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arXiv:2609.30285

Non-abelian quantum cellular automata: $1{+}1$-dimensional $SU(2)$ Yang--Mills with fermions

Dogukan Bakircioglu, Pablo Arrighi

quant-phhep-latSubmitted Sep 3, 2026• Announced Sep 7, 2026• 0 citations• Revised Sep 30, 2026 (v2)
v1 · Sep 3, 2026v2 · Sep 30, 2026
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Abstract

This work provides a digital quantum simulation scheme for $1{+}1$-dimensional $SU(2)$ Yang--Mills theory with Dirac fermions. It takes the form of a quantum circuit, infinitely repeating across space and time with $Δ_t=Δ_x=\varepsilon$, whose wires follow lightlike propagation. The construction mirrors the logic of the standard quantum field theory approach, transposed to the discrete setting. Namely, we start from the Dirac quantum walk, restore $SU(2)$ gauge symmetry by introducing the gauge field, lift the walk to a multi-particle QCA while preserving fermionic anticommutation, and equip the gauge field with its own dynamics. Rather than relying on Clebsch--Gordan decompositions, we use the pointwise-product structure of gauge-link updates, which yields self-contained proofs of unitarity and gauge covariance in quantum-computing notation. The construction provides an explicit algorithmic formulation of the theory, whose continuum limit we discuss.

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Non-abelian quantum cellular automata: 1+11{+}11+1-dimensional SU(2)SU(2)SU(2) Yang--Mills with fermions
This work provides a digital quantum simulation scheme for 1+11{+}11+1-dimensional SU(2)SU(2)SU(2) Yang--Mills theory with Dirac fermions. It takes the form of a quantum circuit, infinitely repeating across space and time with Δt=Δx=εΔ_t=Δ_x=\varepsilonΔt​=Δx​=ε, whose wires follow lightlike propagation. The construction mirrors the logic of the standard quantum field theory approach, transposed to the discrete setting. Namely, we start from the Dirac quantum walk, restore SU(2)SU(2)SU(2) gauge symmetry by introducing the gauge field, lift the walk to a multi-particle QCA while preserving fermionic anticommutation, and equip the gauge field with its own dynamics. Rather than relying on Clebsch--Gordan decompositions, we use the pointwise-product structure of gauge-link updates, which yields self-contained proofs of unitarity and gauge covariance in quantum-computing notation. The construction provides an explicit algorithmic formulation of the theory, whose continuum limit we discuss.
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 Δ t = Δ 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.
A quantum computer, i.e. utilizing the resources of quantum physics, superposition of states and entanglement, could furnish an exponential gain in computing time. A simulation using such resources is called a quantum simulation. The advantage of quantum simulations over classical ones is well established at the theoretical, i.e. software level. Their practical benefit requires their implementation on a quantum hardware. The quantum computer, i.e. the universal one (see below), has not seen the light of day yet, but the efforts in this direction are both growing and diverse. Also, quantum simulation has already been illustrated by numerous experimental proofs of principle, thanks too small-size and specific-task quantum computers or simulators. Quantum walks are particularly-studied quantum-simulation schemes, being elementary bricks to conceive any quantum algorithm, i.e. to achieve so-called universal quantum computation. The present thesis is a step more towards a simulation of quantum field theories based on discrete-time quantum walks (DTQWs). Indeed, it is shown, in certain cases, how DTQWs can simulate, in the continuum, the action of Yang-Mills gauge fields on fermionic matter, and the retroaction of the latter on the gauge-field dynamics. The suggested schemes preserve gauge invariance on the spacetime lattice, i.e. not only in the continuum. In the (1+2)-dimensional Abelian case, consistent lattice equivalents to both Maxwell's equations and the current conservation are suggested. In the (1+1)-dimensional non-Abelian case, a lattice version of the non-Abelian field strength is suggested. Moreover, it is shown how this fermionic matter based on DTQWs can be coupled to relativistic gravitational fields of the continuum, i.e. to curved spacetimes, in 1+2 dimensions.
A family of discrete-time quantum walks (DTQWs) on the line with an exact discrete U(N)\text{U}(N)U(N) gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual U(N)\text{U}(N)U(N) gauge fields in two-dimensional spacetime. A discrete generalization of the usual U(N)\text{U}(N)U(N) curvature is also constructed. An alternate interpretation of these results in terms of superimposed U(1)\text{U}(1)U(1) Maxwell fields and SU(N)\text{SU}(N)SU(N) gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e., nonquantum) motions in classical SU(2)\text{SU}(2)SU(2) fields. The results presented in this paper constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.
We construct and analyze an explicit one-dimensional quantum cellular automaton (QCA) that provides a controlled lattice regularization of the free Dirac equation in (1+1) dimensions, and we extend it to a locally U(1) gauge-covariant version. The microscopic model is strictly local, causal in the Lieb–Robinson sense, and unitarily evolves a chain of finite-dimensional quantum systems by a translation invariant, discrete-time rule. At the representation-theoretic level, it realizes a chiral spin-1/2 degree of freedom per site. We prove that, in the joint limit of small lattice spacing and small time step with fixed physical scales, the QCA dynamics converges to the continuum Dirac evolution, with an explicit bound on the error valid for smooth, band-limited initial states. We then couple the QCA to a compact U(1) gauge field living on links, in a way that is exactly gauge-covariant at the microscopic level. A local Gauss constraint defines a gauge-invariant code subspace, and we show that the QCA update preserves this subspace and implements a gauge-covariant discrete Dirac dynamics. This provides a concrete example of a gauge-coded QCA with a controlled continuum limit, which can serve as a microscopic building block for more elaborate SU(3) × SU(2) × U(1) constructions. All results are stated with modest but explicit hypotheses and proofs, so that the framework is mathematically well-defined and amenable to further generalization.
Simulations of one quantum system by an other has an implication in realization of quantum machine that can imitate any quantum system and solve problems that are not accessible to classical computers. One of the approach to engineer quantum simulations is to discretize the space-time degree of freedom in quantum dynamics and define the quantum cellular automata (QCA), a local unitary update rule on a lattice. Different models of QCA are constructed using set of conditions which are not unique and are not always in implementable configuration on any other system. Dirac Cellular Automata (DCA) is one such model constructed for Dirac Hamiltonian (DH) in free quantum field theory. Here, starting from a split-step discrete-time quantum walk (QW) which is uniquely defined for experimental implementation, we recover the DCA along with all the fine oscillations in position space and bridge the missing connection between DH-DCA-QW. We will present the contribution of the parameters resulting in the fine oscillations on the Zitterbewegung frequency and entanglement. The tuneability of the evolution parameters demonstrated in experimental implementation of QW will establish it as an efficient tool to design quantum simulator and approach quantum field theory from principles of quantum information theory.
Quantum walks on lattices can give rise to one-particle relativistic wave equations in the long-wavelength limit. In going to multiple particles, quantum cellular automata (QCA) are natural generalizations of quantum walks. In one spatial dimension, the quantum walk can be ``promoted'' to a QCA that, in the long-wavelength limit, gives rise to the Dirac quantum field theory (QFT) for noninteracting fermions. This QCA/QFT correspondence has both theoretical and practical applications, but there are obstacles to similar constructions in two or more spatial dimensions. Here we show that a method of construction employing distinguishable particles confined to the completely antisymmetric subspace yields a QCA in two spatial dimensions that gives rise to the two-dimensional (2D) Dirac QFT. Generalizing to 3D will entail some additional complications, but no conceptual barriers. We examine how this construction evades the ``no-go'' results in earlier work.