Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond

Abstract

We construct a novel three-dimensional quantum cellular automaton (QCA) based on a system with short-range entangled bulk and chiral semion boundary topological order. We argue that either the QCA is nontrivial, i.e., not a finite-depth circuit of local quantum gates, or there exists a two-dimensional commuting projector Hamiltonian realizing the chiral semion topological order (characterized by U (1)2 Chern-Simons theory). Our QCA is obtained by first constructing the Walker-Wang Hamiltonian of a certain premodular tensor category of order four, then condensing the deconfined bulk boson at the level of lattice operators. We show that the resulting Hamiltonian hosts chiral semion surface topological order in the presence of a boundary and can be realized as a non-Pauli stabilizer code on qubits, from which the QCA is defined. The construction is then generalized to a class of QCAs defined by non-Pauli stabilizer codes on 2 n -dimensional qudits that feature surface anyons described by U (1)2n Chern-Simons theory. Our results support the conjecture that the group of nontrivial three-dimensional QCAs is isomorphic to the Witt group of non-degenerate braided fusion categories.

Topological phases of unitary dynamics: Classification in Clifford category

“…The state is believed to exhibit a nontrivial SPT under time-reversal symmetry [14], where we quoted the statement conjectural because it is not proved that the state is topologically trivial, i.e., disentangled by a locally generated unitary in the absence of the time-reversal symmetry; the QCA of [11] disentangles every eigenstate of a Hamiltonian whose ground state is a particular Walker-Wang state and is believed to be not locally generated, but we do not know any construction or an existence proof of a locally generated unitary that disentangles the one particular eigenstate that is the Walker-Wang groundstate. This subtlety on the disentanglability is observed in all examples to date [11,10,15] of putative nontrivial QCA (modulo locally generated unitaries and translations) in three dimensions. In four dimensions, there is a construction of a putative nontrivial QCA [16] that disentangles the ground state of an invertible state that does not require any symmetry [16,17].…”

“…Put verbosely, we do not know if the map (1) is injective and we do not know if (1) is surjective. Note that the reductions of the nontriviality (with translations deemed trivial) of the cited QCA [11,10,16,15] in d dimensions to certain conjectures on Hamiltonians in d − 1 dimensions, assume the strict locality.…”

“…However, we have a map {QCA = Strictly locality-preserving causal unitaries} {shallow quantum circuits and shifts} which is likely not surjective. Indeed, if we believe in Walker's conjecture [18] that any strictly local commuting Hamiltonian in two dimensions whose ground state subspace is locally indistinguishable can only realize a Levin-Wen model [19], then the QCA in [15] does not belong to the image of (2). Also, the QCA in [16] does not belong to the image of (2) assuming a different conjecture.…”

Topological invariants for SPT entanglers

“…In this paper, we will consider U that are more general symmetric LPUs, which may not be FDQCs even after forgetting about the symmetry. This is because for some SPT phases, the entangler can only be made symmetric if it is a nontrivial LPU [14][15][16].…”

“…Interestingly, broad classes of SPT phases can be entangled by FDQCs that, though containing gates that break the symmetry, respect the symmetry as a whole [1,[11][12][13]. Sometimes, an SPT cannot be entangled by an FDQC that respects the symmetry as a whole, but can be entangled by a more general locality preserving unitary (LPU) which respects the symmetry [14][15][16]. When the locality is strict, without exponentially decaying tails, these nontrivial LPUs are also known as quantum cellular automata (QCA) [17][18][19].…”