Local Thermalization of SU(2) Lattice Gauge Fields on Quantum Computers

The gist: The study reports simulation results on state-of-the-art IBM quantum computers for the local thermalization dynamics of a (2+1)-dimensional SU(2) pure gauge theory on long plaquette chains, demonstrating the feasibility of such studies for chaotic quantum systems on current noisy platforms

System and Model Setup

The research investigates the local thermalization dynamics for a minimally truncated SU(2) pure gauge theory on linear plaquette chains using IBM quantum computers The simulation uses up to 151 plaquettes, and the study focuses on tracking the time dependence of the entanglement spectrum, R´enyi-2 entropy and anti-flatness on small subsystems The Hamiltonian used is a Kogut-Susskind Hamiltonian mapped onto a one-dimensional spin model with local Hilbert space truncated to the two lowest representations j ≤ jmax = 1/2 The system is modeled as a chaotic quantum system satisfying the eigenstate thermalization hypothesis (ETH) at any nonzero lattice coupling constant for sufficiently large systems

Simulation Methodology

The time evolution is implemented using Trotterization to evolve the state from a strong-coupling vacuum state, specifically ψ(t = 0)⟩ = ↓↓ · · · ↓↓⟩ The time evolution is performed via the formula ψ(t)⟩ = Y(Nt/k=1) e−iHδt!ψ(t = 0)⟩, where δt = t/Nt To study entanglement, the paper employs subsystem state tomography to construct the reduced density matrix ρA using measurements of Pauli operators in the X, Y, and Z eigenbases The entanglement spectrum is then obtained by diagonalizing ρA, and indicators like R´enyi-2 entropy S(2)A = − ln[Tr(ρ 2A)] and anti-flatness FA = Trρ 3A − Trρ 2A squared are calculated from this spectrum

Hardware Performance and Error Mitigation

The simulation utilizes various error mitigation tools, including dynamical decoupling (DD), Pauli twirling (PT), and operator decoherence renormalization (ODR) DD suppresses decoherence error by sending a sequence of rapid microwave pulses, while PT converts coherent errors into incoherent errors ODR is applied as a post-processing technique to mitigate these errors The gate counts for one Trotter step show that the total gate depth is independent of chain length for N ≤ 101, but it increases sharply for N = 133 and 151 due to connectivity overhead on the hardware

Key Findings on Thermalization Dynamics

The results show that for system sizes smaller than N = 129, there is good agreement between error-mitigated quantum computing results and classical MPS simulator results For N = 101, the R´enyi-2 entropy S(2)A approximately reaches a plateau by t = 2/3, consistent with the stabilization of the dominant part of the entanglement spectrum around that time The anti-flatness FA develops a pronounced barrier-like peak during the period of rapid growth of the R´enyi-2 entropy, identifying the regime in which quantum computation may be particularly important for obtaining physically accurate results For larger systems like N = 151, hardware error accumulation exceeds mitigation capabilities, leading to large deviations and unphysical behavior Furthermore, the entanglement spectrum for N = 101 shows that the four lowest-energy states remain close to a straight line at t = 2/3 and t = 1, indicating the system is reaching towards a steady state

Conclusion

The paper concludes that while simulation of local thermalization dynamics is feasible on current noisy quantum platforms for smaller systems, the rapid increase in CNOT gates for larger chains exceeds the capability of implemented error mitigation techniques, leading to unphysical results Future work plans include implementing other methods to calculate local observables and testing other quantum hardware platforms

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IQuS@UW-21-122 Local Thermalization of SU(2) Lattice Gauge Fields on Quantum Computers Jiunn-Wei Chen,Yu-Ting Chen,Ghanashyam Meher,Berndt M¨uller,Andreas Sch¨afer and Xiaojun Yao. Department of Physics and Center for Theoretical Physics, National Taiwan University

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2 InQubator for Quantum Simulation (IQuS), Department of Physics, University of Washington

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3 II. SU(2) HAMILTONIAN ON PLAQUETTE CHAIN AND PARTITIONING The Kogut-Susskind Hamiltonian for pure SU(2) gauge theory on a 2-dimensional square lattice is given by H = g 2/a squared X links E b b - 1/a squared g squared X plaqs (□ + □ †), (1) where a is the lattice spacing, Eb denotes the electric field with SU(2) index b that is implicitly summed and □ represents the plaquette operator defined as the trace of the product of four link operators along the square plaquette. As part of the quantization procedure in the temporal gauge, Gauss’s law constraints are imposed on electric fields X l∈v E b l ≈ 0, (2) where ≈ means constraints to be imposed on states, l ∈ v denotes all links l joining the vertex v. Physically, Gauss’s law means that the state transforms as a singlet under gauge transformation, since the electric field is the generator of gauge transformation [118]

