Scalable quantum computation of Quantum Electrodynamics beyond one spatial dimension

TL;DR

This paper tackles the problem of real-time quantum simulation of Quantum Electrodynamics in more than one spatial dimension, where classical methods fail due to exponential Hilbert-space growth and sign problems. It presents a scalable, gauge-invariant quantum algorithm built on a unitary gauge-link representation and a Weyl-commutation-based digitisation, enabling efficient mapping to qubits and exact preservation of Gauss's law during evolution. The authors implement 2+1D and 3+1D QED on current quantum hardware, benchmark several quantum error mitigation strategies (notably calibration-based methods), and demonstrate that larger lattice spaces can be tackled as hardware improves. The approach provides a pathway to reliable, fully quantum simulations of large-scale QED dynamics and offers a foundation for extending to non-Abelian gauge theories on future quantum platforms.

Abstract

In the Hamiltonian formulation, Quantum Field Theory calculations scale exponentially with spatial volume, making real-time simulations intractable on classical computers and motivating quantum computation approaches. In Hamiltonian quantisation, bosonic fields introduce the additional challenge of an infinite-dimensional Hilbert space. We present a scalable quantum algorithm for Quantum Electrodynamics (QED), an Abelian gauge field theory in higher than one spatial dimensions, designed to address this limit while preserving gauge invariance. In our formulation, Gauss's law is automatically satisfied when the implementation remains fully gauge invariant. We demonstrate how gauge invariance is maintained throughout the lattice discretisation, digitisation, and qubitisation procedures, and identify the most efficient representation for extending to large Hilbert space dimensions. Within this framework, we benchmark several quantum error mitigation techniques and find the calibration method to perform most effectively. The approach scales naturally to larger lattices, and we implement and test the 2+1 and 3+1 dimensional setups on current quantum hardware. Our results indicate that next-generation quantum platforms could enable reliable, fully quantum simulations of large-scale QED dynamics.

Figures (6)

• Figure 1: (a) Hamiltonian quantisation requires an additional truncation of the infinite Hilbert space compared with the Path Integral. (b) A unique $N\to\infty$ limit restores the full Hilbert space in the Hamiltonian approach. (c) Main schemes for quantum simulation of Abelian gauge theory; the gray path indicates our economical, gauge-invariant method using a unitary (cyclic ladder) link and the Quantum Fourier Transform. (d) The unitary treatment extends scalar-field algorithms to gauge fields, while ladder-based methods remain gauge invariant at large-$N$ and can generalise to non-Abelian theories.
• Figure 2: An illustration of the quantum circuits that are used in the method. (a) Quantum Fourier Transform. (b) multi-Z rotations. (c) The plaquette rotation in the "stator" method: (L) the four links and the ancilla; (R) the W-operation illustrated with 2-qubit for the gauge degree. (d) The gauge-fermion rotation: (Upper) the unitary transformation which decouples the gauge degree; (Lower) the two-fermion rotation.
• Figure 3: (a) The illustration of (Upper) the 3D cube and (Lower) the 2D square. The qubit arrangements are also displayed separately, illustrated with 2-qubit for each gauge degree. An explanation of the fermion occupation and the electric basis is also present. (b) The observables, including electric fields and fermion charges, evolve along time on the 2D square. The solid lines stand for the results from Exact Diagonalization, while the dots refer to Qiskit simulations of ${\rm d} t=0.5$. (c) The distance (or the bit-flip number) plot with varying quantum error rate in the trivial runs, simulated via Qiskit. (d) The histogram of the final states in the target runs, simulated via Qiskit. The right-most plot is the select rate versus error rate.
• Figure 4: (a) Results for $n=1,2,3$ (Left, Middle, Right) on IBM quantum computers: the distance (Upper) and the observables (Lower), where the blue, red and orange error bars denote respectively the results from the direct, calibrated and post-selected measurements, and for comparison, the exact values are drawn as the gray bars. (b) Results for 1, 2, 3, and 4 Trotter steps, of $n=2$. (c) Results from different platforms, with $n=2$ and one Trotter step. (Upper) the distance plot; (Lower) the histogram of the final measurements, with the count on x-axis and the state on y-axis.
• Figure 5: On 3D cube: (a) Results for $n=1$ on IBM quantum computers: the histogram of the final states (Left), the distance (Right) and the observables (Middle), where the blue, red and orange error bars denote respectively the results from the direct, calibrated and post-selected measurements, and for comparison, the exact values are drawn as the gray bars. (b) Simulations with initial state of $\Pi_{AB}=-1$. (Left) $n=1$ with one- (Upper) and two- (Lower) Trotter steps; (Right) $n=2$ (Upper) and $n=3$ (Lower) with one Trotter step. (c) The distance plot of results from different platforms, with $n=2$ and one Trotter step.