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Onset of thermalization of q-deformed SU(2) Yang-Mills theory on a trapped-ion quantum computer
Tomoya Hayata, Yoshimasa Hidaka, Yuta Kikuchi
TL;DR
Real-time simulation of genuinely nonabelian gauge dynamics remains limited, especially beyond low-dimensional or abelian settings. This work uses a Fibonacci-anyon truncation of q-deformed SU(2)_3 Yang-Mills theory on a trapped-ion quantum computer, explicitly implementing F-moves to study thermalization. The experiment simulates the dynamics and identifies idling errors as the dominant source, with dynamical decoupling and parallelized F-moves improving performance.
Problem
Real-time dynamics of genuinely nonabelian gauge theories has not been simulated in a simple toy model, despite the importance of such dynamics and the limitations of classical methods.
Method
The paper uses a simplified Fibonacci-anyon model preserving the nonabelian fusion rule of Wilson lines and simulates its real-time evolution with explicit F-move circuits on a trapped-ion computer.
Results
The experiment successfully simulates thermalization dynamics, with dynamical decoupling suppressing deviations from noiseless simulation and improving agreement with classical results.
Takeaways & Limitations
The results provide lessons for scaling F-move-based simulations of nonabelian lattice gauge theories toward regimes beyond the minimal truncation and classical reach.
Takeaways & Limitations
Scaling to more complex categories, larger systems, and dynamical matter will require more efficient F-move and Trotter-circuit decompositions.
Abstract
from arXiv · showhide
Nonequilibrium dynamics of quantum many-body systems is one of the main targets of quantum simulations. This focus - together with rapid advances in quantum-computing hardware - has driven increasing applications in high-energy physics, particularly in lattice gauge theories. However, most existing experimental demonstrations remain restricted to (1+1)-dimensional and/or abelian gauge theories, such as the Schwinger model and the toric code. It is essential to develop quantum simulations of nonabelian gauge theories in higher dimensions, addressing realistic problems in high-energy physics. To fill the gap, we demonstrate a quantum simulation of thermalization dynamics in a (2+1)-dimensional $q$-deformed $\mathrm{SU}(2)_3$ Yang-Mills theory using a trapped-ion quantum computer. By restricting the irreducible representations of the gauge fields to the integer-spin sector of $\mathrm{SU}(2)_3$, we obtain a simplified yet nontrivial model described by Fibonacci anyons, which preserves the essential nonabelian fusion structure of the gauge fields. We successfully simulate the real-time dynamics of this model using quantum circuits that explicitly implement $F$-moves. In our demonstrations, the quantum circuits execute up to 47 sequential $F$-moves. We identify idling errors as the dominant error source, which can be effectively mitigated using dynamical decoupling combined with a parallelized implementation of $F$-moves.
I. INTRODUCTION
The paper addresses the difficulty of simulating real-time nonabelian gauge dynamics, where classical methods face a sign problem and prior quantum demonstrations largely lacked genuine nonabelian structure. It introduces a trapped-ion simulation using a simplified Fibonacci-anyon model and explicitly implemented F-moves, with error mitigation for memory noise.
- Classical lattice methods face a sign problem that limits first-principles real-time studies of gauge theories, including thermalization dynamics relevant to quark-gluon plasma.
- Most prior quantum studies used abelian theories, lower-dimensional systems, or simplifications whose dynamics reduced to abelian or spin models.
- Nonabelian dynamics are implemented with Trotterized circuits based on F-moves acting as local unitary transformations between string networks.
- The experiment uses a simplified Fibonacci-anyon model that retains the essential nonabelian fusion rule of Wilson lines.
- Memory noise during F-move operations is identified as dominant and can be substantially mitigated by dynamical decoupling.
- The experiment demonstrates real-time nonabelian gauge-theory dynamics on noisy quantum hardware when circuits and error mitigation are carefully designed.
II. q-DEFORMED YANG-MILLS THEORY ON A HONEYCOMB LATTICE
The model starts from q-deformed Yang–Mills theory on a honeycomb lattice and restricts the gauge-field representations to Fibonacci anyons. Its gauge-constrained dynamics are implemented through F-moves, electric operators, and plaquette operators, while circuit cost motivates a modified prescription.
- The integer-spin sector of q-deformed SU(2)_3 contains two particle types, 1 and τ, mapped to qubit states and obeying the Fibonacci fusion rule τ × τ = 1 + τ.
- Allowed string-network states satisfy vertex fusion constraints that serve as Gauss-law constraints, and Hamiltonian evolution preserves them when the initial state satisfies them.
- F-moves are local unitary transformations between string networks, and their constraint-preserving property makes the Hamiltonian commute with the gauge constraints.
- Each original honeycomb plaquette term requires ten F-moves per Trotter step, producing excessive two-qubit-gate cost for present noisy devices.
- The authors modify the Hamiltonian by applying plaquette and electric interactions only to a graph subset, preserving gauge symmetry through a topologically equivalent decorated lattice.
- The plaquette operator is implemented as a controlled unitary, while electric evolution uses single-qubit Rz rotations after basis transformations.
A. Model
The model uses an 18-qubit (2+1)-dimensional system with periodic boundaries, evolved through a second-order Trotter decomposition and measured using a self-unitary Wilson loop.
- A. Model: The simulated system has 18 qubits, periodic boundary conditions, four plaquette terms, and six electric terms.
- A. Model: The model may represent a minimal (2+1)-dimensional system.
- A. Model: The Hamiltonian evolution is implemented with a second-order Trotter decomposition using time interval t, N steps, and dt = t/N.
