Tuning entanglement phases and topological memory in the measurement-only Kitaev model with single and multi-qubit checks

TL;DR

The paper addresses how measurement-only dynamics in the Kitaev honeycomb model realize distinct entanglement phases and topological memory by introducing parity-check operators that either do or do not commute with plaquette fluxes. The authors deploy a stochastic circuit with bond measurements augmented by single-, three-, and four-qubit terms, analyzed via tripartite mutual information $I_3$ and entanglement entropies, within stabilizer (Clifford) simulations. They find that a single-qubit, magnetic-field-like term $p_h$ induces a volume-law phase after removing flux protection and eventually leads to a trivial state; a commuting three-qubit term $p_3$ broadens the critical-law regime by preserving flux sectors; and a commuting four-qubit term $p_4$ yields a distinct volume-law phase that coexists with topological memory. Together, these results reveal a rich, operator-algebra–driven phase diagram for the measurement-only Kitaev model and offer guidelines for engineering dynamical quantum memory via tailored measurements.

Abstract

Quantum circuits provide an emerging controllable platform to realize novel dynamical non-equilibrium phases including topologically ordered states. The Kitaev model has become a cornerstone of quantum magnetism due to its quantum spin liquid ground state and rich phase diagram. The Kitaev model has also been treated in the monitored circuit setting, giving rise to topological area-law and critical-law entanglement entropy phases. In this article, we study the evolution of its phase diagram under the addition of new terms, motivated by their effects in the Kitaev model. We find that a single-qubit term, analogous to a magnetic field, leads to a trivial state in the high field limit, but with an additional intermediate volume-law phase. A three-qubit operator that commutes with the flux operators has the opposite effect: it stabilizes the critical-law phase against the short ranged area-law entanglement. We also employ a four-qubit plaquette commuting operator that simultaneously measures two opposite identical-type bonds on a plaquette. This generates a distinct volume-law phase and preserves the plaquette fluxes and associated topological order, yielding extensive entanglement while coexisting with the topological memory characteristic of the area-law phase. We quantitatively locate phase boundaries using stabilizer (Clifford) simulations together with tripartite mutual information and entanglement entropy measures. Our results highlight the rich phase diagram accessible from the measurement-only Kitaev model as well as suggesting rules relating the newly added operators to the phases they promote.

Figures (9)

• Figure 1: Schematic of our circuit model. (a) Two-qubit operators act along bonds (red, green, blue) with single-qubit operator (black). The plaquette operator is marked by $W_p$ with its corresponding sites $i-n$. (b) Circuit schematic: at each timestep operators are chosen at random to be measured. Depending on the model, these can have weights one through four. (c) At each site three three-qubit operators can be defined (d) Four-qubit operators are created by a pair of same flavor opposite bonds within a plaquette.
• Figure 2: Phase diagram of $p_h,p_z$ (note logarithmic axis for $p_h$) showing area-law, volume-law, critical, and trivial phases.
• Figure 3: Fraction of plaquettes in span of stabilizer generators as a function of $p_h$ for selected $p_z$. Inset shows data near area/volume-law transition for $p_z=0.7$, with transition point marked by vertical line.
• Figure 4: Phase diagram in the presence of the three-qubit operator. The critical-law region expands as a function of $p_3$. (a) Tetrahedron representing the full parameter space. (b) Normalized radius of the circular phase boundary plotted as a function of $p_3$.
• Figure 5: (a) Tetrahedral phase diagram of the four-qubit operator. The critical-law region in the $p_4=0$ plane turns to a volume-law region. (b) Normalized radius of the circular phase boundary plotted as a function of $p_4$.