Abstract In this paper, we study matrix-product unitary operators (MPUs) for fermionic one-dimensional chains. In stark contrast to the case of 1D qudit systems, we show that (i) fermionic MPUs (fMPUs) do not necessarily feature a strict causal cone and (ii) not all fermionic quantum cellular automata (QCA) can be represented as fMPUs. We then introduce a natural generalization of the latter, obtained by allowing for an additional operator acting on their auxiliary space. We characterize a family of such generalized MPUs that are locality-preserving, and show that, up to appending inert ancillary fermionic degrees of freedom, any representative of this family is a fermionic QCA (fQCA) and vice versa. Finally, we prove an index theorem for generalized MPUs, recovering the recently derived classification of fQCA in one dimension. As a technical tool for our analysis, we also introduce a graded canonical form for fermionic matrix product states, proving its uniqueness up to similarity transformations.
In this note, we present some results on the classification of quantum cellular automata (QCA) in 1D under strong equivalence rather than stable equivalence. Under strong equivalence, we only allow adding ancillas carrying the original on-site representation of the symmetry, while under stable equivalence, we allow adding ancillas carrying any representation of the symmetry. The former may be more realistic, because in physical systems especially in AMO/quantum computing contexts, we would not expect additional spins carrying arbitrary representations of the symmetry to be present. Ref.~\onlinecite{mpu} proposed two kinds of symmetry-protected indices (SPIs) for QCA with discrete symmetries under strong equivalence. In this note, we show that the more refined of these SPIs still only has a one-to-one correspondence to equivalence classes of
ZN symmetric QCA when
N is prime. We show a counter-example for
N=4. We show that QCA with
Z2 symmetry under strong equivalence, for a given on-site representation, are classified by
Zpq where
p is the number of prime factors of the on-site Hilbert space dimension and
q is the number of prime factors of the trace of the nontrivial on-site
Z2 element. Finally, we show that the GNVW index has a formulation in terms of a
Z2 SPI in a doubled system, and we provide a direct connection between the SPI formulation of the GNVW index and a second Renyi version of the mutual information formula for the GNVW index.
We present an exact renormalisation scheme for fermionic cellular automata on hypercubic lattices. By grouping neighbouring cells into tiles and selecting subspaces within them, multiple evolution steps on the original system correspond to a single step of an effective automaton acting on the subspaces. We derive a necessary and sufficient condition for renormalisability and fully characterise the renormalisation flow for two-cell tiles and two time steps of nearest-neighbour fermionic automata on a chain of spinless modes, identifying all fixed points.
We prove that the three-dimensional chiral
Zp Clifford quantum cellular automata of Haah and of Sun--Yang--Wang--Tantivasadakarn--Chen are, for every odd prime
p, not finite-depth quantum circuits, even after tensoring with arbitrary ancillary degrees of freedom and composing with translations. Previously such statements were known only inside the Clifford category. As a consequence, the group of three-dimensional QCA modulo stabilized circuits and translations is not finitely generated. The key ingredient is a new invariant of invertible subalgebras, which we conjecture to be the square of the total quantum dimension of the hosted anyon theory, modulo rational squares.
We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra – the symmetric subalgebra under a finite Abelian group symmetry G . For systems where each site carries a regular representation of G , we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a ( 2 + 1 ) d G gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of G -symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a Z 2 symmetric subalgebra, demonstrating that it maps to the e - m permutation in the two-dimensional toric code and has an irrational index of 2 . Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.