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Real-time dynamics of lattice gauge theories with a few-qubit quantum computer
E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl, P. Hauke, M. Dalmonte, T. Monz, P. Zoller, R. Blatt
TL;DR
Real-time lattice-gauge-theory dynamics are difficult to compute classically and challenging to encode while preserving gauge constraints. The authors implement an encoded Schwinger model on a few-qubit trapped-ion system, finding qualitative agreement with expected vacuum-decay and finite-size dynamics.
Problem
Experimental quantum simulation of gauge-theory dynamics remained limited because encoded formulations had primarily served analytical or numerical calculations.
Method
The authors eliminate gauge fields through a gauge transformation and realize the resulting spin model with non-local long-range interactions in an ion-trap system.
Results
The measured rate function λ(t) showed good qualitative agreement with ν(t), while N = 4 reproduced larger-system particle-number and entanglement dynamics qualitatively.
Takeaways & Limitations
The results support using few-qubit encoded dynamics to study Schwinger-model particle creation and entanglement qualitatively.
Takeaways & Limitations
The simulations consider only zero background field, with ε0 = 0.
Abstract
from arXiv · showhide
Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. In the spirit of Feynman's vision of a quantum simulator, this has recently stimulated theoretical effort to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented. Here we report the first experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realising 1+1-dimensional quantum electrodynamics (Schwinger model) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields in favour of exotic long-range interactions, which have a direct and efficient implementation on an ion trap architecture. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulating high-energy theories with atomic physics experiments, the long-term vision being the extension to real-time quantum simulations of non-Abelian lattice gauge theories.
Methods
The methods encode the lattice Schwinger model as a pure spin Hamiltonian by eliminating gauge fields, producing long-range Coulomb interactions, and implement its dynamics digitally with trapped-ion-compatible operations.
- Encoding: Gauge degrees of freedom are eliminated in two steps: a gauge transformation removes θ̂_n, followed by iterative elimination of L̂_n using the spin Gauss law.This produces a pure spin Hamiltonian for the simulation.
- Encoding: The resulting spin description omits explicit gauge fields, which instead generate non-local long-range interactions representing Coulomb interactions between charged particles.The approach is applied here as a quantum simulation scheme rather than only for analytical or numerical calculations.
- Encoding: The simulation assumes zero background electric field by setting the boundary field parameter ϵ0 = 0.ϵ0 represents the electric field on the link left of the first lattice site.
- Digital simulation: Digital dynamics are generated through time-coarse graining, dividing the total simulation time into windows of duration T and repeating a three-section cycle.The cycle separately implements ĤZZ, nearest-neighbour Ĥ±, and single-particle rotations ĤZ.
- Digital simulation: The protocol uses local rotations and an infinite-range entangling operation, with the latter routinely implemented in trapped ions using Mølmer–Sørensen gates.The relative couplings J, w, and m are tunable through elementary-window durations or the underlying interaction strength J0.
Long-range interactions ˆHZZ
The electric-field energy becomes an asymmetric long-range two-body spin interaction: each spin couples constantly to spins on its left, while right-side couplings decrease linearly with distance. The protocol implements this interaction efficiently by selectively entangling only participating ions and hiding the others with laser pulses.
- Origin: The HZZ term originates from the electric-field energy contribution in the lattice gauge theory Hamiltonian.
- Interaction structure: Each spin interacts with constant strength with all spins to its left, while coupling to spins on its right decreases linearly with distance.
- Interaction structure: N^2 coupling-matrix elements make brute-force digital simulation require N^2 time steps.
- Implementation: The HZZ interaction is implemented using a Hamiltonian combined with local rotations and a protocol for realizing MSZ.
- Implementation: At time step n, only ions 1 through n + 1 participate in entangling interactions, while ions n + 2 through N are decoupled with hiding pulses.The hiding pulses transfer population to electronic levels unaffected by the global beam interaction.
Nearest neighbour terms ˆH±
The nearest-neighbour Hamiltonian terms H± implement particle–antiparticle pair creation and annihilation. They are realized through sequential pairwise interactions, with their relative strength tunable against long-range couplings.
- Physical role: H± corresponds to the creation and annihilation of particle–antiparticle pairs.These terms form the second part of Eq. (4).
- Trotterized implementation: The H± implementation modifies the interaction’s range and coupling type by dividing its time window into N −1 elementary slots of length ∆tII.Each slot engineers a pairwise interaction between neighboring spins.
- Pairwise gate sequence: Hiding pulses isolate selected ion pairs while a gate sequence transforms the σx-type coupling into the required σ+σ− + H.c. interaction.The sequence uses single-qubit operations and two-qubit gates around evolution under the selected-pair Hamiltonian.
- Interaction-strength control: The relative strength w/J of H± and the long-range HZZ couplings is adjusted by tuning the elementary-window ratio ∆II/∆I.This tuning controls the balance between nearest-neighbour and long-range terms.
Single-particle terms ˆHZ
The single-particle Hamiltonian contribution contains the fermion rest-mass term and an effective mass-shift term induced by eliminating the electric fields. These local terms are implemented using AC–Stark shifts from far-red-detuned laser pulses.
- Single-particle terms ˆHZ: The first term represents the rest masses of the fermions.
- Single-particle terms ˆHZ: The second term is an effective single-particle contribution that changes the effective fermion masses after eliminating the electric fields.
- Single-particle terms ˆHZ: The local Hamiltonian terms are implemented through AC–Stark shifts induced by laser pulses far red-detuned from the qubit transition.
Extended data
The extended data present the time evolution of particle number density, vacuum-decay rate, and entanglement for varied masses, electric-field energies, and system sizes. They also document the experimental pulse sequence used for four evolution steps.
- Particle production and vacuum persistence: Particle number density ν(t) and rate function λ(t) characterize particle production and vacuum persistence decay during real-time evolution.λ(t) characterizes the decay of the vacuum persistence probability.
- Particle production and vacuum persistence: Varying particle mass m at fixed electric-field energy J = w yields time evolutions of ν(t) and λ(t).Here, w is the rate of particle-antiparticle creation and annihilation.
- Particle production and vacuum persistence: Varying J at fixed particle mass m = 0 shows ν(t) and λ(t) as functions of dimensionless time wt.These evolutions are shown for different values of J and fixed m = 0.
- Entanglement dynamics: For different system sizes N, the particle number density ν and logarithmic negativity E_n are tracked versus dimensionless time wt at J = m = w.E_n quantifies entanglement between the two halves of the spin chain when evaluated across its middle cut.
- Experimental pulse sequence: The pulse sequence contains 222 pulses for 4 evolution steps, with 51 repeated pulses per step plus initial and final operations.The total is 12 + 51 × 4 + 6 = 222 pulses.