Quantum Cellular Automata via Algebraic K-Theory / Fermionic QCA in 2d are trivial

Speakers: Mattie Ji (Penn) and Jeongwan Haah (Stanford)

Mattie Ji

Title: Quantum Cellular Automata via Algebraic K-Theory

Abstract: Algebraic K-theory, on a very high level, is the study of how to break apart and assemble objects linearly, which makes the field amenable to classification questions. In this work, we apply this methodology to study the classification of quantum cellular automata (QCA). Over an arbitrary commutative ring R and a general class of metric spaces X, we construct a space of QCA that depends only on the large-scale (coarse) geometry of X. We explain how QCA classification groups (QCA modulo circuits) either arise naturally as or are refined by this space in most cases of interest.

Motivated by negative K-theory, we also show the classification of QCA on Euclidean lattices is given by an $\Omega$-spectrum indexed by the dimension. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya R-algebras, whose negative homotopy groups are the QCA classification groups. When R is the complex numbers, our method can be adapted to yield an $\Omega$-spectrum for QCA of $C^*$-algebras with unitary circuits. This talk is based on joint work with Bowen Yang.

Jeongwan Haah

Title: Fermionic QCA in 2d are trivial

Abstract: We consider bounded spread automorphisms of Z/2-graded algebra (fermionic QCA) on the two-dimensional lattice and prove that every fQCA is a unitary circuit followed by fermionic shifts when stabilized by Majorana modes. This is an analog of a theorem by Freedman and Hastings for the case of ungraded algebras. The overall argument follows a similar line in that we show invertible subalgebras in 1d is trivial, but the stabilization is used crucially. By an existing argument, this triviality of fQCA in 2d implies that the 3d (bosonic) QCA that disentangles the Walker-Wang model with three-fermion theory is nontrivial. The latter was known to be nontrivial against Clifford gates but remained conjectural against more general unitary gates. To my knowledge, this gives the only example ungraded QCA that is proved to be nontrivial against general unitary circuits and shifts, and the only example ungraded invertible subalgebra that is not isomorphic to any tensor product algebra. I will explain elements new to the fermionic setting and give an overview of the nontriviality argument. (Based on an upcoming work with Jeffrey Kwan and David Long)

Zoom link: https://harvard.zoom.us/j/93617252012?pwd=xMGNfbK3c9s4fuqpaZIUC68LjCMkI9.1