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Register a new userFermionic and bosonic quantum field theories from quantum cellular automata in three spatial dimensions
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Quantum walks on lattices can give rise to relativistic wave equations in the long-wavelength limit, but going beyond the single-particle case has proven challenging, especially in more than one spatial dimension. We construct quantum cellular automata for distinguishable particles based on two different quantum walks, and show that by restricting to the antisymmetric and symmetric subspaces, respectively, a multiparticle theory for free fermions and bosons in three spatial dimensions can be produced. This construction evades a no-go theorem that prohibits the usual fermionization constructions in more than one spatial dimension. In the long-wavelength limit, these recover Dirac field theory and Maxwell field theory, i.e., free QED.
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A classical local cellular automaton can describe an interacting quantum field theory for fermions. We construct a simple classical automaton for a particular version of the Thirring model with imaginary coupling. This interacting fermionic quantum field theory obeys a unitary time evolution and shows all properties of quantum mechanics. Classical cellular automata with probabilistic initial conditions admit a description in the formalism of quantum mechanics. Our model exhibits interesting features as spontaneous symmetry breaking or solitons. The same model can be formulated as a generalized Ising model. This euclidean lattice model can be investigated by standard techniques of statistical physics as Monte Carlo simulations. Our model is an example how quantum mechanics emerges from classical statistics.
We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three fermion model. We show that if this QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional commuting projector Hamiltonian which realizes the three fermion topological order which is widely believed not to be possible. We conjecture in accordance with this belief that this QCA is not a quantum circuit of constant depth, and we provide two further pieces of evidence to support the conjecture. We show that this QCA maps every local Pauli operator to a local Pauli operator, but is not a Clifford circuit of constant depth. Further, we show that if the three-dimensional QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional QCA acting on fermionic degrees of freedom which cannot be realized by a quantum circuit of constant depth; i.e., we prove the existence of a nontrivial QCA in either three or two dimensions. The square of our three-dimensional QCA can be realized by a quantum circuit of constant depth, and this suggests the existence of a $mathbb{Z}_2$ invariant of a QCA in higher dimensions, totally distinct from the classification by positive rationals (i.e., by one integer index for each prime) in one dimension. In an appendix, unrelated to the main body of this paper, we give a fermionic generalization of a result of Bravyi and Vyalyi on ground states of 2-local commuting Hamiltonians.
In a series of recent papers it has been shown how free quantum field theory can be derived without using mechanical primitives (including space-time, special relativity, quantization rules, etc.), but only considering the easiest quantum algorithm encompassing a countable set of quantum systems whose network of interactions satisfies the simple principles of unitarity, homogeneity, locality, and isotropy. This has opened the route to extending the axiomatic information-theoretic derivation of the quantum theory of abstract systems to include quantum field theory. The inherent discrete nature of the informational axiomatization leads to an extension of quantum field theory to a quantum cellular automata theory, where the usual field theory is recovered in a regime where the discrete structure of the automata cannot be probed. A simple heuristic argument sets the scale of discreteness to the Planck scale, and the customary physical regime where discreteness is not visible is the relativistic one of small wavevectors. In this paper we provide a thorough derivation from principles that in the most general case the graph of the quantum cellular automaton is the Cayley graph of a finitely presented group, and showing how for the case corresponding to Euclidean emergent space (where the group resorts to an Abelian one) the automata leads to Weyl, Dirac and Maxwell field dynamics in the relativistic limit. We conclude with some perspectives towards the more general scenario of non-linear automata for interacting quantum field theory.
It is shown how a Doubly-Special Relativity model can emerge from a quantum cellular automaton description of the evolution of countably many interacting quantum systems. We consider a one-dimensional automaton that spawns the Dirac evolution in the relativistic limit of small wave-vectors and masses (in Planck units). The assumption of invariance of dispersion relations for boosted observers leads to a non-linear representation of the Lorentz group on the $(omega,k)$ space, with an additional invariant given by the wave-vector $k=pi /2$. The space-time reconstructed from the $(omega,k)$ space is intrinsically quantum, and exhibits the phenomenon of relative locality.
We introduce a quantum cellular automaton that achieves approximate phase-covariant cloning of qubits. The automaton is optimized for 1-to-2N economical cloning. The use of the automaton for cloning allows us to exploit different foliations for improving the performance with given resources.
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