A relativistic discrete spacetime formulation of 3+1 QED

Nathanaël Eon1, Giuseppe Di Molfetta1, Giuseppe Magnifico2,3,4,5, and Pablo Arrighi6

1Aix-Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France
2Dipartimento di Fisica e Astronomia “G. Galilei”, Universita` di Padova, I-35131 Padova, Italy
3Padua Quantum Technologies Research Center, Universita` degli Studi di Padova
4Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, I-35131 Padova, Italy
5Dipartimento di Fisica, Universita` di Bari, I-70126 Bari, Italy
6Université Paris-Saclay, Inria, CNRS, LMF, 91190 Gif-sur-Yvette, France

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This work provides a relativistic, digital quantum simulation scheme for both $2+1$ and $3+1$ dimensional quantum electrodynamics (QED), based on a discrete spacetime formulation of theory. It takes the form of a quantum circuit, infinitely repeating across space and time, parametrised by the discretization step $\Delta_t=\Delta_x$. Strict causality at each step is ensured as circuit wires coincide with the lightlike worldlines of QED; simulation time under decoherence is optimized. The construction replays the logic that leads to the QED Lagrangian. Namely, it starts from the Dirac quantum walk, well-known to converge towards free relativistic fermions. It then extends the quantum walk into a multi-particle sector quantum cellular automata in a way which respects the fermionic anti-commutation relations and the discrete gauge invariance symmetry. Both requirements can only be achieved at cost of introducing the gauge field. Lastly the gauge field is given its own electromagnetic dynamics, which can be formulated as a quantum walk at each plaquette.

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Cited by

[1] Edoardo Centofanti, Paolo Perinotti, and Alessandro Bisio, "Massless interacting fermionic cellular automaton exhibiting bound states", Physical Review A 109 5, 052421 (2024).

[2] Ugo Nzongani and Pablo Arnault, "Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks", Quantum Information Processing 23 5, 193 (2024).

[3] Ugo Nzongani, Julien Zylberman, Carlo-Elia Doncecchi, Armando Pérez, Fabrice Debbasch, and Pablo Arnault, "Quantum circuits for discrete-time quantum walks with position-dependent coin operator", Quantum Information Processing 22 7, 270 (2023).

[4] Nicolás Medina Sánchez and Borivoje Dakić, "Reconstruction of Quantum Particle Statistics: Bosons, Fermions, and Transtatistics", arXiv:2306.05919, (2023).

[5] Ugo Nzongani, Nathanaël Eon, Iván Márquez-Martín, Armando Pérez, Giuseppe Di Molfetta, and Pablo Arrighi, "Dirac quantum walk on tetrahedra", arXiv:2404.09840, (2024).

[6] Ugo Nzongani and Pablo Arnault, "Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks", arXiv:2304.10460, (2023).

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