Renormalisation of Quantum Cellular Automata

The gist The study derives necessary and sufficient conditions for renormalizability in quantum cellular automata by applying a Kadanoff-like coarse-graining procedure, exhaustively solving this problem for one-dimensional QCAs with qubit cells.

How it works

The core idea involves grouping neighboring cells into tiles and selecting a subspace within each tile, such that multiple evolution steps on this subspace are equivalent to a single step of a new quantum cellular automaton where the cells are these subspaces The authors derive a necessary and sufficient condition for renormalizability based on this coarse-graining procedure. This procedure is adapted from Kadanoff's renormalization, which involves progressively erasing degrees of freedom deemed irrelevant to the large-scale picture.

Renormalisation Flow and Fixed Points

The process defines a renormalisation flow in the space of QCAs, where the model describing the large-scale behavior must itself be a QCA. The authors investigate fixed points of this flow, which are highlighted as significant outcomes. A key finding is that very few qubit QCAs are renormalisable in this sense, and those that are have a trivial evolution due to the limited choice of degrees of freedom.

Coarse-graining and Algebra

The coarse-graining procedure is formulated by considering a regular tessellation of the lattice into cubes, where each tile is an Ns-dimensional cube. The problem is reformulated using the wrapping lemma to reduce the coarse-graining problem on an infinite lattice to one over all possible wrappings L. This leads to defining a coarse-grained algebra AΠ(L) over L, which is isomorphic to the algebra A(L') on a new lattice L'. The map V relates the algebras A(L') and AΠ(L), satisfying V†V = I and VV† = P.

Renormalisability Conditions

A QCA T is defined as renormalisable through the coarse-graining (Π0,Λ0) to S if the condition T N w ◦ V = V ◦ S w holds for any wrapping Tw on L. The authors prove that this condition is equivalent to a finite dimensional equation in terms of the unitary matrix defining the QCA on a wrapped lattice. Specifically, Proposition 3 establishes that renormalisability is equivalent to either the existence of a QCA S which is a size-N renormalisation of T through (Π0,Λ0), or there existing a lattice L = Z s N f such that the regular neighborhood condition is satisfied for T N and [UT N, Π] = 0.

One-Dimensional Qubit Automata

When specializing to one-dimensional QCAs with qubit cells, the analysis focuses on size-2 coarse-graining, mapping two cells and two time steps into a single cell and one step. The analysis reveals that for this special case, the index cannot be changed upon renormalisation. Furthermore, most QCAs are not renormalisable unless they are those where information does not actually propagate. The fixed points with non-trivial dynamics occur in a few specific cases related to the parameters of the evolution T, such as when T 2 = I and is renormalised in a local unitary.

Conclusion

The analysis concludes that for the special case of one-dimensional qubit QCAs, renormalisability is highly restrictive, with most models not being renormalisable unless they exhibit specific properties like trivialisation after two steps or propagation of information. The necessary and sufficient condition for a QCA to admit a coarse-graining coincides with the one identified in [52] for a Quantum Walk to be Lorentz covariant. The study suggests that more interesting cases might arise from larger coarse-grainings or by generalizing the classification of Margolus partitioning schemes. The necessary and sufficient condition derived is expected to be exploited to study renormalisation problems for QCAs on higher-dimensional graphs, potentially leading to fixed points representing phase transitions.

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Renormalisation of quantum cellular automata Lorenzo Siro Trezzini Alessandro Bisio and Paolo Perinotti 1Dipartimento di Fisica dell’Università di Pavia, via Bassi 6, 27100 Pavia Istituto Nazionale di Fisica Nucleare, Gruppo IV, via Bassi 6, 27100 Pavia We study a coarse-graining procedure for quantum cellular automata on hypercubic lattices that consists in grouping neighboring cells into tiles and selecting a subspace within each tile. This is done in such a way that multiple evolution steps applied to this subspace can be viewed as a single evolution step of a new quantum cellular automaton, whose cells are the subspaces themselves. We derive a necessary and sufficient condition for renormalizability and use it to investigate the renormalization flow of cellular automata on a line, where the cells are qubits and the tiles are composed of two neighboring cells. The problem is exhaustively solved, and the fixed points of the renormalization flow are highlighted.

One-Dimensional Qubit Automata

When specializing to one-dimensional QCAs with qubit cells, the analysis focuses on size-2 coarse-graining, mapping two cells and two time steps into a single cell and one step. The analysis reveals that for this special case, the index cannot be changed upon renormalisation. Furthermore, most QCAs are not renormalisable unless they are those where information does not actually propagate.

Conclusion

The analysis concludes that for the special case of one-dimensional qubit QCAs, renormalisability is highly restrictive, with most models not being renormalisable unless they exhibit specific properties like trivialisation after two steps or propagation of information. The necessary and sufficient condition derived is expected to be exploited to study renormalisation problems for QCAs on higher-dimensional graphs, potentially leading to fixed points representing phase transitions.

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Quantum Cellular Automata QCA have been developed in the years since the very advent of quantum computation [1] as the quantum counterpart of classical cellular automata [2, 3]. The notion of a QCA was made more and more rigorous over the years [4, 5, 6, 7, 8], until the algebraic formulation that is still presently used [9, 10].

1. Quantum simulation via renormalised QCA dynamics can be performed by implementing size-2 coarse-graining procedures, which allows a single step of S to mimic the behaviour of T N if reduced to the chosen degrees of freedom. This enables error correction for quantum simulations through QCAs, allowing a fine-grained QCA where we do not have full control can be used to run a simulation on its renormalised QCA.

2. Development of an operator algebraic approach to Wilson’s renormalization group allows for the derivation of necessary and sufficient conditions for renormalisability, which are amenable to further analysis. This framework can be applied to study the renormalisation flow of cellular automata on a line, where the cells are qubits and the tiles are composed of two neighboring cells.

3. Implementation of a size-2 renormalisation for a one-dimensional qubit QCA allows for a size-2 renormalisation that results in either a local unitary evolution or an evolution with specific phase factors derived from the parameters of T, such as exp(−iσzθ′) with (ϕ′ = 2ϕ, θ′ = −ϕ).

4. Classification of renormalisable QCAs yields three main classes: those that trivialise after two steps, those that apply a local unitary transformation, and those that are diagonal on a factorised basis without reducing to a local unitary. This classification helps determine the fixed points of the renormalization flow for QCA dynamics.

5. Computation of the renormalised evolution S(B) over a finite lattice L' can be done using Lemma 16, which provides The matrix implementing the renormalised local rule is VS =O x∈L′ J˜†Λx U˜2 TO x∈L′ J˜Λx. This allows for the explicit computation of dynamics on a lattice of size 6 based on the evolution over a lattice of size 8.

6. Identification of the index and its consequences for QCA dynamics can be utilized to check renormalisability, as ind(T) = s dim[L2x] dim[A2x] does not depend on x. This index theory provides tools to understand the structure of renormalised evolutions and their relationship to Margolus partitioning schemes.

We study a coarse-graining procedure for quantum cellular automata on hypercubic lattices that consists in grouping neighboring cells into tiles and selecting a subspace within each tile. This is done in such a way that multiple evolution steps applied to this subspace can be viewed as a single evolution step of a new quantum cellular automaton, whose cells are the subspaces themselves. We derive a necessary and sufficient condition for renormalizability and use it to investigate the renormalization flow of cellular automata on a line, where the cells are qubits and the tiles are composed of two neighboring cells. The problem is exhaustively solved, and the fixed points of the renormalization flow are highlighted.