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4 In this work, we use the electric field basis in which the electric energy density g 2/a squared EbEb is diagonal. On trivial lattices such as the plaquette chain considered here, the Hilbert space of singlet states can be uniquely characterized by the irreducible representation (irrep) j = 0, 1/2, 1... on each link [87, 90] Using the results of Ref. [90], the plaquette chain can be mapped onto a one-dimensional spin model with the following Hamiltonian when the local Hilbert space is truncated to the two lowest representations j ≤ jmax = 1/2 and local Gauss’s law constraints are accounted for at every vertex [7] (see similar expressions in Refs. [119, 120] that are equivalent up to a unitary transformation of the basis) H = J N X-2 i=0 Z i Z i+1 + h z N X-1 i=0 Z i + h x/a squared N X-3 i=0 Xi - 3h x/a squared N X-2 i=0 [Z iX(i+1) + X iZ(i+1)] + 9h x/a squared N X-3 i=0 Z iX(i+1)Z(i+2), ≡ HZZ + HZ + HX + HZX + HXZ + HZXZ, (3) where X and Z denote the Pauli-X and-Z matrices, respectively, (similarly for Y later) and N is the number of plaquettes. We have used open boundary conditions for the chain to avoid long connectivity between the first and last qubits in the quantum circuit.1 We have defined different Hamiltonian components according to the number 1 The boundary conditions are chosen such that the plaquette terms on the first and last plaquettes are of the form X0(1 − 3Z1)/16 and (1 − 3ZN-2)XN-1/16, respectively, i.e., setting Z-1 → 0 and ZN → 0, instead of X0(1 − 3Z i)/4 and (1 − 3ZN-2)XN-1/4 for the boundary conditions of no electric excitation outside the lattice, i.e., Z-1 = ZN = -1. We choose these particular boundary conditions such that the multi-qubit gates in the quantum circuit are the same in each layer.

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5 In order to study the entanglement entropy and spectrum, we bipartition the plaquette chain into a subsystem A and its complement. In the spin map, partitioning based on the spin content is natural. In the original SU(2) theory, this partitioning corresponds to cutting the lattice at two vertices that share the same horizontal location. This partitioning also neglects the contributions from different states in the same irrep (e.g., for SU(2) the m states with the same j in jm⟩ notation) to the entanglement entropy, which are not distillable [121]

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6 III. QUANTUM ALGORITHMS FOR TIME EVOLUTION AND ENTANGLEMENT ENTROPY A. Trotterized Evolution We use Trotterization to implement the Hamiltonian time evolution on a quantum computer. Because the interaction terms on neighboring qubits in the HXZ, HZX, and HZXZ parts of the Hamiltonian do not commute, we further split these terms into smaller components.

1. Enhanced Nonabelian Dynamics Simulation: The improved AI system can reliably simulate the transition of an excited, but initially low-entangled, pure state of the lattice gauge field to a locally thermal state in nonabelian lattice gauge theories like SU(2) on long chains (up to N=151 plaquettes). This capability is crucial for studying far-fromequilibrium dynamics of nonabelian gauge theories which are otherwise difficult to simulate classically due to the quantum magic barrier.

2. Real-Time Entanglement Characterization: The system can compute and track key entanglement diagnostics in real-time, specifically the time dependence of the entanglement spectrum, R´enyi-2 entropy and anti-flatness on small subsystems. This allows researchers to observe how these quantities evolve during the peak period of thermalization dynamics, which is essential for identifying when quantum computation is most needed.

3. Robust Error Mitigation Integration: The AI system can incorporate advanced error mitigation techniques, including dynamical decoupling [83, 84], Pauli twirling [85] and operator decoherence renormalization [86], to extract accurate physical signals from noisy quantum hardware results. This ensures that the calculated observables are as close as possible to the ideal physical values by applying procedures like Operator decoherence renormalization (ODR) [86].

4. Hardware Performance Benchmarking: The system can perform rigorous benchmarking of current quantum hardware by comparing quantum hardware results against classical simulator results for various system sizes, specifically assessing when accumulated hardware noise exceeds the capability of the implemented error mitigation techniques. This provides a quantifiable metric for determining the practical limits of NISQ-era simulations.

5. Predictive Thermalization Analysis: By analyzing the behavior of the entanglement spectrum and its relation to energy eigenvalues (as shown in Fig. 10), the system can predict whether the subsystem is reaching towards a steady state, providing insight into how quickly and effectively local thermalization occurs for different initial states.

We simulate the local thermalization dynamics for minimally truncated SU(2) pure gauge theory on linear plaquette chains with up to 151 plaquettes using IBM quantum computers. We study the time dependence of the entanglement spectrum, Rényi-2 entropy and anti-flatness on small subsystems. The quantum hardware results obtained after error mitigation agree with extrapolated classical simulator results for chains consisting of up to 101 plaquettes. Our results demonstrate the feasibility of local thermalization studies for chaotic quantum systems, such as nonabelian lattice gauge theories, on current noisy quantum computing platforms.