $A$ and $C$ with trivial support on $B$ that contribute to the mutual information in the area-law phases for the honeycomb and square lattices respectively.
• Figure 5: Effect of single site term (a) critical phase (orange) and area-law (gray) in $p_x,p_y,p_z$ triangle for selected values of $p_h$ (b) phase diagram of radius vs $p_h$ in honeycomb lattice (c) critical phase sphere in $p_x,p_y,p_z,p_I$ tetrahedra (d) phase diagram of radius vs $p_h$ in square lattice.