- A. Model: The last half-step of the magnetic evolution can be removed because it commutes with the Wilson loop operator being measured.
- A. Model: The regular Wilson loop is normalized and transformed into a self-unitary operator satisfying (tr U_P)^2 = 1.
B. Gate decomposition
The gate decomposition implements Fibonacci-anyon F-moves with controlled operations, ancilla-assisted Toffoli decompositions, and circuit simplifications for the model’s thermalization dynamics.
- B. Gate decomposition: An F-move is implemented as a four-controlled gate representing the nontrivial Fibonacci-anyon unitary.
- B. Gate decomposition: Ancilla qubits decompose five-qubit and four-qubit Toffoli gates into smaller Toffoli operations for F-move implementation.
- B. Gate decomposition: Special F-moves with repeated arguments are obtained by setting a = d and use reduced controlled-gate constructions.
- B. Gate decomposition: The Wilson-loop implementation uses Pauli-Z representations for relevant operators and includes additional gates for measurement.
- B. Gate decomposition: Classical noiseless simulations identify fast thermalization at K = 0.5 around t = 3.0.
IV. EXPERIMENTS
Experiments on a trapped-ion processor combine circuit optimization, parallel F-move scheduling, and dynamical decoupling to simulate thermalization while suppressing errors from qubit idling.
- IV. EXPERIMENTS: The experiment uses Quantinuum’s H2-1 trapped-ion quantum computer, which provides 56 qubits and native single-qubit and two-qubit gates.
- IV. EXPERIMENTS: Dynamical decoupling inserts single-qubit pulses on idling qubits to reduce coherent-noise accumulation.
- IV. EXPERIMENTS: Circuit optimization reduces the first Trotter step from 680 to 50 ZZPhase gates after compilation and initial-state-aware simplification.
- IV. EXPERIMENTS: Parallel F-move scheduling reduces ZZPhase depth from 140 to 112 for two Trotter steps and from 271 to 238 for three steps.
- IV. EXPERIMENTS: Experimental dynamics follow thermalization, while dynamical decoupling suppresses deviations that appear in raw data at t = 3.0.
V. OUTLOOK
The study demonstrates nonabelian gauge-theory simulation with explicit F-moves and identifies circuit-depth reduction, dynamical decoupling, and improved decompositions as routes toward larger and more complex systems.
- V. OUTLOOK: The demonstrated model preserves Wilson-line nonabelian fusion rules and realizes real-time evolution through explicit F-moves.
- V. OUTLOOK: Memory noise from long multi-qubit controlled operations, rather than two-qubit gates, is identified as the dominant circuit error source.
- V. OUTLOOK: A modular variational circuit compresses individual plaquette or electric-term blocks, reducing optimization overhead and enabling reuse in larger systems.
- V. OUTLOOK: Extending to SU(2)_k or SU(3)_k with k > 3 may require qudits, deeper F-move circuits, and additional qubits for fusion multiplicities.
Appendix A: Measurement of the Wilson loop operator
The appendix describes measuring the Wilson loop operator tr U_P by changing to its eigenbasis, analogous to measuring Pauli X with a Hadamard gate. It also relates the regular Wilson loop to the measured expectation value.
- F-moves provide the basis transformation needed to measure tr U_P on the tadpole graph.The procedure measures tr U_P in the chosen basis, analogously to measuring Pauli X after applying a Hadamard gate.
- The regular Wilson loop is obtained as tr U_reg = (1 + tr U_P)/2.
- The measurement circuit uses qubits |b1⟩ and |b2⟩ associated with edges of the tadpole graph.
1. Explicit construction of the first Trotter step
The first Trotter step is constructed by explicitly evaluating electric-term diagrams and implementing the resulting operations as circuits on the lattice’s anyon qubits. Circuit simplifications reduce costly controlled operations, which is important for experiments on noisy devices.
- 1. Explicit construction of the first Trotter step: The F-move circuit on four faces eliminates the associated Toffoli gates.This removal is likewise described as crucial for experiments using noisy quantum devices.
- 1. Explicit construction of the first Trotter step: The first Trotter step computes electric-term diagrams explicitly before translating them into quantum circuits.The construction proceeds through diagrammatic evaluation followed by circuit implementation.
- 1. Explicit construction of the first Trotter step: A circuit acting on six edges reduces the naive twelve-Toffoli implementation to one controlled-controlled phase gate.The reduction is identified as crucial for experiments on noisy quantum devices.
2. Explicit construction of the general Trotter steps
The general Trotter construction separates magnetic and electric evolution and exploits the state reached after the first step to simplify later magnetic F-moves. Variational approximation and exact boundary F-moves are then combined to reduce circuit depth while retaining the intended evolution accuracy.
- 2. Explicit construction of the general Trotter steps: After the first Trotter step, equal labels allow the general F-move to be replaced by a simpler one for the second magnetic-evolution step.This initial-condition simplification cannot be reused for subsequent Trotter gates.
- 2. Explicit construction of the general Trotter steps: Magnetic evolution is expressed through F-move basis transformations surrounding an evolution operator, while electric evolution uses a separate operator.The evolution operator uses S for magnetic evolution and Z for electric evolution.
- 2. Explicit construction of the general Trotter steps: Variational circuits approximate shallow-time evolution because small time steps make much of the basis transformation nearly trivial.The approximation is retained when its difference from the original circuit is negligible compared with other errors such as Trotter error.
- 2. Explicit construction of the general Trotter steps: The first and last full F-moves are implemented exactly with ancilla qubits, while the entire electric evolution operator is approximated variationally.This combines exact treatment of the boundary transformations with a shallower approximation for electric evolution.