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arXiv:2509.16758

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

Aayush Vijayvargia, Ezra Day-Roberts, Onur Erten

Published Sep 20, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2509.16758

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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=4d=4d=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.
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.
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Monitored quantum circuits allow for unprecedented dynamical control of many-body entanglement. Here we show that random, measurement-only circuits, implementing the competition of bond and plaquette couplings of the Kitaev honeycomb model, give rise to a structured volume-law entangled phase with subleading L ln L liquid scaling behavior. This interacting Majorana liquid takes up a highly symmetric, spherical parameter space within the entanglement phase diagram obtained when varying the relative coupling probabilities. The sphere itself is a critical boundary with quantum Lifshitz scaling separating the volume-law phase from proximate area-law phases, a color code or a toric code. An exception is a set of tricritical, self-dual points exhibiting effective ( 1 + 1 ) d conformal scaling at which the volume-law phase and both area-law phases meet. From a quantum information perspective, our results define error thresholds for the color code in the presence of projective error and stochastic syndrome measurements. Published by the American Physical Society 2024
Quantum circuits offer a versatile platform for simulating digital quantum dynamics and uncovering novel states of nonequilibrium quantum matter. One principal example are measurement-induced phase transitions arising from nonunitary dynamics in monitored circuits, which employ midcircuit measurements as an essential building block next to standard unitary gates. Although a comprehensive understanding of the dynamics in generic circuits is still evolving, we contend that monitored quantum circuits give rise to robust phases of dynamic matter, which---akin to Hamiltonian ground-state phases---yield emergent universal behavior, which can be categorized based on circuit symmetries and spatial dimensionality. To illustrate this concept, we focus on measurement-only quantum circuits within symmetry classes BDI and D, which are measurement-only circuit adaptations of the paradigmatic Kitaev and Yao-Kivelson models, embodying particle-hole-symmetric Majorana fermions with or without time reversal. We establish a general framework (Majorana loop models) for both symmetry classes (in arbitrary spatial dimensions) to provide access to the phenomenology of the entanglement dynamics in these circuits, displaying both an area-law phase of localized Majorana loops and a delocalized, highly entangled Majorana liquid phase. The two phases are separated by a continuous transition displaying quantum Lifshitz scaling, albeit with critical exponents of two distinct, non-Hamiltonian universality classes. The loop model framework provides not only analytical understanding of these universality classes in terms of nonlinear sigma models but also allows for highly efficient numerical techniques capable of simulating excessively large circuits with up to 108{10}^{8}108 qubits. We utilize this framework to accurately determine universal probes that distinguish both the entangled phases and the critical points of the two symmetry classes. Our work thereby further solidifies the concept of emergent circuit phases and their phase transitions.
Majorana fermions arise in a broad range of systems in condensed matter physics, not only as a useful abstraction but also as emergent quasiparticles, e.g., at the boundaries of topological insulators. In this dissertation, we consider two distinct settings wherein Majorana fermions play a central role: (i) non-abelian Kitaev spin liquids and (ii) entanglement dynamics in monitored quantum circuitsThe Kitaev spin liquid is a paradigmatic model featuring non-abelian anyons and a chiral Majorana edge mode when time-reversal symmetry is broken. To this end, it presents an alluring platform for realizing topologically protected quantum information. However, experimental efforts to identify this putative phase have been hampered both by the difficulty of probing the charge neutral quasiparticles and by the presence of phonons and other deviations from the idealized toy model. Here we examine how judicious device design may suppress bulk phonon contributions to thermal transport, allowing for refined observation of transport mediated by the chiral Majorana edge mode. Moreover, we show that the bulk topological order can be revealed through universal features in the tunneling conductance across a point contact. Looking forward, we develop a roadmap for near-term experimental devices to demonstrate robust control of topologically protected qubits encoded in the bulk anyons.Much as Majorana models form the building blocks for symmetry-protected topological order, Majorana circuits provide a powerful framework for studying the entanglement dynamics and phases of monitored quantum circuits. Here we show that the entanglement dynamics of Majorana circuits can be mapped exactly to the statistical mechanics of classical loop models. Harnessing this framework, we undertake a precise numerical characterization of the universality of entanglement transitions in symmetry classes BDI and D in both (1+1) and (2+1)-dimensional circuits. Furthermore, we employ duality relations for loop models to construct an analytically solvable interacting circuit which features not only the usual percolation criticality but also Berezinskii-Kosterlitz-Thouless critical lines and a tricritical point with Ising universality. Lastly, by generalizing to colored loop models for Majorana circuits with classical control, we provide a blueprint for engineering even more diverse entanglement dynamics which may include Hilbert-space fragmentation and subdiffusive transport.
Quantum spin liquids (QSLs) are long-range entangled phases of matter and a natural setting for exploring how quantum correlations generate quantum complexity. Motivated by the emergence of magic, or nonstabilizerness, as a diagnostic of many-body complexity beyond entanglement, we study magic of QSLs by computing the stabilizer Rényi entropy (SRE) of the Kitaev honeycomb model. We derive a correspondence between Pauli strings and products of itinerant Majorana operators in the model's free-fermion description; this enables the sampling of SRE using an optimized algorithm for Gaussian states in systems with thousands of spins. Our results show that magic is largest in the gapless phase, where subleading volume-law corrections indicate nonlocal contributions, while in the gapped phases, SRE decreases in quantitative agreement with a perturbative expansion that we develop in the anisotropic limit. At the topological phase transition, SRE exhibits universal scaling dictated by critical exponents.